Harmonics and graded Ehrhart theory
Abstract.
The Ehrhart polynomial and Ehrhart series count lattice points in integer dilations of a lattice polytope. We introduce and study a -deformation of the Ehrhart series, based on the notions of harmonic spaces and Macaulay’s inverse systems for coordinate rings of finite point configurations. We conjecture that this -Ehrhart series is a rational function, and introduce and study a bigraded algebra whose Hilbert series matches the -Ehrhart series. Defining this algebra requires a new result on Macaulay inverse systems for Minkowski sums of point configurations.
1. Introduction
A lattice polytope in is the convex hull of a finite set of points in . With their deep connections to commutative algebra and toric geometry, lattice polytopes are a central object in algebraic combinatorics. Our goal is to introduce in a canonical way an extra -parameter into two important algebro-combinatorial objects associated to a lattice polytope : the Ehrhart polynomials/series and the affine semigroup ring associated to .
1.1. Classical Ehrhart theory and a -analogous conjecture
The classical story to be generalized is one of lattice point enumeration. Given a -dimensional lattice polytope with boundary and (relative) interior , for , one has its -fold dilate , along with its interior . Define these integer point enumerators , and their generating functions in :
| (1) | |||||
| (2) |
We summarize here some of the results from the original work of Ehrhart [15], as well as Macdonald [37] and Stanley [46, 48]; see also expositions and surveys in Braun [8], Beck and Robins [5], Beck and Sanyal [6], Stanley [51, Ch. 1] [53, §4.6].
Classical Ehrhart Theorem. Let be a -dimensional lattice polytope.
-
(i)
Both and are polynomial functions of of degree . Both generating functions and are rational functions which can be written over the denominator , and whose numerator polynomials lie in , with numerator of degree at most in the case of and degree exactly in the case of .
-
(ii)
The integer coefficients defined uniquely by
are nonnegative, with and the normalized -volume .
-
(iii)
For a lattice -simplex with vertices , one can interpret the as
where is this semi-open parallelepiped spanned by :
(3) -
(iv)
(Ehrhart-Macdonald reciprocity) The series determine each other uniquely by
Our goal is to introduce a natural -parameter into this picture111Note that Chapoton [11] introduced a different, less canonical -analogue, discussed in Remark 3.20 below.. Let be a field and let be a finite locus of points in affine -space over . The point-orbit method (also known as the orbit harmonics method) is a machine which produces from a graded quotient of the polynomial ring . Let be the vanishing ideal of given by
| (4) |
Since is finite, by Lagrange interpolation one has an identification of the space of functions and the quotient ring . We let
| (5) |
where is the associated graded ideal of , generated by the top degree homogeneous components of nonzero polynomials . The ideal is homogeneous, so the quotient ring is graded. Geometrically, the construction is a flat deformation of the reduced subscheme deforming linearly to a zero dimensional subscheme of degree supported at the origin, as illustrated by the picture below.
The construction is a canonical way to put a graded structure on the finite locus . One has an isomorphism of -vector spaces
| (6) |
In particular, introducing a grading variable , one finds that the Hilbert series
is a polynomial in with nonnegative integer coefficients giving a geometrically motivated and canonical -analogue of the cardinality of , that is,
| (7) |
Furthermore, when is stable under the action of a finite subgroup (as suggested by the previous picture where and the transpositions in act by reflections in the three lines), the vector space isomorphism (6) is also a Brauer isomorphism of -modules, that is, and share the same composition multiplicities for all simple -modules. In particular, when is invertible in , such as when characteristic zero, then (6) is a -module isomorphism.
However, the deformation does not have perfect memory: since is not reduced for , the rings and are almost never isomorphic. Despite this amnesia, we will exhibit an unexpected multiplicative structure on direct sums of point-orbit rings (or rather their associated harmonic spaces) coming from dilates of a lattice polytope.
Introduced by Kostant [33] in his study of coinvariant rings of finite reflection groups, the rings have proven to be an enormously fruitful source of graded quotients of . The rings present the cohomology of Springer fibers [21], have ties to Coxeter-Catalan theory [3], give generalized coinvariant rings with ties to Macdonald-theoretic delta operators [28, 25], give graded modules over (0-)Hecke algebras [31, 32], prove cyclic sieving results [40], and encode the Viennot shadow line construction of the Schensted correspondence [43, 35]. The rings have also been used to understand [41] Donaldson-Thomas invariants of symmetric quivers when consists of lattice points in certain zonotopes. In all of these examples, it is crucial to choose a point locus with strategic organization.
Here we will take and consider the case where consists of the integer points of an arbitrary lattice polytope , together with its dilates. Define two -Ehrhart series for , analogous to those in (1), (2)
| (8) | ||||
| (9) |
Thus lie in , and (7) implies they specialize to when . The following is our main conjecture, on the form of .
Conjecture 1.1.
Let be a -dimensional lattice polytope in . Then both of the series (8),(9) lie in , and are expressible as rational functions
over the same denominator of the form , necessarily with . Furthermore, there exists such an expression with all of these properties:
-
(i)
the numerators lie in ,
-
(ii)
if is a lattice simplex, and , then both numerators have nonnegative coefficients as polynomials in .
Lastly, one has this -analogue of Ehrhart-Macdonald reciprocity:
-
(iii)
the two series determine each other via
Remark 1.2.
Conjecture 1.1 requires since , and part (i) of the Classical Ehrhart Theorem implies has a pole at of order . Similarly, letting be the -degree of the denominator , the numerator polynomials and would necessarily have -degrees at most and exactly , respectively.
Example. Consider a -dimensional lattice polytope of volume , that is, a line segment for some integer . Thus for , one has and
Therefore one can calculate
| (10) |
where we are using here a standard -analogue of the positive integer
Note that , and hence, as expected, one has
Since has interior lattice points
a similar calculation shows that
| (11) |
with , as expected. Furthermore, the line segment is a lattice simplex, and one can check that (10), (11) are consistent with all parts (i),(ii),(iii) of Conjecture 1.1.
1.2. The affine semigroup ring and the harmonic algebra
For lattice -polytopes , work of Stanley (see, e.g., [51]) explains the Classical Ehrhart Theorem by reinterpreting as the Hilbert series of a certain commutative graded algebra associated to , reviewed here.
One embeds in as . The cone over is
It is an affine polyhedral -dimensional cone in , and its lattice points form a semigroup under addition. The affine semigroup ring of over a field is
| (12) |
The first coordinate on endows with the structure of a graded algebra, having Krull dimension . Inside , one has the (homogeneous) interior ideal , the -span of all monomials corresponding to the interior lattice points . Then one has
Although is not always generated in degree one, there is always a tower of integral ring extensions
where is the subalgebra of generated by the monomials corresponding to the vertices of , and is any choice of a linear (degree one) system of parameters for ; their existence is guaranteed by Noether’s Normalization Lemma [39]. Most of the assertions (i),(ii),(iii),(iv) from the Classical Ehrhart Theorem are then explained by these four facts, respectively:
This might inspire one to approach Conjecture 1.1 by introducing a bigraded deformation of . Since where comes from the slice of
| (13) |
one might look for such a bigraded object by defining a graded multiplication on the direct sum
| (14) |
That is, one would like a way to ‘multiply’ elements and to produce a new element in a fashion which respects the polynomial degrees of and . Geometrically, this corresponds to deforming the lattice points in linearly onto the coning axis and defining a multiplication between fat point loci. The relevant picture is shown below when is a square in .
The naïve approach of looking at the ring has two problems:
-
•
the transformation deforms a locus to the origin, but we want to deform to the coning axis, and
-
•
if is full-dimensional, then is Zariski-dense and . Consequently, the deformation for remembers only the affine span of , not itself.
We achieve the task of defining a multiplication on the direct sum (14) in Section 5 by replacing each summand with its harmonic space (or Macaulay inverse system). The resulting bigraded ring is the harmonic algebra attached to . The bigraded Hilbert series of equals the -Ehrhart series . Although not isomorphic to in general, we conjecture that enjoys analogous algebraic properties (finite generation, Cohen-Macaulayness, identification of the canonical module) that would explain much of Conjecture 1.1.
In spite of all of these properties remaining conjectural, we are able to show that the -Ehrhart series and harmonic algebra behave in a predictable fashion when performing three well-studied operations on lattice polytopes :
-
•
dilation by an positive integer factor , sending to ,
-
•
Cartesian product, sending and to ,
-
•
free join, sending to defined by
(15)
The next result is proven in Section 6, and relates the above operations to the constructions of Veronese subalgebras, Segre products and graded tensor products for harmonic algebras .
Theorem 1.3.
Let be lattice polytopes.
-
(i)
For positive integers , the dilation has given by
-
(ii)
The Cartesian product has given by the Hadamard product of series
-
(iii)
The free join has given by
A further pleasant feature of the harmonic algebra arises when one considers Stanley’s two poset polytopes [50] associated to a finite poset: its order polytope and its chain polytope. Although these two lattice polytopes look very different, Stanley showed that they share the same Ehrhart series. It will turn out (see Section 5.6 below) that all of our conjectures hold for both of these families of polytopes, that they share the same -Ehrhart series, and even share the same harmonic algebras. This is in contrast to the fact that their affine semigroup rings are generally not isomorphic.
1.3. Harmonic spaces and Minkowski addition
The fact that the harmonic algebra is closed under multiplication is not obvious, and rests upon a surprising new property of harmonic spaces for arbitrary finite loci . Define their Minkowski sum to be the finite point locus
| (16) |
The point-orbit rings and are graded quotients of . When has characteristic zero, the partial differentiation action of on itself gives rise to a -linear perfect pairing on each homogeneous component of , and we may replace by their harmonic spaces
| (17) |
If is a homogeneous ideal, the harmonic space is a graded subspace with the same Hilbert series as . Since elements of are honest polynomials , whereas elements of are cosets , working in avoids coset-related issues which arise in proving, e.g., linear independence results. On the other hand, unlike the graded ring , the subspace has the defect of not being closed under multiplication. Nevertheless, in Section 4 we prove the following.
Theorem 1.4.
For any pair of finite point loci in over any field , one has
This containment may be interpreted as saying that the rings ‘remember’ the structure of Minkowski sums via multiplication of their harmonic spaces. As the deformation does not respect ring structure, we find Theorem 1.4 a bit unexpected222It is reminiscent of another unexpected fact, about standard monomials for with respect to a chosen monomial ordering on , observed by F. Gundlach [26]; see Remark 2.8 below..
Note Theorem 1.4 is stated for finite point loci inside where is any field, not just or a field of characteristic zero. This requires defining harmonic spaces over an arbitrary field , which occurs already in the theory of Macaulay’s inverse systems over all fields discussed, e.g., in Geramita [23], and reviewed in Section 4 below. These more general harmonic spaces are defined not inside a polynomial ring over , but rather in the divided power algebra over . When has characteristic zero, these two rings are the same, and the definitions of coincide.
The remainder of the paper is structured as follows.
Section 2 reviews commutative algebra of associated graded ideals and rings, Gröbner bases, harmonic spaces and Macaulay’s inverse systems (first in characteristic zero, and then over all fields). It then briefly reviews some aspects of groups acting on rings and representation theory.
