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Weak Bruhat interval modules of the 0-Hecke algebra
Abstract.
The purpose of this paper is to provide a unified method for dealing with various 0-Hecke modules constructed using tableaux so far. To do this, we assign a -Hecke module to each left weak Bruhat interval, called a weak Bruhat interval module. We prove that every indecomposable summand of the -Hecke modules categorifying dual immaculate quasisymmetric functions, extended Schur functions, quasisymmetric Schur functions, and Young row-strict quasisymmetric Schur functions is a weak Bruhat interval module. We further study embedding into the regular representation, induction product, restriction, and (anti-)involution twists of weak Bruhat interval modules.
Key words and phrases:
-Hecke algebra, representation, weak Bruhat order, quasisymmetric characteristic2020 Mathematics Subject Classification:
20C08, 05E10, 05E051. Introduction
The -Hecke algebra is a degenerate Hecke algebra obtained from the generic Hecke algebra by specializing to . The representation theory of is very complicated, as can be inferred from the fact that it is not representation-finite for (see [7, 8]). Nevertheless, it has attracted the attention of many mathematicians because of its close connection with quasi-symmetric functions. This link was discovered by Duchamp, Krob, Leclerc, and Thibon [9], who constructed an isomorphism called the quasisymmetric characteristic between the Grothendieck ring associated to -Hecke algebras and the ring of quasisymmetric functions. In particular, since the mid-2010s, there have been many attempts to construct -modules categorifying important quasisymmetric functions using tableau models, rather than simply adding irreducible modules (for instance, see [1, 2, 21, 24, 26]).
The purpose of the present paper is to provide a method to treat these modules in a uniform manner. We start with the observation that every indecomposable direct summand of these modules has a basis isomorphic to a left weak Bruhat interval of when it is equipped with the partial order defined by
This leads us to consider the -module for each weak Bruhat interval , called the weak Bruhat interval module associated to , whose underlying space is the -span of and whose action is given by
In a similar point of view, we also consider the -module for each weak Bruhat interval , called the negative weak Bruhat interval module associated to , whose underlying space is the -span of and whose action is given by
Here . It should be pointed out that Hivert, Novelli, and Thibon [13] introduced semi-combinatorial -modules associated to Yang-Baxter intervals to study the representation theory of -Ariki-Koike-Shoji algebras, and our and can also be recovered by the and specialization of these modules, respectively. Here, is the longest element of .
The family of weak and negative weak Bruhat interval modules is very adequate to our purpose in that it contains many -modules of our interest such as projective indecomposable modules, irreducible modules, the specializations of the Specht modules of in [8], and all indecomposable direct summands of the -modules in [1, 2, 21, 24, 26]. What is more appealing is that weak and negative weak Bruhat interval modules can be embedded into the regular representation of and they behave very nicely with respect to induction product, restriction, and (anti-)involution twists. Let be the full subcategory of the category of finite dimensional -modules whose objects are direct sums of weak and negative weak Bruhat interval modules up to isomorphism. From a categorical point of view, is a good subcategory in the sense that
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•
the Grothendieck group of is isomorphic to the Grothendieck group of , and
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•
the subcategory of is closed under induction product, restriction, and (anti-)involution twists.
In Section 3, we study structural properties of weak Bruhat interval modules. In the first two subsections, we present background material for weak and negative weak Bruhat interval modules and then construct an -module isomorphism
(Theorem 3.5). As an immediate consequence of this isomorphism, we see that projective indecomposable modules and irreducible modules appear as weak Bruhat interval modules up to isomorphism. By a slight modification of , we also see that the specializations of the Specht modules of are certain involution twists of weak Bruhat interval modules (Remark 3.7). Although not covered here, the isomorphism and its modification reveal interesting connections between certain -modules and weak Bruhat interval modules (Section 5 (3)).
In the third subsection, we study restriction of weak Bruhat interval modules. As for induction product of weak Bruhat interval modules, the following formula can be derived from [13]:
| (1.1) |
(see Lemma 3.8). We provide an explicit formula concerning restriction of weak Bruhat interval modules. Let be the set of -element subsets of . Given , set
With this notation, our restriction rule appears in the following form:
| (1.2) |
(Theorem 3.12). For undefined notations , , , and , see (3.11) and (3.13). The elements in parametrize the direct summands appearing in the right hand side of (1.2). It would be nice to find an easy description of this set, but it is not available at the current stage. Combining (1.1) with (1.2) yields a Mackey formula for weak Bruhat interval modules. We prove that this is a natural lift of the Mackey formula due to Bergeron and Li [3], which works for elements of the Grothendieck ring of -Hecke algebras, to weak Bruhat interval modules (Theorem 3.14).
