Involution Matrix Loci and Orbit Harmonics

Let $\mathcal{Z}\subseteq{\mathbb{C}}^{N}$ be a finite locus of points in affine $N$-space. As in the introduction, the vanishing ideal ${\mathbf{I}}(\mathcal{Z})$ is given by

Multivariate Lagrange interpolation gives the identification of vector spaces

where ${\mathbb{C}}[\mathcal{Z}]$ is the space of all functions $\mathcal{Z}\to{\mathbb{C}}$. If $\mathcal{Z}$ is stable under the action of a finite matrix group $G\subseteq GL_{N}({\mathbb{C}})$, the ideal ${\mathbf{I}}(\mathcal{Z})$ is also $G$-stable and (2.2) is an isomorphism of $G$-modules.

Given a nonzero polynomial $f\in{\mathbb{C}}[{\mathbf{x}}_{N}]$, write $f=f_{0}+f_{1}+\dots+f_{d}$, where $f_{i}$ is homogeneous of degree $i$ and $f_{d}\neq 0$. Let $\tau(f)$ be the top degree homogeneous part of $f$, that is, $\tau(f):=f_{d}$. Given an ideal $I\subseteq{\mathbb{C}}[{\mathbf{x}}_{N}]$, the associated graded ideal of $I$ is the homogeneous ideal

The identification of ungraded ${\mathbb{C}}$-vector spaces in Equation (2.2) extended to a vector space isomorphism

Since ${\mathrm{gr}}\,{\mathbf{I}}(\mathcal{Z})\subseteq{\mathbb{C}}[{\mathbf{x}}_{N}]$ is a homogeneous ideal, $R(\mathcal{Z})$ has the additional structure of a graded ${\mathbb{C}}$-vector space. If $\mathcal{Z}$ is stable under the action of a finite matrix group $G\subseteq GL_{n}({\mathbb{C}})$, we may regard (2.4) as an isomorphism of ungraded $G$-modules, where $R(\mathcal{Z})$ has the additional structure of a graded $G$-module.

We will need a slightly more refined statement than the isomporphism (2.4). If $V=\bigoplus_{d\geq 0}V_{d}$ is a graded vector space and $m\geq 0$, we write $V_{\leq m}:=\bigoplus_{d=0}^{m}V_{d}$. In particular, we write ${\mathbb{C}}[{\mathbf{x}}_{N}]_{\leq m}$ for the vector space of polynomials in ${\mathbb{C}}[{\mathbf{x}}_{N}]$ of degree $\leq m$. If $I\subseteq{\mathbb{C}}[{\mathbf{x}}_{N}]$ is an ideal, we have a subspace $I\cap{\mathbb{C}}[{\mathbf{x}}_{N}]_{\leq m}\subseteq{\mathbb{C}}[{\mathbf{x}}_{N}]_{\leq m}$. The following result is standard; see e.g. [RhoadesViennot, Lem. 3.15].

Now suppose $\mathcal{Z}\subseteq{\mathbb{C}}^{N}$ is a finite locus which is stable under the action of a finite matrix group $G\subseteq GL_{N}({\mathbb{C}})$. For $m\geq 0$, the subspace ${\mathbb{C}}[{\mathbf{x}}_{N}]_{\leq m}\subseteq{\mathbb{C}}[{\mathbf{x}}_{N}]$ is $G$-stable, as is the ideal ${\mathbf{I}}(\mathcal{Z})$. The quotient ${\mathbb{C}}[{\mathbf{x}}_{N}]_{\leq m}/(I\cap{\mathbb{C}}[{\mathbf{x}}_{N}]_{\leq m})$ is therefore an ungraded $G$-module. The following result refines the isomorphism (2.4).

Lemma 2.3 is well-known in orbit harmonics theory, but we include a proof for the benefit of the reader.

The above proof works over any field $\mathbb{F}$ in which the order of $G$ is nonzero. For arbitrary fields $\mathbb{F}$, the $\mathbb{F}[G]$-modules $(\mathbb{F}[{\mathbf{x}}_{N}]/{\mathrm{gr}}\,{\mathbf{I}}(\mathcal{Z}))_{\leq m}$ and $\mathbb{F}[{\mathbf{x}}_{N}]_{\leq m}/({\mathbf{I}}(\mathcal{Z})\cap\mathbb{F}[{\mathbf{x}}_{N}]_{\leq m})$ have the same composition series (i.e. these modules are Brauer-isomorphic).