Section 3 defines the -Ehrhart series , reviews Conjecture 1.1, and then examines several families of examples. It also incorporates symmetries of in an equivariant -Ehrhart series , and computes some highly symmetric examples, such as simplices and cross-polytopes.
Acknowledgements
The authors thank Ben Braun, Winfried Bruns, Sarah Faridi, Takayuki Hibi, Katharina Jochemko, Martina Juhnke-Kubitzke, Sophie Rehberg and Raman Sanyal for helpful conversations. They thank Ian Cavey for help in streamlining the proof of Theorem 1.4, and thank Christian Haase for pointing them to Balletti’s database [4]. They are especially grateful to Vadym Kurylenko for his computations appearing in Remark 3.6, equation (62) and ExtraData.pdf. Authors partially supported by NSF grants DMS-1745638 and DMS-2246846, respectively.
2. Background
2.1. Commutative algebra
Let be a field, let , and let be a list of variables. We write for the polynomial ring in over with its standard grading induced by for all .
We will consider various graded -subspaces and quotients of , as well as other rings. If is a graded -vector space with each piece finite-dimensional and is variable, the Hilbert series of is the formal power series
| (18) |
More generally, if is a bigraded vector space, the bigraded Hilbert series is
| (19) |
Given a nonzero polynomial, write for the top degree homogeneous component of . That is, if with homogeneous of degree and , we have . If is an ideal, the associated graded ideal is given by
| (20) |
The ideal is homogeneous by construction, so that is a graded ring. In fact, we wish to explain why it is isomorphic to the associated graded ring
| (21) |
for the ascending filtration on
| (22) |
where is the image of the polynomials of degree at most under the surjection . Note that this filtration satisfies , so that the graded multiplication in is well-defined.
Proposition 2.1.
Define a -algebra map sending in .
-
(i)
The map is surjective, with kernel , inducing an -graded -algebra isomorphism
-
(ii)
Consequently, any homogeneous polynomials whose images give a -basis of will also have their images giving a -basis of .
-
(iii)
In particular, whenever is Artinian, that is, is finite, one can view as a -analogue of in this sense:
Proof.
For (i), the surjectivity of holds because is generated by . Hence one has , and therefore .
To show , we check for . If and , then
so that in .
To prove , it suffices to show every homogeneous in lies in . If , then implies , say with . But then has , so .
For (ii), it suffices to check that for each , the set is a -basis for . However, this follows by induction on , since our hypotheses imply that is a -basis for .
Assertion (iii) then follows immediately from (ii). ∎
We have another consequence in the case : the quotient will be determined by the intersection with the first summand in the -vector space direct sum decomposition
| (23) |
Lemma 2.2.
Assume the ideal has finite.
-
(i)
The inclusion is an isomorphism for all .
-
(ii)
The graded ring vanishes in degrees , that is, .
Proof.
Recall that for the filtration on from (22), in which is the image of the composite . The composite has kernel , so , and hence
| (24) |
Note that since is a graded -algebra generated in degree one, its nonzero graded components form an initial segment of degrees. Since , one concludes that for all , and consequently, for all , proving assertion (ii). For assertion (i), note (24) also shows that for all , and therefore the inclusion must be an isomorphism. ∎
If is an ideal with generating set , we have , but this containment is strict in general. A finite generating set for may be computed using graded term orderings and Gröbner theory as follows; see Cox, Little, O’Shea [13] for more background.
Definition 2.3.
A total order on the monomials of is a term order if for all monomials , and whenever one also has for all monomials .
For a term order and , write for the -largest monomial appearing in .
Example 2.4.
The lexicographical term order is defined by if there exists such that and . The graded lex term order is defined by
if or ( and )
where .
A term order is graded if whenever . Equivalently, is graded if and only if one has for all that
| (25) |
Example 2.5.
Lexicographic order is not graded for , but is always graded.
Let be an ideal and let be a term order. The initial ideal of is the monomial ideal
| (26) |
Definition 2.6.
A finite subset is a Gröbner basis of if . Equivalently, for every in there exists some in with dividing .
One can show that a Gröbner basis for always generates as an ideal. We also have the following useful -basis for . Say that monomial in is a standard monomial of (with respect to ) if . Equivalently, this means that does not divide for all , where is a Gröbner basis of with respect to . Then the set
| (27) |
is a -basis of the quotient ring . It is uniquely determined by the term order , and called the standard monomial basis of . The following can then be proven easily using (25).
Proposition 2.7.
Fix a graded term order on . Then for any ideal , a Gröbner basis for (with respect to ) gives rise to a Gröbner basis with respect to
for the homogeneous ideal . Consequently, also generates as an ideal:
Furthermore, share the same set of standard monomials with respect to , which descend to -bases and for and , respectively.
Remark 2.8.
When is the vanishing ideal in for a finite point set , Gundlach [26, Lem. 4] gives a very interesting alternate characterization of the -standard monomials for . Let denote the -vector space of all functions , with pointwise addition and -scaling. Endow with a nondegenerate -bilinear form given by
Let denote perp with respect to for -subspaces . Nondegeneracy of implies . By multivariate Lagrange interpolation, the map restricting polynomials to functions on is surjective. Since its kernel is , it gives a -vector space isomorphism . This implies, that for any , there must exist some monomials in for which . Consequently, having fixed the monomial order , for each , there will be a -smallest such monomial associated to , since is a well-ordering:
Proposition 2.9.
[26, Lem.4] For any finite subset , and for any choice of monomial order on , one has this equality of sets:
Proof.
For any monomial in one has the following:
2.2. Homogeneous harmonics in characteristic zero
A good reference for much of this material is Geramita [23, §2]. Let be a field of characteristic zero. We wish to set up two polynomial rings over , one of which acts on the other by partial derivatives. Let and its -dual have dual ordered -bases and with respect to the usual -bilinear pairing of functionals and vectors
| (28) |
so that . If one considers the polynomial algebras
then one can extend this to a -valued pairing
| (29) |
by requiring that each act on as a derivation. That is, acts as , and for polynomials , one has
In this way, one obtains an -module structure on . This -module structure on is degree-lowering for the usual gradings on in which , in the sense that it restricts to a map This lets one extend the pairing from (28) to a -bilinear pairing defined by
| (30) |
Employing an exponential notation for monomials in , where lies in , and similarly for monomials in , one can check that
One sees that pairs orthonormally the -dual bases and hence restricts to a perfect -linear pairing on the (finite-dimensional!) spaces
Note that and are perpendicular with respect to the pairing whenever .
Definition 2.10.
For any homogeneous ideal , the harmonic space (or Macaulay inverse system) is the graded vector space
| (31) | ||||
| (32) |
The equality of the two sets on the right in (31), (32) is justified as follows. If then , showing the set from (32) is contained in the set from (31). For the reverse inclusion, note that if for some in , say with some , then the ideal contains with .
Note that in each degree , the perfect pairing gives a -vector space isomorphism sending . This induces a -vector space isomorphism
| (33) |
showing that . Hence as graded -vector spaces one has
| (34) |
2.3. Homogeneous harmonics for all fields: divided powers
In order to define harmonic spaces over arbitrary fields, as needed in Theorem 1.4, we will need to replace with a divided power algebra over a field . We therefore review divided power algebras here; a reader who is content with seeing Theorem 1.4 stated and/or proven only in characteristic zero can mostly skip this section. Useful references for this material are Eisenbud [16, A2.4] and Geramita [23, §9].
Definition 2.11.
Let be any field. For , let be a list of variables, thought of as the -basis for . Then the divided power algebra of rank over is defined as a -vector space with “monomial” -basis given by the symbols for . One can then define a multiplication on which is -bilinear and determined on monomials by the rule
| (35) |
where the binomial coefficients are regarded as elements of in the natural way; some will vanish when has positive characteristic. This makes an associative, commutative -algebra with unit . It has a grading where has -basis .
Roughly speaking, the symbol plays the role of , even when in . When has characteristic zero, the -vector space isomorphism given by
is a ring isomorphism. In fact, with conventions and , the set map for extends to what is called a system of divided powers on : a collection of maps for
satisfying these axioms (modeled on properties of the maps that exist whenever ):
| (36) | ||||
| (37) | ||||
| (38) | ||||
| (39) | ||||
| (40) |
For example, the reader might wish to check that iterating (37) implies , so that one has no choice but to define whenever . Similarly, if one iterates (40), which Eisenbud [16, A2.4] calls the “beginner’s binomial theorem”, one obtains the “beginner’s multinomial theorem”:
| (41) |
It is also not hard to check that one has a graded -algebra isomorphism
| (42) |
For the sake of defining harmonics and inverse systems, let as before. The algebra attains a unique -module structure by having act on as a derivation, extending the rule
| (43) |
This gives rise to a -bilinear pairing given by
| (44) |
under which
| (45) |
In particular, the -bilinear pairing generalizes the one from (86) when . It again leads to perfect -bilinear pairings on these finite-dimensional spaces:
This leads to the following generalization of Definition 2.10.
Definition 2.12.
For any field and any homogeneous ideal , the harmonic space (or Macaulay inverse system) is the graded -vector space
2.4. Symmetry
The natural action of on and a given basis induces a left-action on the polynomials by linear substitutions. The contragredient action on precomposes functionals with , that is, sending , thereby acting on , as well as on the polynomials . Explicitly, if in acts in the basis via the matrix in , then
Note that the pairing between functionals and vectors in (28) satisfies a certain invariance with respect to these -actions: for any linear functional in , vector in , and in , one has
Consequently, the pairings and are similarly invariant with respect to the -action:
| (48) | ||||
| (49) |
Remark 2.13.
For arbitrary fields when one replaces the polynomial algebra with the divided power algebra , it is still true the -action on extends to an action via graded -algebra automorphisms on all of . This fact is more apparent when one constructs multiplication in as the (graded) dual of the coalgebra structure on in which , that is, each is primitive. See Akin, Buchsbaum and Weyman [2, §I.4], Eisenbud [16, A2.4], Geramita [23, §9] for more on this alternate construction of .
2.5. Representation theory
We will be interested in polytopes and point loci in with symmetry, and wish to keep track of the representations of their symmetry groups on the various -vector spaces that we construct. We review one way to do such bookkeeping, using the language of -modules and representation rings.
Definition 2.14.
For a field and finite group , define its representation ring as follows.
-
•
As a -module, is the quotient of free -module with -basis elements indexed by all isomorphism classes finite-dimensional -modules , in which one mods out by the submodule -spanned by all relations
(50) -
•
As a -algebra, its multiplication is induced by the rule
-
•
The operation of taking the contragredient -module leads to a -automorphism and involution on
Whenever lies in , Maschke’s Theorem asserts that -modules are completely reducible, which shows that is a free -module on the -basis where are the non-isomorphic simple/irreducible -modules.
For any field , the map sending to its character defined by
becomes an algebra map from into the ring of class functions , that is, functions which are constant on -conjugacy classes. The ring of class functions is given pointwise addition, multiplication, and the involution defined by . Whenever has characteristic zero, this algebra map is injective, and in particular, two -modules are isomorphic (that is, ) if and only if they have the same character .