In the final subsection, we describe the (anti-)involution twists of weak Bruhat interval modules for the involutions and the anti-involution due to Fayers [10] and then demonstrate the patterns how these (anti-)involution twists act with respect to induction product and restriction. Here, , , and for . Given an -module and an (anti-)automorphism , we denote by the -twist of . The precise definition can be found in (3.18) and (3.19). We prove that
Here, is the conjugation of by . Using these isomorphisms, we can also describe the twists for the compositions of , and . For the full list of (anti-)involution twists, see Table 3.1 and Table 3.2. Next, we explain the patterns how the (anti-)involution twists act with respect to induction product and restriction. Let , , and be weak Bruhat interval modules of , , and respectively. We see that
and
(Corollary 3.18).
Section 4 is devoted to show that every indecomposable direct summand arising from the -modules in [1, 2, 21, 24, 26] are weak Bruhat interval modules up to isomorphism. We begin with reviewing the results in these papers. Let be a composition of .
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Tewari and van Willigenburg [24] construct an -module by defining a -Hecke action on the set of standard reverse composition tableaux of shape . Its image under the quasisymmetric characteristic is the quasisymmetric Schur function attached to .
- •
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Berg et al. [2] construct an indecomposable -module by defining a -Hecke action on the set of standard immaculate tableaux of shape . Its image under the quasisymmetric characteristic is the dual immaculate quasisymmetric function attached to .
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•
Searles [21] constructs an indecomposable -module by defining a -Hecke action on the set of standard extended tableaux of shape . Its image under the quasisymmetric characteristic is the extended Schur function attached to .
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Bardwell and Searles [1] construct an -module by defining a -Hecke action on the set of standard Young row-strict tableaux of shape . Its image under the quasisymmetric characteristic is the Young row-strict quasisymmetric Schur function attached to .
The modules and are indecomposable, whereas , and are not indecomposable in general. The problem of decomposing into indecomposables have been completely settled out by virtue of the papers [5, 15, 24, 26]. Indeed, has the decomposition of the form
where represents the set of equivalence classes of under a certain distinguished equivalence relation and is the indecomposable submodule of spanned by .
To achieve our purpose, we first show that all of , , and have the source and sink, which are written as follows:
|
For the definitions of source and sink, see Definition 4.1. Then, using the essential epimorphisms constructed in [6], we introduce two readings and , where the former is defined on and and the latter is defined on each class . With this preparation, we prove that
(Theorem 4.4 and Theorem 4.8). It should be remarked that a reading on different from ours has already been introduced by Tewari and van Willigenburg [24]. They assign a reading word to each standard reverse composition tableau and show that is isomorphic to as graded posets. The weak Bruhat interval module , however, is not isomorphic to in general (see Remark 4.9). Concerned with , we consider its permuted version instead of itself. We introduce new combinatorial objects, called permuted standard Young row-strict tableaux of shape and type , and define a -Hecke action on them. The resulting module turns out to be isomorphic to the -twist of . This enables us to transport various properties of to in a functorial way.
In the final section, we provide some future directions to pursue.
2. Preliminaries
Given any integers and , define to be the interval whenever and the empty set else. For simplicity, we set . Unless otherwise stated, will denote a nonnegative integer throughout this paper.
2.1. Compositions and their diagrams
A composition of , denoted by , is a finite ordered list of positive integers satisfying . We call () a part of , the length of , and the size of . And, we define the empty composition to be the unique composition of size and length .
Given and , let and . The set of compositions of is in bijection with the set of subsets of under the correspondence (or ). The reverse composition of is defined to be , the complement of be the unique composition satisfying , and the conjugate of be the composition .
2.2. Weak Bruhat orders on the symmetric group
Every element of the symmetric group may be written as a word in with . A reduced expression for is one of minimal length. The number of simple transpositions in any reduced expression for , denoted by , is called the length of . Let
It is well known that if in one-line notation, then
| (2.1) |
The left weak Bruhat order (resp. right weak Bruhat order ) on is the partial order on whose covering relation (resp. ) is defined as follows: if and only if (resp. if and only if ). Equivalently, for any ,
| if and only if and for some , and | ||
Although these two weak Bruhat orders are not identical, there exists a poset isomorphism
To avoid redundant overlap, our statements will be restricted to the left weak Bruhat order.
Given and , the closed interval is called the left weak Bruhat interval from to and denoted by . It can be represented by the colored digraph whose vertices are given by and -colored arrows given by
Let us collect notations which will be used later. We use to denote the longest element in . For , let be the parabolic subgroup of generated by and the longest element in . For , let . Finally, for , we let .
2.3. The -Hecke algebra and the quasisymmetric characteristic
The -Hecke algebra is the associative -algebra with generated by the elements subject to the following relations:
Frequently we use another set of generators .
For any reduced expression for , let
It is well known that these elements are independent of the choices of reduced expressions, and both and are -bases for .