More generally, for graded -modules , with each a finite-dimensional -module, we will track the representation with a power series in :
| (51) |
In particular, when is a homogeneous ideal which is stable under the action of a finite subroup of , both the quotient and the harmonic space inherit the structure of graded -modules. Then (46) implies a graded -module isomorphism , and hence in .
On the other hand, we will also consider potentially inhomogeneous ideals that are stable under the action of a finite subgroup , e.g., where is a -stable locus.
Proposition 2.15.
Assume lies in . For any ideal which is stable under a finite subgroup of , if finite, then one has a -module isomorphism .
Proof.
Recall (21) gave an isomorphism , where the sum on the right is finite here due to the finiteness assumption on . This isomorphism is easily seen to be -equivariant. Since the action of preserves degree, the filtration of is -stable, and its filtration factors match the -module structures on the graded components of . Complete reducibility of -modules then shows . ∎
This proof also shows, for any field , the -modules and are Brauer-isomorphic.
3. The -Ehrhart series and Conjecture 1.1
Throughout this section, we take and consider finite point loci . The locus has two associated ideals inside the (inhomogeneous) vanishing ideal
and its associated graded ideal
Within the polynomial ring in the dual variables , the harmonic space of will play a crucial role in our work. To reduce notational clutter, we write
| (52) |
for this harmonic space. This notation emphasizes the role of as a graded subspace of which is almost never closed under multiplication. On the other hand, (34) shows that, as a graded vector space, it has the same Hilbert series (actually a polynomial here) as the quotient ring that was defined in (5)
Note Proposition 2.1(iii) shows this Hilbert series is a -analogue of the cardinality , that is,
Furthermore, when is a finite subgroup of that preserves setwise, it acts via ring automorphisms and (graded) -modules on all of the objects under consideration:
Because finite-dimensional -modules are all self-contragredient (), one can check that (33) implies graded -module isomorphisms
| (53) |
and then Proposition 2.15 implies a further ungraded -module isomorphisms
| (54) |
which are all three isomorphic to the -permutation module on the points .
3.1. Definition of -Ehrhart series and the conjecture
As in the introduction, a lattice polytope is the convex hull of a finite set of points in the lattice . For each integer one obtains a finite point locus . For each , one has the interior point locus , where is the (relative) interior where one removes the union of all boundary faces of .
Definition 3.1.
For a lattice polytope , define two -Ehrhart series in :
| (55) | ||||
| (56) | ||||
Note the series reduce to the classical Ehrhart series at .
Example 3.2.
Recall the Example in the Introduction looked at the general -dimensional lattice polytope with and volume , finding that
| (57) | ||||
| (58) |
where .
Note that Example 3.2 shows that both for lattice polytopes depend only upon a single parameter, which one could take either to be the volume , or the number of lattice points , or the -vector entry . This illustrates a certain affine-lattice invariance that one might expect, similar to classical Ehrhart theory. Recall that the group of affine transformations of is a semidirect product , where is the subgroup fixing the origin, and is the translation subgroup. This restricts to a semidirect product decomposition .
Proposition 3.3.
For any , one has and .
Proof.
We give the argument for ; the argument for is similar.
Note that for each , the point locus is an affine transformation of the locus , namely by an affine transformation whose translation vector is scaled by from the translation vector of . It therefore suffices to check for finite point loci and for any , that one has
| (59) |
Note that acting on variables by for some has
Therefore . Since acts via a graded -algebra automorphism on , this means it induces a graded -algebra isomorphism implying (59):
Before discussing more examples, recall the main conjecture from the Introduction.
Conjecture 1.1. Let be a -dimensional lattice polytope in . Then both of the series (8),(9) lie in , and are expressible as rational functions
over the same denominator of the form , necessarily with . Furthermore, there exists such an expression with all of these properties:
-
(i)
The numerators lie in .
-
(ii)
If is a lattice simplex, and , then both numerators have nonnegative coefficients as polynomials in .
-
(iii)
The two series determine each other via
3.2. Examples: lattice polygons
Having computed for all one-dimensional lattice polytopes (line segments) in the Introduction and Example 3.2, one might wish to see data for lattice polygons. Proposition 3.3 allows one to consider them only up to the action of . If one bounds the normalized volume of a -dimensional lattice polytope in , a well-known result of Lagarias and Ziegler [34] shows that there are only finitely many such up to the action of . Work of Balletti [4] gives an algorithm to list them, including an online database at https://github.com/gabrieleballetti/small-lattice-polytopes listing equivalence classes of lattice polytopes up to dimension of relatively small normalized volumes. In particular, it includes lattice triangles up to normalized area .
Using this data, Figures 1, 2, 3 present the -Ehrhart series of -equivalence classes of lattice polygons of normalized volume at most . These were first guessed using Macaulay2 to compute for up to some reasonably large values of . The guesses were then proven correct via some extra computation in Macaulay2 that uses our results on the harmonic algebras in Section 5; see Remark 5.10 below for an explanation.
Note the last column of the tables, indicating whether the polygon is -equivalent to one in the tamer subclass of antiblocking lattice polytopes, discussed in Section 3.4.
With the kind assistance of V. Kurylenko, we also have computed guesses for for all lattice polygons of area , as well as a selection of lattice tetrahedra. These can be found tabulated in the supplementary data file ExtraData.pdf in the arXiv version of this paper.
All the data in Figures 1, 2, 3 (and ExtraData.pdf) is consistent with Conjecture 1.1. However, we close this subsection with remarks on some cautionary features.
Remark 3.4.
The specialization sometimes has interesting numerator and denominator cancellations when .
Remark 3.5.
Note that, for fixed , the classical Ehrhart series for a lattice -polytope can be expressed as an affine-linear function of the real parameters :
One has no such affine-linear formula for , already at , since the formulas (57)
give affine-linearly independent functions of where . On the other hand this formula does express for as a function (which is not affine-linear) of the one real parameter . When , there can be no such function of the two parameters , since one can have two lattice polygons with the same classical series but different -Ehrhart series ; this happens for two area lattice polygons in Figure 1.
Remark 3.6.
Within ExtraData.pdf, one finds a guess for for a particular lattice triangle that we found difficult to compute. Eventually, this guess was kindly computed for us by V. Kurylenko, using data up through -degree :
with a numerator polynomial in having nonnegative coefficients, and -degree . We found the -power in its denominator surprisingly large compared to other examples333This example is one reason that we revised our main Conjectures 1.1, 5.5 from arXiv version 1 of this paper..
3.3. Examples: a few Reeve tetrahedra
An important family of examples in Ehrhart theory are Reeve’s tetrahedra [5, Example 3.22], defined by
The parameter is their normalized volume, and they have these Ehrhart polynomial and series:
Thus has , and for , gives examples where the -vector has internal zeroes. This can only happen for lattice polytope lacking the following property.
Definition 3.7.
Say that a lattice polytope has the integer decomposition property (IDP) if
| (60) |
See Braun [8] and Cox, Haase, Hibi and Higashitani [12] for more on the IDP. All lattice polygons have the IDP. The smallest non-IDP lattice polytope is the Reeve tetrahedron shown here:
It is not IDP since .
We have either computations or guesses for the -Ehrhart series for :
-
•
is a unimodular tetrahedron, -equivalent to from Section 3.6 below, so
- •
-
•
V. Kurylenko computed this guess for the -Ehrhart series for , correct up to -degree :
(62)
Note that, in this guess for , there are denominator factors, but . This is the first instance where this occurs for a lattice simplex444This example is another reason that we revised our Conjectures 1.1, 5.5 from arXiv version 1 of this paper., raising the following question.
Question 3.8.
For which lattice -simplices can one express
with denominator factors, and with in ?
-
(a)
Does this occur for all lattice triangles?
-
(b)
Does it occur more generally for all lattice simplices with the IDP property?
3.4. Examples: antiblocking polytopes
For certain polytopes introduced by Fulkerson [19, 20] in the context of combinatorial optimization, both -Ehrhart series have a simpler description as classical weighted lattice point enumerators, avoiding harmonic spaces. This will allow us to verify Conjecture 1.1 for such polytopes.
In this section, abbreviate the nonnegative reals by , the nonnegative orthant by , and the nonnegative integers . Define the componentwise partial order on via if for all .
Definition 3.9.
Say that a convex polytope is antiblocking (or of antiblocking type) if , and forms a (lower) order ideal in the componentwise order on , that is, whenever with then also .
Here is an example of an antiblocking polygon inside the orthant .
Intersecting antiblocking polytopes with gives rise to point loci with a restrictive property that vastly simplifies the ring and harmonic space .
Definition 3.10.
Call a subset shifted if and forms a (lower) order ideal in the componentwise order on , that is, whenever and then also . Equivalently, is shifted if and only if the -vector space forms a monomial ideal inside .
Lemma 3.11.
For any finite shifted point locus ,
-
(i)
the ideal is monomial, with -basis ,
-
(ii)
the quotient ring of has -basis ,
-
(iii)
the harmonic space has -basis inside .
Proof.
Since is shifted, the -vector space is a monomial ideal. Note that all assertions in the lemma would follow from showing (i), that is, .
To see this, we first show the inclusion . Since is shifted, we claim that for any , the polynomial
lies in the vanishing ideal for . To see this claim, assume not, so there exists some point having . But then for each one must have , implying , and contradicting . Hence contains showing .
To show the opposite inclusion , note that the surjection must be an isomorphism by dimension-counting, since . ∎
If one defines the following -weight enumerator for finite point loci
| (63) |
then the next corollary is immediate from Lemma 3.11
Corollary 3.12.
For finite shifted point loci , one has
We next explain how antiblocking polytopes give rise to shifted point loci. For , let
Proposition 3.13.
Let be an antiblocking -polytope in .
-
(i)
The finite point locus is shifted, and the same holds for for all .
-
(ii)
There exists a re-indexing of the coordinates in such that the translated set
is shifted, and the same holds for the set for all .
Proof.
First note that whenever is antiblocking, then so is . Thus one only needs to check the first part of both assertions (i),(ii) for .
Assertion (i) for is straightforward from the definitions of antiblocking and shiftedness.
To prove assertion (ii) for , consider the support sets for . Since is convex and lies in , whenever , the points in the interior of the line segment between them have . Consequently, there exists with Reindexing coordinates so that , one concludes that contains the -dimensional parallelepiped of vectors componentwise between and , and also is contained in the -dimensional coordinate subspace . Since is -dimensional, this forces , and the last coordinates vanish on all points of .
Therefore in the rest of the proof that is shifted, it suffices to project away the last coordinates which all vanish on , and assume . As a full -dimensional antiblocking polytope , starting with any inequality description
where have all entries in , one can describe as
Then the points of have this description:
Hence the points of have this description, which defines a shifted subset of :
Proposition 3.13 lets us re-cast the -Ehrhart series for an antiblocking polytope in terms of the following simpler and more classical weight enumerators, involving from (63).
Definition 3.14.
For any lattice polytope , define two -weight enumerator series
Corollary 3.15.
For any -dimensional lattice polytope which is antiblocking, one has
| (64) | ||||
| (65) |
Classical results now apply to the weight enumerators , showing that they have rational expressions with predictable properties, for any lattice polytopes .
Proposition 3.16.
Let be any -dimensional lattice polytope, with vertices .
-
(i)
Both have rational expressions with the same denominator
and numerators .
-
(ii)
For a lattice -simplex with vertices , one has these numerators in
where is the semi-open parallelepiped from (3), and its “opposite”, defined by
-
(iii)
The two series determine each other via
Proof.
These are all bivariate specializations of results of Stanley on multivariable lattice point enumerators for rational polyhedral cones, implicit in [45], and more explicit in [53, §4.5]. One must apply them to which is defined to be the -dimensional cone nonnegatively spanned by the vectors . In variable set , Stanley considers
He proves [53, Thm. 4.5.11] that one has rational expressions of the form
| (66) |
for some polynomials in . He also proves [53, Cor. 4.5.8] that a -simplex has
| (67) |
And he shows in [53, Thm. 4.5.14] (often called Stanley’s Reciprocity Theorem) that
| (68) |
Assertions (i),(ii),(iii) are the specializations at of (66),(67),(68). ∎
Corollary 3.17.
Conjecture 1.1 holds for all lattice polytopes which are antiblocking.
Proof.
Remark 3.18.
See Section 5.6 below for a discussion of an interesting family of antiblocking polytopes mentioned in the Introduction: the chain polytope associated to a finite poset . It will be shown there that shares the same -Ehrhart series as the order polytope associated to , generalizing a result of Stanley [50] on their classical Ehrhart series.
Remark 3.19.
Not all lattice polytopes are equivalent via to antiblocking polytopes. In particular, sometimes is not a monomial ideal in , and has no monomial -basis. For example, the lattice triangle shown here with vertices
is equivalent to the second triangle of area in Figure 1, with vertices . It has
Via hand calculation or using Macaulay2, one can check that
| (69) |
and this ideal cannot be generated by monomials. Again hand calculation or Macaulay2 shows that
which has no -basis of monomials.
Remark 3.20.
The -weight enumerators in Definition 3.14 bear some resemblance to work of Chapoton [11]. He introduced -analogues of that first require a choice a -linear functional which is nonconstant along edges of and has for all vertices of . Having made such a choice, he defined
Therefore Corollary 3.15 shows that if a lattice polytope is antiblocking, and happens to have that the functional is nonconstant along its edges, then . This -analogue is also considered by Adeyemo and Szendrői [1], who calculate it for standard simplices, cross-polytopes and cubes.
3.5. Definition of the equivariant -Ehrhart series
We next incorporate symmetries of subgroups of acting on a lattice polytope. Such groups are often called crystallographic groups.
Definition 3.21.
Let be a lattice polytope in which is stable under the action of a finite crystallographic group . Define the equivariant -Ehrhart series
where recall is the class in of any graded -module . Hence lies in the power series ring , and even lies in its subring .
When , this series specializes to Stapledon’s equivariant Ehrhart series [54] involving the ungraded characters of , or equivalently the permutation -modules .
Example 3.22.
We return to Example 3.2, where is a lattice polytope , but now assume it is stable under the action of a nontrivial crystallographic group . This implies for some and . One can identify the representation ring
where denote the isomorphism classes of the one-dimension trivial and nontrivial -modules, respectively. A calculation using , similar to the one in the Introduction, shows that
and consequently, one has
Since negates the variables in , one concludes (with some algebra) that
| (70) | ||||
| (71) |
This expression (71) specializes to the non-equvariant series by applying the ring homomorphism that sets , giving an answer consistent with (57) at :
On the other hand, setting in (71) should also give Stapledon’s equivariant Ehrhart series , which we calculate directly here. The -orbit structure permuting
has one fixed point orbit with class in and free orbits , each with class in . Hence one calculates that
which agrees with the expression for in (71) at .
3.6. Examples: standard simplices
We compute the equivariant -Ehrhart series for two families of simplices that carry an action of the symmetric group on letters, permuting the standard basis vectors in .
Definition 3.23.
The standard -simplex in is
lying inside the affine subspace of where . There is also a full -dimensional simplex in which is the pyramid having as its base, and apex at the origin :
We compute the equivariant -Ehrhart series of . For this, it helps to have two facts. The first is an easy lemma on power series for commutative rings , and their truncations to , defined as these polynomial partial sums in :
Lemma 3.24.
For any in , one has in .
The second fact is a formula due to Lusztig and Stanley [47, Prop. 4.11] for the -equivariant Hilbert series of the polynomial ring . Recall that the simple -modules are indexed by number partitions of (written ), so that has -basis .
Theorem 3.25.
(Lusztig, Stanley) The graded -module has this class in :
with the fake-degree polynomial, having these sum, product expressions [52, Cor. 7.21.5]
| (72) |
whose notations are explained below.
Here in the summation in (72) runs over all standard Young tableaux of shape , and is the sum of all entries in for which appears weakly southwest of (using English notation for tableaux). The -factorial is . For a partition with parts, we have . For a box in row of the Ferrers diagram of (written in the product), the hooklength is , with the conjugate partition to .
We first compute the equivariant -Ehrhart series for the -simplex .
Proposition 3.26.
For the -simplex , one has
| (73) | ||||
| (74) |
Proof.
We start with (73). Note and all of its dilates are antiblocking, with defined by the inequalities and . Hence Lemma 3.11 shows that
Thus the graded -module is the same as the permutation module on the monomials of degree at most in . Therefore in one has
Applying Lemma 3.24 then gives the equality marked (*) here
| (75) |
with the last equality substituting in the formula from Theorem 3.25. This proves (73).
Now it is easier to deal with the equivariant series for the -simplex .
Proposition 3.27.
For , one has
| (76) | ||||
| (77) |
Proof.
We start by identifying . Since , one has
On the other hand, since lies in the affine hyperplane within , one also has and . Consequently,
and we claim that this inclusion is actually an equality. To see this, note that the surjection
is an isomorphism via dimension-counting: since is the dilation of a standard -simplex,
and if one abbreviates the quotient ring , then
From this one concludes that in one has
and hence, applying Lemma 3.24, one has
On the other hand, we claim that This holds because is an -invariant nonzero divisor in , so that multiplying by on gives rise to an exact sequence of graded -modules . Comparing this with (75) then proves (76).
The proof of (77) is again a parallel, but easier computation. ∎
Remark 3.28.
We note here some notationally convenient reformulations of these results, for readers familiar with the ring of symmetric functions and the Frobenius characteristic isomorphism see [38, 44, 52] for more background. One note of caution: multiplication in corresponds to the internal or Kronecker product on , not the external or induction product corresponding to the multiplication .
This Frobenius characteristic isomorphism maps the Schur function , and the complete homogeneous symmetric function . Using the plethystic notation, explained in Bergeron [7, Ch. 3], Haglund [27], Loehr and Remmel [36], one can show (see [7, eqn. (7.7)]) that the class of the graded -module corresponds to this element of :
The Lusztig-Stanley Theorem 3.25 expanding this plethysm in Schur functions is discussed in Haglund [27, (1.89)]. Consequently, one can recast Propositions 3.27 and 3.26 plethystically:
Remark 3.29.
3.7. Examples: cross-polytopes
The cross-polytope or hyperoctahedron is defined by
Its symmetries are the hyperoctahedral group of all signed permutation matrices, acting by permuting and negating coordinates in . We first analyze the non-equivariant -Ehrhart series for , and then incorporate the group action.
Proposition 3.30.
The -Ehrhart series of the cross-polytope is given by
| (78) |
Proof.
Let . Then for each , the lattice points of the dilate are given by
| (79) |
We wish to identify the ideal . For this, it helps to note that every monomial in can be expressed uniquely in the form that separates even and odd powers of variables
| (80) |
for some pair with and . We then have the following claim.
Claim: is the monomial ideal
To verify the Claim, we first show that for any such monomial in , so , if one defines the set , then is divisible by
where is the following polynomial in :
To see has the appropriate vanishing to lie in , note that for any where , each coordinate with must lie in , and hence have ; similarly, each coordinate with must lie in , and hence have But this shows that , since
On the other hand, has a bijection to the complementary set of monomials
sending with and
The inverse bijection sends defined by for and for . This shows the inclusion is an equality, since the surjection
must be an isomorphism via dimension-counting.
To calculate , it helps to recast the unique expressions in (80) as a graded -vector space isomorphism. Define the polynomial subalgebra within , and introduce the -linear subspace which is spanned by the squarefree monomials, that is, . One then has the following graded -vector space isomorphism:
| (81) |
Endow the tensor product on the left with a bigrading or -grading in which
Then the bigraded Hilbert series in variables that tracks via is
To recover the usual -grading, tracking via , one must set and .
To enhance Proposition 3.30 to a -equivariant calculation, one must recall the parametrization of simple -modules, e.g., from Geissinger and Kinch [22], Macdonald [38, Ch. 1, App. B]. The simple -modules are indexed by ordered pairs of partitions
One can construct using the simple -modules as follows. Introduce
-
•
the operation of induction
for any pairs of -modules,
-
•
the operation of inflation of -modules to -modules, by precomposing with the group quotient map that ignores the signs, and
-
•
the one-dimensional character sending a signed permutation matrix to the product of its signs, that is, .
Then starting with the simple -modules , one builds the -module as follows:
Proposition 3.31.
For the cross-polytope , one has in
Proof.
The group acts on by permuting and negating the variables . As the isomorphism (82) is also an isomorphism of graded -modules, the key is to calculate the class
within the representation ring of -graded -modules.
For the right tensor factor , we claim that its -graded component carries the simple -module . This is because it is a direct sum of the lines which are the -images of the line . This line is stabilized setwise by the subgroup , where the factor acts trivially and the factor acts via . Hence
To analyze as a -module, note that since all variables appear squared as , the sign changes in have no effect; acts via inflation through the surjection . Consequently, one can obtain the class of the graded -module from that of the graded -module given in Theorem 3.25, simply by applying the inflation map
The upshot is that
Finally, as in the proof of Proposition 3.30, is obtained from upon multiplying by and replacing and . ∎
4. Minkowski closure and the proof of Theorem 1.4
Recall from the Introduction that we hope to approach Conjecture 1.1 through the new harmonic algebra of a lattice polytope defined in Section 5 below. The existence of this algebra structure relies on Theorem 1.4, asserting that for finite point loci (such as and ), their harmonic spaces always satisfy
The goal of this section is to prove this result. The first two subsections collect some preparatory material. This includes broadening the notion of the perp or Macaulay inverse system to allow not only homogeneous ideals , but arbitrary ideals.
4.1. Vanishing ideals for finite point loci
For with a field, the vanishing ideal is
It is convenient to have a concrete generating set for for finite, even if it has redundant generators. The following proposition is well-known, but perhaps hard to find in the literature.
Lemma 4.1.
For any field and finite subset , its vanishing ideal has these descriptions:
| (83) | ||||
| (84) | ||||
| (85) |
Proof.
When so that , the assertions all hold, and is the maximal ideal . From this, (83) follows by definition.