According to [20], there are pairwise inequivalent irreducible -modules and pairwise inequivalent projective indecomposable -modules, which are naturally indexed by compositions of . For , let and endow it with the -action as follows: for each ,
This module is the irreducible -dimensional -module corresponding to . And, the projective indecomposable -module corresponding to is given by the submodule
of the regular representation of . It is known that is isomorphic to , where is the radical of , the intersection of maximal submodules of .
Let denote the -span of the isomorphism classes of finite dimensional -modules. We denote by the isomorphism class corresponding to an -module . The Grothendieck group is the quotient of modulo the relations whenever there exists a short exact sequence . The irreducible -modules form a free -basis for . Let
Let us review the beautiful connection between and the ring of quasisymmetric functions. For the definition of quasisymmetric functions, see [23, Section 7.19].
For a composition , the fundamental quasisymmetric function , introduced in [11], is defined by
It is known that forms a -basis for . When is an -module and is an -module, we write for the induction product of and , that is,
Here, is viewed as the subalgebra of generated by .
It was shown in [9] that, when is equipped with this product as multiplication, the linear map
| (2.2) |
called quasisymmetric characteristic, is a ring isomorphism. Indeed it is not only a ring isomorphism, but also a Hopf algebra isomorphism.
3. Weak Bruhat interval modules
In this section, we present background material for weak Bruhat interval modules and then investigate their structural properties extensively.
3.1. Definition and basic properties
Let us start with the definitions of weak and negative weak Bruhat interval modules.
Definition 3.1.
Let .
-
(1)
The weak Bruhat interval module associated to , denoted by , is the -module with as the underlying space and with the -action defined by
(3.1) -
(2)
The negative weak Bruhat interval module associated to , denoted by , is the -module with as the underlying space and with the -action defined by
Hivert, Novelli, and Thibon [13] introduced a semi-combinatorial -module associated to the interval of a Yang-Baxter basis for each . We omit the proof of the well-definedness of and since they can be recovered as the semi-combinatorial modules associated to and , respectively.
The following properties are almost straightforward. In particular, the last one can be obtained by mimicking [24, Section 5].
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Given , the linear map , sending , is an injective -module homomorphism.
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Given , the linear map , sending if and else, is a surjective -module homomorphism.
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, where .
Finally, we briefly remark the classification of weak Bruhat interval modules up to isomorphism. As pointed out in Subsection 2.2, every weak Bruhat interval can be viewed as a colored digraph. One sees from (3.1) that if there is a descent-preserving (colored digraph) isomorphism between and . For instance,
is such an isomorphism, thus . On the other hand, it is not difficult to show that there is no descent-preserving isomorphism between and although they are isomorphic as colored digraphs. Indeed, is indecomposable (Remark 4.9), whereas can be decomposed into as seen in Figure 3.1. It would be very nice to characterize when a descent-preserving isomorphism between two intervals exists and ultimately to classify all weak Bruhat interval modules up to isomorphism.
3.2. Embedding weak Bruhat interval modules into the regular representation
The purpose of this subsection is to see that every weak Bruhat interval modules can be embedded into the regular representation. More precisely, we prove that is isomorphic to .
For ease of notation, we use to denote an arbitrary element in for each . The following relations among ’s and ’s, which will be used significantly, can be easily verified:
| (3.2) | |||||
One sees that , and are missing in the list of (3.2). The following lemma tells us the reason for this.
Lemma 3.2.
For with , let and be arbitrary reduced expressions of and , respectively. Let be the word
from the alphabet , where are viewed as letters not as elements of . Then every word obtained from by applying only the following six braid relations and three commutation-relations
| (3.3) | ||||
| (3.4) |
contains no subwords of the form
| (3.5) |
Proof.
Let denote the number of occurrences of the relations in (3.3) in the middle of obtaining from by applying the relations in both (3.3) and (3.4). We will prove our assertion using the mathematical induction on . When , our assertion is obvious since is obtained from by applying the relations in (3.4) only.
Given a positive integer , assume that our assertion holds for all whenever . Now, let , which means that there is a word such that and is obtained from by applying the relations in (3.4) and one of the relations in (3.3) only once. By the first paragraph, we have only to consider the case where is obtained from by applying a relation in (3.3) to for some . Suppose that contains a subword of the form (3.5). We observe that no words in (3.5) appear in the six relations in (3.3), so . In case where , is of the form (3.5). But, this is absurd since . In case where , has a consecutive subword of the form or . For , let be the subindex given by . Using the notation instead of and , one can see that the relations in both (3.3) and (3.4) coincide with braid and commutation relations of . It says that is obtained from by applying braid and commutation relations of , thus is a reduced expression for . This, however, cannot occur since contains a consecutive subword of the form or . Consequently, contains no subwords of the form (3.5), so we are done. ∎
Given a word from the alphabet , we can naturally see it as an element of . For clarity, we write it as . If a word is obtained from by applying the relations in (3.3) and (3.4), then by (3.2). Combining (3.2) with Lemma 3.2 yields the following lemma.