Equality (84) then follows by induction on , once we check that the two ideals and are coprime in the sense that : this implies the inclusion becomes an equality. Coprimeness of follows from multivariate Lagrange interpolation (see e.g. [31, top of p. 13]) which provides the existence of a polynomial that has and for all , so : in the quotient ring , one has , and hence .
The equality (85) also follows by induction on , since the product of two ideals generated as can be generated as . ∎
4.2. Harmonic spaces for inhomogeneous ideals: completions and exponentials
It will be much easier to identify the harmonic spaces of the inhomogeneous ideals , rather than their homogeneous deformations . However, the harmonic spaces for inhomogeneous ideals naturally live in the power series completion of , or more generally, a completion of the divided power algebra . Within these completions, the harmonic spaces will turn out to have a simple basis of “exponentials” indexed by ; see Lemma 4.6 below.
Recall the set-up from Sections 2.2, 2.3 for harmonic spaces of homogeneous ideals in . For a characteristic zero field, consider the polynomial algebra
More generally, over any field or any commutative ring with , consider the divided power algebra
making the identification whenever has characteristic zero. One has an -module structure on given by where acts on as the derivation (so that ), with these formulas in the characteristic zero and arbitrary ring cases:
This leads to a -bilinear pairing defined by
| (86) |
This pairs the -bases as , giving a perfect pairing in each degree, identifying and , for all . In other words, are what are sometimes called graded (or restricted) -duals.
When working with inhomogeneous ideals , we will include in a larger ring which is the inverse limit of the projections . In other words, with multiplication defined -linearly extending the rule in (35). We can therefore extend the pairing and -module structure to to via and extend the -bilinear pairing
| (87) |
via the same formula in which is the constant term of . This pairing identifies isomorphically with the (unrestricted) -linear dual space:
As a consequence, for any subspace , its annihilator or perp space
has a -linear isomorphism
In particular, when is a field and is finite-dimensional, then so is , since .
Definition 4.2.
For any field , and for any (possibly inhomogeneous) ideal , define its harmonic space as .
The preceding discussion shows that, for a field and whenever is finite, one has
| (88) |
The associated graded ideal has harmonic space . Before turning our attention to ideals coming from point loci, we give a general relationship between the harmonic spaces and whenever is Artinian.
Given a nonzero element , let be the bottom degree homogeneous component of . That is, if with homogeneous of degree and , we have . Also define . The map is not -linear; if and one has whereas .
Proposition 4.3.
Let be any -linear subspace of . The subspace of generated by its image under is a -linear subspace of of the same dimension as .
Proof.
Let be the -subspace consisting of series of the form where is homogeneous of degree . We have a descending filtration on
with . This induces a similar filtration
in which where . One therefore has
| (89) |
Let and consider the composite map
| (90) |
where the map is the canonical projection. Despite the fact that is not -linear, it is not hard to see that is -linear and satisfies .
Write for the subspace generated by . Then is graded, and its degree graded piece equals . Since , the linear map induces a linear map , and it is not hard to see that this latter map is bijective. Consequently, we have a linear isomorphism and
| (91) |
Given with homogeneous of degree and , define the valuation . Also define . The relationship between the set map and harmonic spaces may be stated as follows.
Corollary 4.4.
Let be any ideal (not necessarily homogeneous) such that is a finite-dimensional -vector space, so that . Then is the -linear subspace of spanned by .
Proof.
We apply Corollary 4.4 to ideals of the form for finite loci . The harmonic spaces of these ideals have simple -bases.
Definition 4.5.
For any field , given , define the exponential in the completion
where the right equality above used (41). It can be rewritten more familiarly when as
It is not hard to check that one has
| (92) |
where for and , one has
Lemma 4.6.
For a field and a finite subset, the harmonic space has -basis
Proof.
Note that (88) shows that the set in the lemma has the correct size
Hence it suffices to check that its elements lie in , and are -linearly independent.
To check that for each , by Lemma 4.1 it suffices to check that for any one has where
Thus it suffices to check . Letting , one has
Thus in the end it suffices to check for , which is not hard:
We next check that are -linearly independent within . We first deal with the case. When , if we let , so that , we note that inside , one has
These exponentials are -linearly independent because are -linearly independent and the Vandermonde matrix is invertible.
When , it is helpful to note that the factorization (92) lets one identify with the element lying within the proper555Even in characteristic zero, one has , without forming a completed tensor product. subspace
Letting for denote the coordinate projections, the case proven above shows each subset is -linearly independent. Therefore the set
| (93) |
is -linearly independent inside . Since the set in (93) contains
as a subset, the latter must therefore also be -linearly independent subset, as desired. ∎
Recall the familiar formula
| (94) |
which holds in the power series ring ; it is an easy exercise using (40) to check that it remains valid in for any field . Combining (94) with Lemma 4.6 immediately gives the following easier and more precise inhomogeneous ideal version of Theorem 1.4.
Proposition 4.7.
For any field and finite subsets , one has inside that
| (95) |
We have all of the tools necessary to prove the Minkowski Closure Theorem. Let us recall its statement.
Theorem 1.4 For any pair of finite point loci in over any field , one has
Proof.
Recall that and similarly for and . Let and be homogeneous elements. We show that as follows.
We conclude this subsection with a few remarks. The first are some interesting examples of Theorem 1.4, including two small examples with point loci in . Another example discusses the relation between Theorem 1.4 for point loci in , and the additive combinatorics of sumsets and the Cauchy-Davenport Theorem. In the next subsection, we give another proof of Theorem 1.4 which connects to deformation geometry and remark on a similar-sounding result of F. Gundlach.
Example 4.8.
One can easily have a proper inclusion . E.g., inside take
Although , the two loci are equivalent under , with
On the other hand, has dimension , and one can compute that
so that properly contains the -dimensional space in this case.
Example 4.9.
A somewhat nontrivial instance of Theorem 1.4 occurs when
where is the lattice triangle discussed in Remark 3.19. Here one can check that
as shown below:
Remark 3.19 already mentioned one can calculate by hand or Macaulay2 that
with spacing to indicate segregation of -basis elements by degree. Similarly one can calculate that
It can be checked that multiplying any two basis elements of and results in an element of . In this example, one has equality .
Example 4.10.
Let us see what Theorem 1.4 says in a -dimensional space , about the cardinalities of two finite loci , versus the cardinality of their sumset . As in the Example from the Introduction, one can easily check for finite that one has
From this one can calculate
Consequently, the assertion from Theorem 1.4 holds if and only if
In other words, Theorem 1.4 says that for , one has
| (97) |
When has characteristic zero, one has , and the sumset lower bound (97) is not hard to show directly; it is also easily seen to be sharp. When is a finite field, this lower bound is a result of Eliahou and Kervaire [17, Thm. 2.1], generalizing both the case for known as the Cauchy-Davenport Theorem, as well as the case for due to work of Yuzvinsky [55] on quadratic forms. Eliahou and Kervaire show [18, Thm. 2.2] that this lower bound (97) is also sharp for . Interestingly, their proof method for (97) uses associated graded ideals and is close in spirit to our results.
4.3. An alternative proof of Theorem 1.4
The proof of Theorem 1.4 given above was concise, but does not directly relate to the point-orbit geometry of linearly deforming a locus to the origin. We describe a method for proving Theorem 1.4 which makes this geometry apparent by introducing a parameter to situate the locus in a flat family over as follows.
Let be the univariate polynomial ring over and consider the divided power algebra with coefficients in and its completion . If , we have an -module structure and an -bilinear pairing as before in which extracts the constant term of . For , we again define the vanishing ideal in , and its annihilator/perp -submodule of :
Here we emphasize the dependence on the ring via the subscript in the notation.
Now consider for a field and finite locus , the “rescaled locus” for , along with its vanishing ideal and annihilator/perp -module
Although the ideal annihilates under the -action for , the -module contains elements which are not in the -span of . In contrast with Lemma 4.6, to obtain the full -module we must saturate with respect to as follows.
Lemma 4.11.
The above -submodule has this description:
Proof.
(Sketch) Naming the right side in the lemma as , the inclusion is not hard, and follows similarly to the proof of Lemma 4.6. To show , one strengthens the saturation property of , to show this
Claim: If and have , then .
By replacing with its algebraic closure, it is enough to establish this claim when is algebraically closed. To do this, start with a relation of the form
| (98) |
for some . If is a nonzero root of , substituting and using the -linear independence of we get for all , so we may cancel a factor of from both sides of (98). We reduce to the case where for some and the claim follows easily.
Note that in , one still has
and hence Lemma 4.11 immediately implies the following analogue of Proposition 4.7.
Corollary 4.12.
For any two finite loci , as -submodules of one has
| (99) |
We remark that one can produce small examples with showing both that the -saturation in the statement of Lemma 4.11 is necessary, and in contrast to Proposition 4.7, that the inclusion in Corollary 4.12 can be proper.
Lemma 4.13.
Let be the map which evaluates . For any finite locus , the map restricts to a surjection .
Proof.
(Sketch) One checks that indeed maps into . Row reduction over the field shows that the -span of contains elements with the following property:
each for some nonnegative power of times another element (so also lies in by Lemma 4.11) and the -images are -linearly independent in .
Since the space has -dimension , this shows that surjects. ∎
Lemma 4.13 is closely related to Corollary 4.4. Indeed, the images appearing in the above proof are nothing but the bottom degree components of the .
Second proof of Theorem 1.4.
The geometric interpretation of this second proof is as follows. As mentioned in the Introduction, one can view as a flat deformation of the reduced subscheme deforming linearly to a zero dimensional subscheme of degree supported at the origin.
The extension of scalars from to in the second proof corresponds to viewing as a flat family over via , with generic fibers over and special fiber over . Taking the limit corresponds to applying the map or (equivalently) the functor , focusing on the special fiber. Passage to the fraction field , as in the proof of Lemma 4.11, corresponds to applying , which geometrically is localization over the general point . The containment of harmonic spaces in Corollary 4.12 (in contrast to the equality in Proposition 4.7) philosophically holds because the flat family sees behavior over its special fiber which its general fiber does not.
Remark 4.14.
Theorem 1.4 has a resemblance to another interesting observation of Gundlach [26]. Suppose has characteristic 0. For any finite point set , and any choice of monomial ordering on , his result quoted as Proposition 2.9 earlier shows the set of all -standard monomials for has the following alternate description: He then uses this to fairly quickly prove the following fact [26, eqn. (5)] : for any two finite point sets , one has
| (100) |
It is not clear how closely this is related to the similar-sounding Theorem 1.4.
Remark 4.15.
An earlier version of this paper [42, p. 36-41] gives yet another proof of Theorem 1.4. Extending scalars if necessary, one assumes that is a metric topological field and considers degree-truncating surjections to finite-dimensional subspaces for sufficiently large. Theorem 1.4 is deduced from a convergence statement [42, Lem. 4.17] within Grassmannians of subspaces of these finite-dimensional vector spaces.
5. Harmonic algebras
Our motivation for Theorem 1.4 was to define here the harmonic algebra as an approach toward our main Conjecture 1.1 generalizing the Classical Ehrhart Theorem. Before defining , we explain why a less algebraic approach to Conjecture 1.1 seems elusive.