Lemma 3.3.
For with , if , then there exists a word from the alphabet such that and .
Proof.
Let be a reduced expression of and a reduced expression of . Since , by the exchange property of Coxeter groups, there exists such that , where denotes the permutation obtained from by removing . Viewing just as a word from the alphabet , due to (3.2) and Lemma 3.2, the relations in (3.3) and (3.4) play the same role as the braid and commutation relations of . This implies that Let if and if . Note that, while applying the relations in (3.3) and (3.4) to , the values of ’s under are staying unchanged as seen in the following figure:
Thus, we conclude that . ∎
Given , let us say that appears at if . The following lemma plays a key role in describing the -action on .
Lemma 3.4.
For any , the following hold.
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(1)
is nonzero if and only if .
-
(2)
If , then is a unique element of maximal length in .
-
(3)
if and only if .
Proof.
Let be a reduced expression of and a reduced expression of .
Let us prove the “if” part of (1) and (2) simultaneously. For each , let be the integer defined by
It is clear that for all with . Thus, is a unique element of maximal length in and .
For the “only if” part of (1), we prove that if , then . Since , there exists such that
that is, . Let . Since is a reduced expression of , we have that . By Lemma 3.3, there exists a word such that and . But, since for all , this implies that
To prove (3), note that for any , and . This implies that if and only if . Now our assertion is obvious from . ∎
Now we are ready to state the main theorem of this subsection.
Theorem 3.5.
Let .
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(1)
The set forms a -basis for .
-
(2)
For any and ,
Moreover, if , then if and only if .
-
(3)
The linear map defined by
is an -module isomorphism.
Proof.
The assertion (2) follows from the definition of and Lemma 3.4 (1).
For the assertion (1), we first observe that is spanned by . But, Lemma 3.4 (1) and Lemma 3.4 (3) ensure that unless . To prove that is linearly independent, we suppose that , but not all coefficients are zero. Let and choose a permutation which is of maximal length in . Again, Lemma 3.4 (1) and Lemma 3.4 (3) ensure that is nonzero for all . Combining this fact with Lemma 3.4 (2) shows that cannot appear at for any . It means that appears at , which is absurd.
Finally, let us prove the assertion (3). Suppose that and . For our purpose, by virtue of (1) and (2), we have only to prove that is equivalent to . Let us first show that implies that . Since , there exists such that
| (3.6) |
From the second equality in (3.6) together with the assumption , we deduce that . Applying this to the first equality in (3.6) says that , equivalently, or equally . Now, is obvious since by the first equality in (3.6). Next, let us show that implies that . The assumption says that there exists a permutation such that and , thus . ∎
In Subsection 2.3, we introduced projective indecomposable modules and irreducible modules . To be precise, and is isomorphic to and one-dimensional. By applying Theorem 3.5, we derive the following isomorphisms of -modules:
Example 3.6.
Since with , by Theorem 3.5, we have an -module isomorphism . This is illustrated in the following figure:
Remark 3.7.
It is remarked in [8, Definition 4.3] that all the specializations of the Specht modules of are of the form for some and . On the other hand, following the way as in Theorem 3.5, we can deduce that the -linear map
is an -module isomorphism. Combining these results, we see that all the specializations of the Specht modules of appear as for some and .
3.3. Restriction and Mackey formula
Throughout this subsection, and denote positive integers.
Hivert, Novelli, and Thibon [13] presented a formula on induction product of semi-combinatorial modules associated with Yang-Baxter intervals. Using this formula, one can derive the following lemma.
Lemma 3.8.
Remark 3.9.
For and , let be the set of permutations satisfying that and are subwords of and . For and , let
It is not difficult to show that for and . Therefore, Lemma 3.8 can be rewritten as . In particular, , which can be regarded as a lift of
Here, and are arbitrary permutations satisfying and .
In contrast to induction product, restriction of semi-combinatorial modules associated with Yang-Baxter intervals has not yet been well studied except for the simple and projective indecomposable modules (see [14]). 222 In fact, in [14], the author considered the induction product and restriction of the simple and projective indecomposable modules over the 0-Hecke algebra of not only type but also type and . The main purpose of this section is to provide an explicit restriction rule for weak Bruhat interval modules. To begin with, let us collect necessary notations. Let be the set of -element subsets of , on which acts in the natural way, that is, . Given and , let
| (3.7) | ||||
For instance, . A simple calculation shows that .
When , write as . Let and be the permutations in given by
respectively. For instance, and . Note that
| (3.10) |
which will be used in the proof of Lemma 3.10. For each , let
| (3.11) |
The following lemma shows that .
Lemma 3.10.
Let . For , we have
Proof.
By a careful reading of the proof of Lemma 3.10, one can derive that
Example 3.11.
Given , let and . Let and be the permutations given by
| (3.13) |
For instance, if , , and , then and .