5.1. The missing valuative property
Recall from the Introduction that the Classical Ehrhart Theorem asserts several properties for the Ehrhart series of a -dimensional lattice polytope in , where . Specifically, one has its rationality with a precise denominator, the nonnegativity of the numerator coefficients , the combinatorial interpretation of the in terms of the semi-open parallelepiped (3) when is a simplex, and the reciprocity relating to .
These properties were originally given elementary proofs that avoid any commutative algebra, via the following strategy: one first proves the result for lattice simplices (and semi-open simplices), and then generalizes to arbitrary lattice polytopes employing triangulations (and sometimes shellings of a triangulation) of . The key tool in such proofs is the following valuative property that comes directly from the definitions of and : when are lattice polytopes and their intersection is a proper face of both , then
| (101) | ||||
| (102) |
One might therefore hope to prove the analogous Conjecture 1.1 on
via similar valuative techniques. The following simple example illustrates some difficulty in identifying an analogous valuative property.
Example 5.1.
Let be coprime positive integers, and let be the lattice triangles
Their union is the lattice rectangle , and their intersection is the line segment . The figure below depicts the case .
Since are equivalent under , and since and are both antiblocking polytopes, one can apply Corollary 3.15 to compute that
| (103) | ||||
| (104) |
Since are coprime, the set consists of only two lattice points, and its dilate consists of collinear lattice points, so that
| (105) |
It is not clear if there is a -analogue of the valuative property (101) that applies here to relate (103),(104),(105). This subtlety is not surprising– for general finite loci , it is difficult to predict the structure of from that of and .
5.2. The harmonic algebra and a conjecture
As explained in the Introduction, work of Stanley (see, e.g., [51]) provided an alternative proof for the assertions in the Classical Ehrhart Theorem, avoiding valuative techniques, and substituting commutative algebra. For a -dimensional lattice polytope , he considered the affine semigroup ring inside the Laurent polynomial ring and its interior ideal , defined as follows:
This ring is the affine semigroup ring associated to the polyhedral -dimensional cone
whose lattice points in form a semigroup under addition. The distinguished zeroth coordinate on endows with the structure of an -graded algebra of Krull dimension , and one can re-interpret
| (106) | ||||
| (107) |
As mentioned in the Introduction, Stanley explained the
-
•
rationality of via being a finitely generated -algebra, shown by Gordan [24],
-
•
denominator of via Noether’s Normalization Lemma [39],
-
•
nonnegativity of the via Cohen-Macaulayness of , a result of Hochster [30], and
- •
Motivated by this, we would like to eventually prove the -analogous assertions of Conjecture 1.1, via the following commutative algebra.
Definition 5.2.
Let be a lattice polytope in , and introduce the polynomial ring
We will consider as a bigraded or -graded -algebra in which
with Hilbert series in tracking a monomial by . Occasionally we will specialize this to a single -grading by sending . Then define the harmonic algebra and its interior ideal as the following two -homogeneous -linear subspaces of :
| (108) | ||||
| (109) |
We first justify the terms “algebra”, “ideal” in the definition, starting with an easy observation.
Lemma 5.3.
For nested finite subsets over any field , one has .
Proof.
implies the opposite inclusion within . This then implies . Taking perps in reverses inclusion: . ∎
Proposition 5.4.
Within the ring , the -linear subspace is an -graded -subalgebra, and is an -graded ideal of , with
| (110) | ||||
| (111) |
Furthermore, any finite subgroup of preserving acts by -graded automorphisms on . Thus in the representation ring of -graded -modules, one has
Proof.
For the first algebra assertion about , given , one must check that
However, note that one can use Theorem 1.4 to deduce the first inclusion here
The second inclusion follows from Proposition 5.3 together with this inclusion
| (112) |
derived from . The ideal assertion for is similar, replacing the last fact with . The remaining assertions follow from the definitions. ∎
We conjecture the following properties for the harmonic algebra , analogous to those known for the affine semigroup ring , which would explain most of Conjecture 1.1.
Conjecture 5.5.
For any lattice polytope , the harmonic algebra is
-
(i)
a Noetherian (finitely generated) -subalgebra of ,
-
(ii)
a Cohen-Macaulay algebra, and
-
(iii)
its canonical module is isomorphic to the ideal , up to a shift in -grading.
5.3. Relation of Conjecture 5.5 to Conjecture 1.1
We explain here the implications of the algebraic Conjecture 5.5 on for the enumerative Conjecture 1.1 on its Hilbert series .
Note that when we specialize the -grading of the harmonic algebra to the -grading, by setting , its Hilbert series becomes the Ehrhart series , matching that of the affine semigroup ring :
| (113) |
where the first equality uses (110) and the last is (106). Consequently, if Conjecture 5.5(i) holds, so is a finitely generated algebra, we know has Krull dimension , the same that of .
Also, one could then choose a finite set of -homogeneous algebra generators, e.g., by taking the set of all -homogeneous components from any finite list of algebra generators. Assume one has chosen such a list of -homogeneous algebra generators for , or even just some satisfying the weaker condition that they generate a subalgebra over which is a finite extension. Name their -degrees for , and create an -graded polynomial ring with variables having . Then the map sending makes a finitely generated -graded -module. Hilbert’s Syzygy Theorem therefore predicts the existence of a finite -graded free -resolution of
| (114) |
Here each free -module in the resolution has the form
for some -graded Betti numbers , meaning has exactly of its -basis elements of -degree . Taking the -graded Euler characteristic of the resolution (114) implies
which shows the first rationality assertion in Conjecture 1.1 on , and also Conjecture 1.1(i). This argument also works for the rationality assertion on : the ideal in would be finitely generated, due to Noetherian-ness of , and hence also a finitely generated -module.
The Cohen-Macaulayness assertion Conjecture 5.5(ii) would imply an important special case of Conjecture 1.1(ii), as follows. Since has Krull dimension , when one picks as in the previous paragraph to generate a subalgebra having as a finite extension, necessarily . Suppose is a simplex, and that , meaning contains an -homogeneous system of parameters666Noether’s Normalization Lemma shows -graded Noetherian -algebras always contain an -homogeneous system of parameters. But some -graded Noetherian -algebras have no -homogeneous system of parameters. E.g., inside with -grading where and , the -graded subalgebra of Krull dimension has no -homogeneous system of parameters , and it has . Then the previous paragraph shows that the hypothesis of Conjecture 1.1(ii) holds, and we claim that Conjecture 5.5(ii) yields the conclusion of Conjecture 1.1(ii) for the numerator of : Cohen-Macaulayness implies is a free -module, so that the resolution (114) stops at , and
In other words, . A similar argument applies to the numerator of in this case, assuming that Conjecture 5.5(iii) holds: this would imply that is isomorphic to the Cohen-Macaulay module , and hence is also a free -module.
Lastly, Conjecture 5.5(iii) could imply the -reciprocity assertion Conjecture 1.1(iii), if the shift in -grading for the isomorphism works out correctly: one knows that the canonical module for an -graded Cohen-Macaulay ring of Krull dimension satisfies
for some choice of -degree ; see Stanley [51, §I.12, p.49].
5.4. The singly-graded rings and are generally not isomorphic
In light of the fact (113) that and have the same singly-graded Hilbert series, one might wonder whether they are isomorphic as -graded algebras. This fails already for small lattice polygons, e.g., this triangle , for which a portion of is shown below:
To see that as -graded algebras, one can check that
| (115) | ||||
| (116) |
Checking (115) means showing which can be done directly. Meanwhile, the proper inclusion (116) follows from examining the first few terms of :
One sees has as its highest nonvanishing -degree , but for it is . This example seems related to the deformation forgetting algebra structure.
Remark 5.6.
In contrast to the above example, it may be interesting to note that there is a very similar-sounding algebra to which actually is isomorphic to . Recall from Section 4.2 that for the inhomogeneous ideal one can still define a perp space inside the power series completion . The space has -basis given in Lemma 4.6 by , and (94) asserts that these basis elements multiply via the usual rule
If one then considers the ring as -graded by and for , one can compile the spaces for to define an -graded subalgebra
Since has -basis , with , the map
is an -graded -algebra isomorphism.
5.5. Example: antiblocking polytopes revisited
In spite of the example in Section 5.4, there is a subfamily of lattice polytopes for which one has not only an -graded, but even an -graded algebra isomorphism , and Conjecture 5.5 holds: the antiblocking polytopes of Section 3.4.
In order to discuss this, we first note that for a lattice polytope , the affine semigroup ring has a finer -grading. This is because it is a homogeneous subalgebra of with respect to the -multigrading in which is the standard basis vector. One can then specialize this to a -grading via and for . In other words, this -grading has , tracked by the Hilbert series monomial .
Proposition 5.7.
For any antiblocking lattice polytope one has an -graded -algebra isomorphism defined by the bijection on -bases for . That is, identifying , one has equality of the two subalgebras.
Consequently, Conjecture 5.5 holds for antiblocking polytopes.
5.6. Example: Chain and order polytopes
Let be a finite poset. Stanley [50] associated two polytopes in the positive orthant of to the poset as follows. The chain polytope consists of functions satisfying
for each chain in .
The order polytope consists of functions satisfying
The chain polytope is antiblocking, so the previous example applies to describe and proves that Conjecture 5.5 holds. Although the order polytope is almost never antiblocking, we will show that Conjecture 5.5 holds for this family of polytopes, as well.
For any , Stanley introduced a piecewise-linear map
| (117) |
given by the formula
| (118) |
where the maximum is taken over elements covered by . If is a minimal element of , this maximum is interpreted to be 0. Stanley proved that is bijective, and restricts to a bijection
| (119) |
Consequently, , and . Observe that the map is only piecewise-linear; indeed, the chain and order polytopes are not in general affine-equivalent. Nevertheless, we have the following result.
Theorem 5.8.
For finite posets , inside , one has equality of the harmonic algebras of its chain and order polytopes, as well as equality for their interior ideals.
Proof.
To prove it suffices to establish for each , the equality of ideals
inside the polynomial ring where . In fact, it suffices to show an inclusion,
| (120) |
since then the equality
would show the surjection is bijective, via dimension-counting.
For any , let in . Since is antiblocking, we have
The desired inclusion (120) is then implied by the following claim.
Claim: Suppose that is not a lattice point of . Then there exists an element such that .
To prove the Claim, first observe that
The required element may be constructed explicitly. Since , there is a chain in such that . Consider the polynomial
One can readily check that It remains to show that for all in . To see this, assume for the sake of contradication that lies in but . Adopting the convention that , one can show by induction on that for each , as follows. The base case follows from . In the inductive step, note that and implies that
| (121) |
On the other hand, the fact that implies that
| (122) |
But then since lies in , together (121), (122) imply , completing the inductive step. However, one then reaches the contradiction .
The proof that is similar. It again suffices to prove for the inclusions
| (123) |
since they would then be equalities via dimension-count: as have the same classical Ehrhart series, then Ehrhart-Macdonald Reciprocity implies .
To argue (123), note that since is antiblocking, the ideal is spanned over by monomials where satisfies for some chain in . If is such a monomial, consider the polynomial
which one can readily check has . Using the description
one similarly checks vanishes on : any in with would have for , and is a contradiction. ∎
In contrast to Theorem 5.8, for most posets the affine semigroup rings of and are not isomorphic. Indeed, if are lattice polytopes, even in the category of ungraded algebras it follows from [10, Chap. 5] that if and only if is unimodularly equivalent to . Hibi and Li proved [29] that for a poset , the chain polytope and the order polytope are unimodularly equivalent if and only if does not contain the 5-element ‘X-shape’ shown below as a subposet. Theorem 5.8 therefore gives an infinite family of lattice polytopes with isomorphic harmonic algebras but nonisomorphic semigroup rings.
Remark 5.9.
We note that it is easy to compute for the two “extreme” posets on elements: chains and antichains. When is the chain on , by definition, is the simplex considered in Section 3.6. Therefore one has
When is an antichain on elements, by definition , the -dimensional cube. Using the fact that these are antiblocking polytopes, from Corollary 3.15 one can conclude that
This then implies that
| (124) |
where the descent number and major index are defined by
The last equality in (124) is a famous result of Carlitz, sometimes also credited to MacMahon; see the historical discussion surrounding Theorem 1.1 in Braun and Olsen [9].
5.7. Case study: an interesting triangle
For lattice polytopes that are not antiblocking, the analysis of can be significantly more complicated. We return to study the lattice triangle from Remark 3.19, Example 4.9, with shown below:
This triangle is not antiblocking, and is -equivalent to the second lattice triangle of area in Figure 1, with these Ehrhart and -Ehrhart series
We explain below how one can use the harmonic algebra to prove the above calculation of is correct. Before doing so, we note two interesting features of .
First, note requires canceling a numerator/denominator factor at .
Second, note that the bidegrees appearing in the denominator factors of are all different, even though the triangle has a -symmetry that swaps the two vertices . Thus even in the special case where both Conjecture 5.5(ii) and Conjecture 1.1(ii) hold and is a lattice triangle with , one should not expect some simple and natural bijection between the three vertices of and the -graded system of parameters .
Expecting that Conjecure 5.5 might hold, and examining , shown color-coded here,
| (125) |
one might expect from the denominator that the harmonic algebra contains a homogeneous system of parameters of -degrees . From the numerator one might expect to find six -basis elements whose -degrees match Using Macaulay2, one can compute -graded -bases for the harmonic spaces with , and hence -graded -bases for . We then rewrote these bases to make the action of apparent in each graded piece, and identified candidates for and the six -basis elements, color-coded in this table:
With these candidates, one can use Macaulay2 to verify (125) as follows. One can check that , which have -degrees , do generate a subalgebra of having -graded Hilbert series . Thus are algebraically independent. And then one can check that together with the six elements colored brown in the table, they generate a subalgebra of having -graded Hilbert series . Since this matches , these nine red and brown elements must generate all of . Furthermore, this shows that the six brown elements must be free -basis elements for , so it is Cohen-Macaulay, as predicted in Conjecture 5.5(ii).
Remark 5.10.
Note that this algorithm computed without finding generators for the ideals for all . We began with the guess (125) for that came from computing the Hilbert series of for small values of . The answer suggested that we might find generators for whose bidegrees all had . And indeed, after computing -bases for for , we were then able to prove that they generate . Such an algorithm is not guaranteed to terminate in all cases, but was successfully used to verify all of the in Figures 1, 2, 3.
Lastly, we use the above description of to compute the equivariant -Ehrhart series in for the action of the group of order two that swaps . As in Example 3.22, the representation ring where denote the isomorphism classes of the one-dimension trivial and nontrivial -modules, respectively. Examining the color-coded elements in the above table, one sees that they were chosen so that the red for are all -fixed, while the brown basis elements are either -fixed (carrying the representation ), or -negated (carrying the representation ), or swapped as a pair (carrying the regular representation ). Consequently, one has
| (126) |
5.8. Some cautionary remarks
We give here a few further cautionary remarks regarding the definition of and Conjecture 5.5.
Remark 5.11.
One might be tempted to define the harmonic algebra differently, in a more general context. Starting with any finite subset , over any field , one could introduce an -graded -subalgebra of the ring defined by
| (127) |
Here we adopt the convention that the 0-fold Minkowski sum of with itself is , so that . Theorem 1.4 shows that this does define a -subalgebra of . However, there are two issues with this definition (127) for , even when .
The first issue is that when one takes for a lattice polytope , the above algebra can be a proper subalgebra of the harmonic algebra . This is because the inclusion
can be strict, namely for lattice polytopes failing to have the integer decomposition property (IDP) discussed in Defintion 3.7. Although not every -dimensional lattice polytope has the IDP, a result of Cox, Haase, Hibi and Higashitani [12] shows that its dilation always has the IDP; see [12]. It follows that the affine semigroup ring is always generated by elements of -degree at most . However, the interesting lattice triangle discussed in Section 5.7 required an algebra generator for its harmonic algebra of -degree , showing that sometimes we must go to higher degrees than to generate .
A second issue with the above definition (127) for is that it is not always finitely generated as an algebra, even when and . For example, let . It is not hard to check that the -fold Minkowski sum for all . This means that, after identifying with , one has , and
| (128) |
Therefore is the semigroup ring for the additive subsemigroup of that may be visualized as follows, with a dot at coordinates representing in :
This illustrates the lack of finite generation: minimal monomial -algebra generators for are
This does not violate Conjecture 5.5(i), since for any lattice polytope . But proving Conjecture 5.5 must involve extra features of harmonic spaces of lattice points inside polytopes.
Remark 5.12.
One can regard as a functor , where is the category of -graded -algebras. Here is a category of lattice polytopes whose morphisms are -linear maps with . The fact that this induces an -algebra map stems from the fact that the point configurations and have . This implies that the map that precomposes will have , and hence . Therefore the adjoint map , when extended to a ring map satisfies
If one instead considers as a functor into the category of -graded -algebras, one can compare it to the similar such functor . These two functors and are not isomorphic; see the example in Section 5.4. The authors do not know if they are equivalent.
Remark 5.13.
One might hope to find some canonical -linear basis for . Although the authors are unaware of such a basis which is -homogeneous, there is at least a canonical -homogeneous -basis, coming from such -bases for harmonic spaces of finite subsets .
To describe these, assume is a field of characteristic zero, and identify and its basis with and its basis . Then mapping gives the middle isomorphism here
This lets one view the subspace as a subspace of .
Proposition 5.14.
For a field of characteristic zero, and any finite point set , there is a unique basis of the harmonic space such that for all .
Proof.
Since , one has a -vector space direct sum decomposition . Consequently, the composite map is a -vector space isomorphism. Thus any set of homogeneous polynomials that give a -basis for the (graded) subspace will descend to a -basis of . Proposition 2.1(ii) implies that the same set of polynomials will also descend to a basis of . Hence this composite is also a -vector space isomorphism:
| (129) |
Now multivariate Lagrange interpolation identifies
| (130) |
and the space on the right has a unique -basis of defined by . The assertion then follows from combining this with (129) and (130). ∎
Note that Proposition 5.14 provides -bases that are generally inhomogeneous. Nevertheless, given a lattice polytope , and taking , one can assemble all of these distinguished -bases for into an -graded -basis for . Note that, even though this -basis for is in bijection with an -basis for the semigroup ring , the bijection does not respect multiplication. This is to be expected, since the example discussed in Section 5.4 shows that these two rings are not isomorphic as -graded -algebras.
6. Dilations, products, and joins: proof of Theorem 1.3
Recall from the Introduction these three basic operations on polytopes:
-
•
dilation by an positive integer factor , sending to ,
-
•
Cartesian product, sending and to ,
-
•
free join, sending to defined by
We wish to understand how these operations interact with -Ehrhart series . In the process we will see how they interact with harmonic algebras , by relating them to the commutative algebra constructions for -graded algebras of Veronese subalgebras, Segre products and graded tensor products; When applying these constructions to the -graded algebra , we will always use the grading specialization to regard it as an -graded algebra. The analysis will also prove Theorem 1.3, whose statement we recall here.
Theorem 1.3. Let be lattice polytopes.
-
(i)
For positive integers , the dilation has given by
-
(ii)
The Cartesian product has given by the Hadamard product
-
(iii)
The free join has given by
We deal with each of the three parts of this theorem in the next three subsections.
6.1. Dilations and the Veronese construction
The first and simplest of these is given by dilation . If is a graded algebra, the Veronese subalgebra is
| (131) |
Then Theorem 1.3(i) along with the following result are immediate from the definitions.
Proposition 6.1.
For a lattice polytope and any one has .
6.2. Cartesian products and the Segre construction
The Cartesian product of lattice polytopes is again a lattice polytope. Since , it is easily seen that
There is a similarly easy relation for their harmonic algebras, which comes from a general lemma on the behavior of orbit harmonic rings and harmonic spaces when one has a Cartesian product of point loci.
Lemma 6.2.
For a field and finite subsets , one has graded isomorphisms
Proof.
Write for the polynomial rings which contain , so that will be an ideal in . We claim one has the following equality of ideals
| (132) |
from which the remaining assertions will follow. To see this claim, first check the containment : for and , one has both and lying in , and if , their top degree components and both lie in . On the other hand, the inclusion must be an equality because the surjection
is an isomorphism via dimension-counting:
For -graded -algebras , their Segre product is the graded algebra
| (133) |
Proposition 6.3.
For any lattice polytopes and , one has
Proof.
This comes via Lemma 6.2 and the definitions, as ∎
6.3. Joins
Recall that for polytopes , their (free) join is
If and are lattice polytopes, the join is again a lattice polytope. The effect of join on harmonic algebras is more complicated than dilation and Cartesian product. We will eventually see, in Theorem 6.6 below, that it is almost, but not quite, the -graded tensor product
We first recall some facts on joins of lattice polytopes in . Let denote the extra first coordinate in the ambient space for the join . Slicing by the hyperplanes gives a disjoint decomposition of the dilates for :
| (134) |
We illustrate (134) for with , when and :
This decomposition (134) implies a well-known fact (see, e.g., Beck and Robins [5, Exer. 3.33]):
| (135) |
This fact (135) is easily seen to be equivalent to the following simple relation of Ehrhart series:
| (136) |
Recalling that , the next result is the -analogue of (136).
Theorem 6.4.
For lattice polytopes , one has the following relation among the -graded Hilbert series for the harmonic algebras :
As mentioned in Remark 3.29, if we take to be a single point, Theorem 6.4 says that the -Ehrhart series of the pyramid is .
Proof.
Letting , we start by describing the ideal
The product structure in each term of (134) leads to some natural elements of and . First translate , without loss of generality, so that and , implying nestings
| (137) |
Suppose have , and one is given and . Then we claim that the nestings (137) imply that the polynomial
| (138) |
vanishes on . To see this, note that a typical point of of will either
-
•
have so vanishes due to one of its factors ,
-
•
or else have , and then vanishes due to the factor , since ,
-
•
or else have , and then vanishes due to the factor , since
Thus lies in , implying that contains its top degree homogeneous component
| (139) |
Note also that vanishes on . Hence contains this ideal:
| (140) |
We wish to show that, in fact, this containment is an equality. To this end, we calculate the perp space within for the ideal . Abbreviate
| (141) |
By Lemma 5.3 we have
| (142) |
The following space of polynomials is annihilated by under the -action, so lies in :
| (143) |
By (148) in Lemma 6.5 below, has the following dimension:
Therefor these inequalities
must all be equalities, implying
| (144) | ||||
| (145) |
The previous proof used the following technical lemma, on Hilbert series arising from graded tensor products of spaces equipped with homogeneous filtrations.