Theorem 3.12.
For , we have
Proof.
For each , we observe that is closed under the -action for . This means, by virtue of Lemma 3.10, that is an -module. Since
it follows that
On the other hand, is in bijection with under , which again induces an -module isomorphism
as required. ∎
Example 3.13.
Next, let us deal with a Mackey formula for weak Bruhat interval modules. Bergeron and Li [3, Subsection 3.1 (5)] provide a Mackey formula working on the Grothendieck ring of -Hecke algebras. It says that for any -module , -module , and ,
where
|
|
is the functor sending . On the other hand, by combining Lemma 3.8 with Theorem 3.12, we can derive a formula working weak Bruhat interval modules:
| (3.14) |
Although it is very naive, one can expect that Bergeron and Li’s Mackey formula lifts to our formula at least for weak Bruhat interval modules.
To prove our result, we need the notion of standardization. For and , let be a unique permutation in satisfying the condition that if and only if for all . For instance, if , then . This standardization preserves the left weak Bruhat order on , in other words,
| (3.15) |
The following theorem shows that (3.14) is a natural lift of Bergeron and Li’s Mackey formula.
Theorem 3.14.
For , , and ,
Proof.
Using Lemma 3.8 and Theorem 3.12, we derive that
where the sum ranges over all pairs in
Let be a map defined by
for all . To begin with, let us verify that is a well-defined bijection.
First, we prove that is well-defined. Given , let
| (3.16) |
Since , there exists a permutation such that . Combining (3.15) with the definition of and yields that
Note that
This tells us that and , thus is well-defined.
Next, we prove that is bijective by constructing its inverse. Let and be nonnegative integers satisfying , , and . Given and , consider the mapping Note that there exist permutations and such that
Let given by
where . Then and , and therefore . In addition, by the definition of . On the other hand, given , letting , one can easily see that . So the inverse of is well-defined and thus is bijective.
For our assertion, it suffices to show that for each ,
where , and are defined as in (3.16). This isomorphism immediately follows from the four equalities:
Let us prove the equality (1). Let . Then,
Assume that and . Set
Since
it follows that
In case where , we have
In case where , we have
Thus, the equality (1) holds.
Next, let us prove the equality (2). Under the same setting with the above paragraph, we have
From the second and third equalities, we have
In case where , we have
In case where , we have
Thus, the equality (2) holds.
Equalities (3) and (4) can be proven in a similar way with (1) and (2) respectively, so we omit the proofs. ∎
Remark 3.15.
For and , let be the set of permutations satisfying that and are subwords of and . For and , let
In the proof of Theorem 3.14, we employ the fact that for and . In fact,
Therefore, by Lemma 3.8, we have that
It is well known that the multiplicative rule for the fundamental quasisymmetric functions are described as follow:
| (3.17) |
Here, and satisfying and . Now, one can see that the multiplicative rule (3.17) lifts to the induction product
3.4. (Anti-)automorphism twists of weak Bruhat interval modules
Let be an isomorphism of associative algebras over . Given an -module , we define by the -module with the same underlying space as and with the action twisted by in such a way that
Let be the category of finite dimensional left -modules. Any isomorphism induces a covariant functor
| (3.18) |
where for every -module homomorphism . We call the -twist.
Similarly, given an anti-isomorphism , we define to be the -module with , the dual space of , as the underlying space and with the action defined by
| (3.19) |
Any anti-isomorphism induces a contravariant functor
where for every -module homomorphism . We call the -twist. In [10], Fayers introduced the involutions and the anti-involution of defined in the following manner:
These morphisms commute with each other. We study the (anti-)involution twists for , , , and their compositions , , , .
The viewpoint of looking at (anti-)involutions as functors is quite useful for many reasons. The primary reason is that using the exactness of the corresponding functors, one can transport various structures of a given -module to their twists in a functorial way. An application in this direction can be found in Subsection 4.2.2. Additional reasons include that some well known functors appear in the context of our (anti-)involution twists. Given any anti-automorphism of , the standard duality appears as , where is the functor induced by the inverse of . In particular, . The Nakayama functor is naturally isomorphic to , which can be derived by combining [10, Proposition 4.2] with [22, Proposition IV.3.13]. To explain in more detail, the former reference implies is a Nakayama automorphism and the latter reference shows the relationship between Nakayama automorphisms and . And, the -dual functor, , is naturally isomorphic to , and therefore is naturally isomorphic to .
Now, let us focus on the main topic of this subsection, (anti-)involution twists of weak Bruhat interval modules. For irreducible modules and projective indecomposable modules, it was shown in [10, 14] that
The following theorem shows how the (anti-)involution twists act on weak Bruhat interval modules.
Theorem 3.16.
For , we have the following isomorphisms of -modules.
-
(1)
.
-
(2)
.
-
(3)
. In particular, .
Proof.
Consider the -linear isomorphisms defined by
where and denotes the dual of with respect to the basis for . Since it can be proven in a similar manner that these maps are -isomorphisms, we here only deal with (3).
Note that, for ,
which yields that
On the other hand,
It immediately follows from (2.1) that if and only if . Moreover, it is trivial that if and only if . Thus, we verified . And, combining the equality with (3) yields . ∎
| -twist | -twist | -twist | -twist | -twist | -twist | -twist | |
|---|---|---|---|---|---|---|---|
| -twist | -twist | -twist | -twist | -twist | -twist | -twist | |
|---|---|---|---|---|---|---|---|
As seen in Table 3.1, various (anti-)involution twists can be obtained from Theorem 3.16 by composing , , and . For the reader’s understanding, we deal with irreducible modules and projective indecomposable modules in a separate table (see Table 3.2).
Example 3.17.
For an (anti-)automorphism and an -module , we simply write for . The subsequent corollary shows that (anti-)involution twists behave nicely with respect to induction product and restriction.
Corollary 3.18.
Let , , and be weak Bruhat interval modules of , , and , respectively. Then we have following isomorphisms of modules.
-
(A1)
-
(A2)
-
(A3)
-
(B1)
-
(B2)
-
(B3)
Proof.
(A1), (A2), (B1), (B2), and (B3) are straightforward from the definitions of , , and .
4. Various -Hecke modules constructed using tableaux
Suppose we have a family of quasisymmetric functions that can be expanded in the basis of the fundamental quasisymmetric functions with positive coefficients. The correspondence (2.2) tells us that each of them appears as the image of the isomorphism classes of certain -modules. Among them, it would be very nice to find or construct one which is nontrivial, in other words, not a direct sum of irreducible modules and has a combinatorial model that can be handled well. Since the mid-2010s, some -modules have been constructed in line with this philosophy, more precisely, in [1, 2, 21, 24]. In this section, we show that all of them are equipped with the structure of weak Bruhat interval modules.
To deal with these modules, we need the notion of source and sink.
Definition 4.1.
Let be a basis for an -module such that is closed under the action of .
-
(1)
An element is called a source of if, for each , there exists such that .
-
(2)
An element is called a sink of if, for each , there exists such that .
Following the way as in [24], one can see that there are at most one source and sink in . In case where is the basis for , is the source and is the sink.
Hereafter, denotes a composition of . To introduce the tableaux in our concern, we need to define the composition diagram of shape . It is a left-justified array of boxes where the th row from the top has boxes for . For a filling of , we denote by the entry in the th row from the top and th column from the left.
4.1. Standard immaculate tableaux, standard extended tableaux, and their -modules
We begin with introducing the definition of standard immaculate tableaux and standard extended tableaux.
Definition 4.2.
([2, 21]) Let be a composition of .
-
(1)
A standard immaculate tableau of shape is a filling of the composition diagram with such that the entries are all distinct, the entries in each row increase from left to right, and the entries in the first column increase from top to bottom.
-
(2)
A standard extended tableau of shape is a filling of the composition diagram with such that the entries are all distinct, the entries in each row increase from left to right, and the entries in each column increase from top to bottom.
We remark that our standard extended tableaux are slightly different from those of Searles [21]. In fact, the former can be obtained by flipping the latter horizontally.
Denote by the set of all standard immaculate tableaux of shape and by the set of all standard extended tableaux of shape . Berg et al. [2] define a -Hecke action on and denote the resulting module by . And, Searles [21] define a -Hecke action on and denote the resulting module by . By the construction of and , it is clear that and are bases for and , respectively.
It is not difficult to show that both and have a unique source and a unique sink. Denote the source of by and the source of by . They are obtained by filling with entries from left to right and from top to bottom. Denote the sink of by and the sink of by . In contrast of and , and have to be constructed separately. The former is obtained from in the following steps:
-
(1)
Fill the first column with entries from top to bottom.
-
(2)
Fill the remaining boxes with entries from left to right from bottom to top.
On the other hand, the latter is obtained by filling with the entries from top to bottom and from left to right.
Definition 4.3.
For a filling of a composition diagram, is defined to be the word obtained from by reading the entries from right to left starting with the top row.
With this definition, we can state the following theorem.
Theorem 4.4.
For any , we have the -module isomorphisms
| (4.1) |
Here, the words in the parentheses are being viewed as permutations in one-line notation.
Proof.
To prove the first isomorphism in (4.1), we need the -module homomorphisms
| and | (4.2) |
where
-
-
the notation denotes the projective indecomposable module spanned by the standard ribbon tableaux of shape in [14, Subsection 3.2],
-
-
the first homomorphism is an isomorphism given in [14, Theorem 3.3 and Proposition 5.1], which is given by reading standard ribbon tableaux from left to right starting with the bottom row, and
-
-
the second homomorphism is an essential epimorphism given in [6, Theorem 3.2].