Lemma 6.5.
Let be graded vector spaces, each with nested sequences of homogeneous subspaces
Then one has the following Hilbert series identity for each :
| (147) |
In particular, setting , one has
| (148) |
Proof.
Since the spaces and nest, the sum of subspaces of is not a direct sum. However, after abbreviating
the left side of the lemma can be rewritten
| (149) |
whose coefficient of is
abbreviating . Meanwhile, the right side of the lemma can be rewritten
| (150) |
Hence the coefficient of on the right side of the lemma is
We are ready to relate the harmonic algebras and , starting with the following suggestive rewriting of Theorem 6.4:
| (151) |
To interpret this identity, it helps to name the “auxiliary” variables in as , and to abbreviate the other variable sets , . This means that the rings , are -subalgebras of the following ambient polynomial algebras
Their -gradings have , and all other variables have . Since the ambient polynomial algebras are integral domains, so are the harmonic algebras inside them. Hence one concludes that
-
•
is a nonzero divisor inside , with -degree , and
-
•
is a nonzero divisor inside , with -degree .
In a -graded algebra with a homogeneous nonzerodivisor of degree , the exact sequence
shows that one has the Hilbert series relation
Thus the left and right sides of (151) can reinterpreted as Hilberts series for these quotient rings:
In this way, the following result is an algebraic strengthening of Theorem 6.4.
Theorem 6.6.
For any lattice polytopes and , one has an -graded algebra isomorphism
Proof.
In light of the preceding discussion on Hilbert series, it suffices by dimension-counting to exhibit an -graded -algebra surjection from the left side to the right side in the theorem.
Consider the -linear map defined via
One can readily check that
-
•
is actually a map of -graded -algebras,
- •
-
•
that lies in its kernel.
Consequently, descends to a map on the quotient
| (152) |
This map is not surjective. However, after post-composing it with the canonical projection map , then we claim that
| (153) |
actually is the surjection that we seek. To see this, note that the image of within is
On the other hand, (143) shows the remaining part of not lying in this image has the form
which is contained in the ideal and therefore vanishes in the quotient . ∎
References
- [1] P. Adeyemo and B. Szendrői. Refined Ehrhart series and bigraded rings. Studia Sci. Math. Hungar., 60(2-3):97–108, 2023.
- [2] K. Akin, D. A. Buchsbaum, and J. Weyman. Schur functors and Schur complexes. Adv. in Math., 44(3):207–278, 1982.
- [3] D. Armstrong, V. Reiner, and B. Rhoades. Parking spaces. Advances in Mathematics, 269:647–706, 2015.
- [4] G. Balletti. Enumeration of lattice polytopes by their volume. Discrete Comput. Geom., 65(4):1087–1122, 2021.
- [5] M. Beck and S. Robins. Computing the continuous discretely. Integer-point enumeration in polyhedra. With illustrations by David Austin. Undergraduate Texts Math. New York, NY: Springer, 2nd edition edition, 2015.
- [6] M. Beck and R. Sanyal. Combinatorial reciprocity theorems, volume 195 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2018. An invitation to enumerative geometric combinatorics.
- [7] F. Bergeron. Algebraic combinatorics and coinvariant spaces. CMS Treatises in Mathematics. Canadian Mathematical Society, Ottawa, ON; A K Peters, Ltd., Wellesley, MA, 2009.
- [8] B. Braun. Unimodality problems in Ehrhart theory. In Recent trends in combinatorics, volume 159 of IMA Vol. Math. Appl., pages 687–711. Springer, [Cham], 2016.
- [9] B. Braun and M. Olsen. Euler-Mahonian statistics and descent bases for semigroup algebras. European J. Combin., 69:237–254, 2018.
- [10] W. Bruns and J. Gubeladze. Polytopes, rings, and K-theory. Springer Science & Business Media, 2009.
- [11] F. Chapoton. -analogues of Ehrhart polynomials. Proc. Edinb. Math. Soc. (2), 59(2):339–358, 2016.
- [12] D. A. Cox, C. Haase, T. Hibi, and A. Higashitani. Integer decomposition property of dilated polytopes. Electron. J. Combin., 21(4), 2012. #P4.28.
- [13] D. A. Cox, J. Little, and D. O’Shea. Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. Undergraduate Texts Math. Cham: Springer, 4th revised ed. edition, 2015.
- [14] V. I. Danilov. The geometry of toric varieties. Uspekhi Mat. Nauk, 33(2(200)):85–134, 247, 1978.
- [15] E. Ehrhart. Sur les polyèdres rationnels homothétiques à n dimensions. CR Acad. Sci. Paris, 254:616, 1962.
- [16] D. Eisenbud. Commutative algebra, volume 150 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. With a view toward algebraic geometry.
- [17] S. Eliahou and M. Kervaire. Sumsets in vector spaces over finite fields. Journal of Number Theory, 71(1):12–39, 1998.
- [18] S. Eliahou and M. Kervaire. Sumsets in vector spaces over finite fields. J. Number Theory, 71(1):12–39, 1998.
- [19] D. R. Fulkerson. Blocking and anti-blocking pairs of polyhedra. Math. Programming, 1:168–194, 1971.
- [20] D. R. Fulkerson. Anti-blocking polyhedra. J. Combinatorial Theory Ser. B, 12:50–71, 1972.
- [21] A. M. Garsia and C. Procesi. On certain graded -modules and the -Kostka polynomials. Advances in Mathematics, 94(1):82–138, 1992.
- [22] L. Geissinger and D. Kinch. Representations of the hyperoctahedral groups. J. Algebra, 53(1):1–20, 1978.
- [23] A. V. Geramita. Inverse systems of fat points: Waring’s problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein ideals. In The Curves Seminar at Queen’s, Vol. X (Kingston, ON, 1995), volume 102 of Queen’s Papers in Pure and Appl. Math., pages 2–114. Queen’s Univ., Kingston, ON, 1996.
- [24] P. Gordan. Ueber die Auflösung linearer Gleichungen mit reellen Coefficienten. Math. Ann., 6(1):23–28, 1873.
- [25] S. Griffin. Ordered set partitions, Garsia-Procesi modules, and rank varieties. Transactions of the American Mathematical Society, 374(4):2609–2660, 2021.
- [26] F. Gundlach. Polynomials vanishing at lattice points in a convex set, 2021.
- [27] J. Haglund. The ,-Catalan numbers and the space of diagonal harmonics, volume 41 of University Lecture Series. American Mathematical Society, Providence, RI, 2008. With an appendix on the combinatorics of Macdonald polynomials.
- [28] J. Haglund, B. Rhoades, and M. Shimozono. Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture. Advances in Mathematics, 329:851–915, 2018.
- [29] T. Hibi and N. Li. Unimodular equivalence of order and chain polytopes. Mathematica Scandinavica, pages 5–12, 2016.
- [30] M. Hochster. Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes. Ann. of Math. (2), 96:318–337, 1972.
- [31] J. Huang and B. Rhoades. Ordered set partitions and the -Hecke algebra. Algebraic Combinatorics, 1(1):47–80, 2018.
- [32] J. Huang, B. Rhoades, and T. Scrimshaw. Hall-Littlewood polynomials and a Hecke action on ordered set partitions. Proceedings of the American Mathematical Society, 147(5):1839–1850, 2019.
- [33] B. Kostant. Lie group representations on polynomial rings. Amer. J. Math., 85:327–404, 1963.
- [34] J. C. Lagarias and G. M. Ziegler. Bounds for lattice polytopes containing a fixed number of interior points in a sublattice. Canad. J. Math., 43(5):1022–1035, 1991.
- [35] M. J. Liu. Viennot shadows and graded module structure in colored permutation groups. arXiv preprint arXiv:2401.07850, 2024.
- [36] N. A. Loehr and J. B. Remmel. A computational and combinatorial exposé of plethystic calculus. J. Algebraic Combin., 33(2):163–198, 2011.
- [37] I. G. Macdonald. Polynomials Associated with Finite Cell-Complexes. Journal of the London Mathematical Society, 2(1):181–192, 1971.
- [38] I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford Classic Texts in the Physical Sciences. The Clarendon Press, Oxford University Press, New York, second edition, 2015. With contribution by A. V. Zelevinsky and a foreword by Richard Stanley, Reprint of the 2008 paperback edition [ MR1354144].
- [39] E. Noether. Der endlichkeitsatz der invarianten endlicher linearer gruppen der charakteristik p. pages 28–35, 1926.
- [40] J. Oh and B. Rhoades. Cyclic sieving and orbit harmonics. Mathematische Zeitschrift, 300(1):639–660, 2022.
- [41] M. Reineke, B. Rhoades, and V. Tewari. Zonotopal algebras, orbit harmonics, and Donaldson–Thomas invariants of symmetric quivers. International Mathematics Research Notices, 2023(23):20169–20210, 2023.
- [42] V. Reiner and B. Rhoades. Harmonics and graded Ehrhart theory. arXiv preprint arXiv:2407.06511v2, 2024. Version 2.
- [43] B. Rhoades. Increasing subsequences, matrix loci, and Viennot shadows. arXiv preprint arXiv:2306.08718, 2023.
- [44] B. E. Sagan. The symmetric group, volume 203 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2001. Representations, combinatorial algorithms, and symmetric functions.
- [45] R. P. Stanley. Combinatorial reciprocity theorems. Advances in Math., 14:194–253, 1974.
- [46] R. P. Stanley. Combinatorial reciprocity theorems. In Combinatorics: Proceedings of the NATO Advanced Study Institute held at Nijenrode Castle, Breukelen, The Netherlands 8–20 July 1974, pages 307–318. Springer, 1975.
- [47] R. P. Stanley. Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc. (N.S.), 1(3):475–511, 1979.
- [48] R. P. Stanley. Decompositions of rational convex polytopes. Ann. Discrete Math, 6(6):333–342, 1980.
- [49] R. P. Stanley. Linear Diophantine equations and local cohomology. Invent. Math., 68(2):175–193, 1982.
- [50] R. P. Stanley. Two poset polytopes. Discrete & Computational Geometry, 1(1):9–23, 1986.
- [51] R. P. Stanley. Combinatorics and commutative algebra, volume 41 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, second edition, 1996.
- [52] R. P. Stanley. Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.
- [53] R. P. Stanley. Enumerative combinatorics. Volume 1, volume 49 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2012.
- [54] A. Stapledon. Equivariant Ehrhart theory. Adv. Math., 226(4):3622–3654, 2011.
- [55] S. Yuzvinsky. Orthogonal pairings of Euclidean spaces. Michigan Math. J., 28(2):131–145, 1981.