Composing with yields a surjective -module homomorphism . In view of the definition of and , one can see that
Composing with the isomorphism
we finally have the surjective -module homomorphism
Next, let us consider the projection
By the definition of , one sees that and . This implies that is a surjective -module homomorphism.
For our purpose, we have only to show . From the definition of the -action on it follows that for any and therefore . On the other hand, using the fact that the -Hecke action on satisfies the braid relations, one can show that every appears as for some . This says that , so we are done.
4.2. Standard permuted composition tableaux and their -modules
We begin with introducing the definition of standard permuted composition tableaux. In this subsection, denotes a permutation in .
Definition 4.5.
([26]) Given and , a standard permuted composition tableau of shape and type is a filling of with entries in such that the following conditions hold:
-
(1)
The entries are all distinct.
-
(2)
The standardization of the word obtained by reading the first column from top to bottom is .
-
(3)
The entries in each rows decrease from left to right.
-
(4)
If and , then and .
Denote by the set of all standard permuted composition tableaux of shape and type . Tewari and van Willigenburg define a -Hecke action on and denote the resulting module by . Contrary to and , this module is not indecomposable in general. In the following, we briefly explain how to decompose into indecomposables.
For , define if for each positive integer , the relative order of the entries in the th column of is equal to that of . This relation is an equivalence relation on . Let be the set of all equivalence classes under . Every class in is closed under the -Hecke action, which gives rise to the decomposition
where is the -module spanned by . All the results in the above can be found in [26]. The decomposition was improved in [5] by showing that every direct summand is indecomposable.
4.2.1. Weak Bruhat interval module structure of
We show that is isomorphic to a weak Bruhat interval module. To do this, we need a special reading of standard permuted composition tableaux. Let us introduce the necessary notations and the results. Given , let
It should be noticed that the set plays the same role as the complement of since
but
Every equivalence class has a unique source and a unique sink. Denote the source by and the sink by . Let be the number of elements in and set
For , let be the horizontal strip occupied by the boxes with entries from to in . For each , let be the subfilling of occupied by in .
Definition 4.6.
For and , let be the word obtained by reading from left to right. The reading word, , of is defined to be the word .
Example 4.7.
Let . We have
Therefore,
Next, we introduce the notion of generalized compositions. A generalized composition of is defined to be a formal sum , where for positive integers ’s with . Given , let and , where is the concatenation of and for . Let .
Choi, Kim, Nam, and Oh [6] found the projective cover of by constructing an essential epimorphism , where is a generalized composition defined by using the source of in a suitable manner and is the projective module spanned by standard ribbon tableaux of shape . From now on, we simply write for since contains information on and . For the details, see [6, Subsection 2.3 and Section 5].
Now, we are ready to prove the following theorem.
Theorem 4.8.
Let and . For each ,
Proof.
To prove our assertion, we need the -module homomorphisms
| and |
where
-
-
the notation denotes the projective module spanned by the generalized ribbon tableaux of shape ,
-
-
the first homomorphism is an isomorphism given in [14, Theorem 3.3 and Proposition 5.1], which is given by reading standard ribbon tableaux from left to right starting with the bottom row, and
-
-
the second homomorphism is an essential epimorphism given in [6, Theorem 5.3].
Composing with yields a surjective -module homomorphism . In view of the definition of and , one can see that
Composing with the isomorphism
we finally have the surjective -module homomorphism
Next, let us consider the projection
By the definition of , and and therefore is a surjective -module homomorphism.
For our purpose, we have only to show that . From the definition of the -action on , it follows that for any and therefore . On the other hand, using the fact that the -Hecke action on satisfies the braid relations, one can show that every appears as for some . This says that , so we are done. ∎
Remark 4.9.
In case where , a different reading from ours in Definition 4.6 has already been introduced in [24, Definition 4.1]. More precisely, for each , they define a reading word , called the column word of . They also introduce a partial order on and prove that is a graded poset isomorphic to (see [24, Theorem 6.18]). In view of Theorem 4.4, one may expect that is isomorphic to . This, however, turns out to be false. For instance, let be the equivalence class given in Example 4.7. Then , , and is not indecomposable as seen in Figure 3.1. Therefore, is not isomorphic to .
4.2.2. Involution twists of
Standard Young row-strict tableaux were first introduced in [18] as a combinatorial model for the Young row-strict quasisymmetric Schur functions . Recently, Bardwell and Searles [1] succeeded in constructing an -module whose quasisymmetric characteristic image equals . It is constructed by defining a -Hecke action on the set of standard Young row-strict tableaux of shape . We here introduce permuted standard Young row-strict tableaux which turn out to be very useful in describing the -module , where .
Definition 4.10.
Given and , a standard permuted Young row-strict composition tableau SPYRT of shape and type is a filling of with entries such that the following conditions hold:
-
(1)
The entries are all distinct.
-
(2)
The standardization of the word obtained by reading the first column from bottom to top is .
-
(3)
The entries in each row are increasing from left to right.
-
(4)
If and , then and .
Denote by the set of all SPYRTs of shape and type . Let be the -span of . Define
| (4.3) |
for and . Here, is obtained from by swapping and .
We claim that (4.3) defines an -action on . For , let be the filling of defined by . Define a -linear isomorphism by letting
then extending it by linearity. Here, is the dual of with respect to the basis for and , where is the equivalence class containing . One can verify that
which proves our claim. In particular, when , our is exactly same to due to Bardwell and Searles. To summarize, we state the following proposition.
Proposition 4.11.
For each and , (4.3) defines an -action on . Moreover, is an -module isomorphism.
Remark 4.12.
In the combinatorial aspect, our SPYRTs are precisely the standard permuted Young composition tableaux (SPYCT) in [6, Definition 4.4]. But, they should be distinguished in the sense that they have different -Hecke actions. For the -Hecke action on SPYCTs, see [6, Subsection 4.2]. The set of SPYCTs is a combinatorial model for an -module which is isomorphic to .
By virtue of Proposition 4.11, one can transport lots of properties of to via the functor . For each , let . Combining Proposition 4.11 with the results in [5, 6, 26], we have the following corollary.
Corollary 4.13.
For each , the following hold:
-
(1)
is indecomposable. In particular, is a decomposition of into indecomposables.
-
(2)
The injective hull of is .
-
(3)
.
Proof.
(1) Since is an isomorphism, it preserves direct sum. Therefore, the assertion can be obtained by combining Proposition 4.11 with [5, Theorem 3.1].
The three commutative diagrams in Figure 4.1 show various (anti-)involution twists of as well as their images under the quasisymmetric characteristic when .
In the first and second diagram, the functors assigned to parallel arrows are all same and the arrows in red are being used to indicate that the domain and codomain have the same image under the quasisymmetric characteristic. In the last diagram, is the Young quasisymmetric Schur function in [17, Definition 5.2.1], is the row-strict quasisymmetric Schur function in [19, Definition 3.2], and are automorphisms of defined by and .
Remark 4.14.
(1) Let be a positive integer and . It was stated in [19, Theorem 5.1] that
On the other hand, the third diagram in Figure 4.1 shows that , thus
As a consequence, we derive that
We add a remark that in some literature such as [17, Subsection 5.2] and [25, Remark 4.4], the identity is incorrectly stated as .
(2) One can also observe in [18, Theorem 12]. This identity, however, should appear as by the third diagram in Figure 4.1. Within the best understanding of the authors, this error seems to have occurred for the reason that the descent sets of Young composition tableaux and that of standard Young row-strict composition tableaux are defined in a different manner. The proof of [18, Theorem 12], with a small modification, can be used to verify .
5. Further avenues
(1) We have studied the structure of weak Bruhat interval modules so far. However, there are still many unsolved fundamental problems including the following:
-
-
Classify all weak Bruhat interval modules up to isomorphism.
-
-
Given an interval , decompose into indecomposables.
-
-
Given an interval , find the projective cover and the injective hull of .
(2) In [24, Section 9 and 10], Tewari and van Willigenburg provide a restriction rule for and ask if there is a reciprocal induction rule for with respect to the restriction rule. By combining Lemma 3.8 with Theorem 4.8, we successfully decompose into weak Bruhat interval modules. But, at the moment, we do not know if it can be expressed as a direct sum of ’s. We expect that a better understanding of the weak Bruhat interval modules appearing in the decomposition would be of great help in solving this problem. In line with this philosophy, it is interesting to find or characterize all intervals such that is isomorphic to , , or .
(3) Let and be arbitrary positive integers and . Using the fact that the left -action on commutes with the right -action on , Krob and Thibon [16] construct -modules
for every composition of . Then they prove that, as ranges over the set of nonempty compositions, ’s form a complete family of irreducible polynomial -modules and ’s a complete family of indecomposable polynomial -modules which arise as a direct summand of for some . They also realize and as the left ideals of
| (5.1) |
Hence, by replacing by in (5.1), one obtains the -modules and from the -modules and , respectively. This relationship seems to work well at the character level as well. In this regard, Hivert [12] shows that the Weyl character of is equal to the quasisymmetric polynomial , where .
In the present paper, we study intensively weak Bruhat interval modules, which are of the form or up to isomorphism (Theorem 3.5). Hence, it would be very meaningful to investigate how our results about weak Bruhat interval modules are reflected on the corresponding -modules, in other words, the -modules of the form and for .
Acknowledgments. The authors would like to thank Sarah Mason and Elizabeth Niese for helpful discussions on Remark 4.14. The authors also would like to thank Dominic Searles for helpful discussions on the -Hecke action on . The authors are grateful to the anonymous referee for careful readings of the manuscript and valuable advice.
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