Let $n$ be a positive integer. A partition of $n$ is a sequence of non-decreasing positive integers $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{k})$ such that $\sum_{i=1}^{k}\lambda_{i}=n$. We write $\lambda\vdash n$ to say that $\lambda$ is a partition of $n$.

Let $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{k})\vdash n$. A Young diagram of shape $\lambda$ is the diagram obtained by placing $\lambda_{i}$ boxes in the $i$-th row. Below on the left is a Young diagram of shape $(4,2,1)\vdash 7$.

&           {ytableau} 1 & 3 6 7 2 5 4

A standard Young tableaux of shape $\lambda$ is a bijective filling of $[n]$ into the boxes of the Young diagram of shape $\lambda$, such that the entries are increasing across rows and down columns. Above on the right is a standard Young tableaux of shape $(4,2,1)$. We write $\mathrm{SYT}(\lambda)$ to denote the collection of standard Young tableau of shape $\lambda$.

The Schensted correspondence [Schensted] is a bijection

which associates each permutation $w\in{\mathfrak{S}}_{n}$ with a pair of standard Young tableau $(P(w),Q(w))$ of the same shape. The Schensted correspondence is usually described by an insertion algorithm, which can be found in [Sagan].

We write $\Lambda=\bigoplus_{n\,\geq\,0}\Lambda_{n}$ for the graded algebra of symmetric functions in an infinite variable set ${\mathbf{x}}=(x_{1},x_{2},\dots)$ over the ground field ${\mathbb{C}}(q)$. For example, the power sum symmetric function of degree $n$ is given by $p_{n}:=\sum_{i\geq 1}x_{i}^{n}\in\Lambda_{n}$. It is well known that $\{p_{1},p_{2},\dots\}$ is an algebraically independent generating set $\Lambda$.

Bases of the $n^{th}$ graded piece $\Lambda_{n}$ of $\Lambda$ are indexed by partitions $\lambda\vdash n$. For example, the power sum basis $\{p_{\lambda}\,:\,\lambda\vdash n\}$ is defined by $p_{\lambda}:=p_{\lambda_{1}}p_{\lambda_{2}}\cdots$ for $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots)$. We will also need the Schur basis $\{s_{\lambda}\,:\,\lambda\vdash n\}$.

A symmetric function $F\in\Lambda$ is Schur-positive if the coefficients $c_{\lambda}(q)$ in its $s$-expansion $F=\sum_{\lambda}c_{\lambda}(q)\cdot s_{\lambda}$ satisfy $c_{\lambda}(q)\in\mathbb{R}_{\geq 0}[q]$. If $F,G\in\Lambda$ we define $F\leq G$ by

$F\leq G$ if and only if $G-F$ is Schur-positive.

Furthermore, if $P$ is some property that partitions may or may not have and $F=\sum_{\lambda}c_{\lambda}(q)\cdot s_{\lambda}$, we define the truncation $\{F\}_{P}$ by

Here $\chi[S]$ is 1 if a statement $S$ is true and 0 otherwise. The truncation operator $\{-\}_{P}$ will typically be used to restrict the first row lengths of a partition, e.g. $\{-\}_{\lambda_{1}=a}$ or $\{-\}_{\lambda_{1}\leq a}$.

In addition to its usual multiplication, the graded ring $\Lambda$ carries an additional binary operation called plethysm. If $F=F(x_{1},x_{2},\dots)\in\Lambda$ and $n>0$, we defined $p_{n}[F]\in\Lambda$ by

Since $\{p_{1},p_{2},\dots\}$ is an algebraically independent generating set of $\Lambda$, the rules

for all $G_{1},G_{2},F\in\Lambda$ and all scalars $c\in{\mathbb{C}}(q)$ uniquely define a symmetric function $G[F]\in\Lambda$ for all $G,F\in\Lambda$. For more details on plethysm, see Macdonald’s book [Macdonald].

Irreducible representations of the symmetric group ${\mathfrak{S}}_{n}$ over ${\mathbb{C}}$ are in one-to-one correspondence with partitions $\lambda\vdash n$. If $\lambda\vdash n$ is a partition, write $V^{\lambda}$ for the corresponding irreducible ${\mathfrak{S}}_{n}$-module. If $V$ is any finite-dimensional ${\mathfrak{S}}_{n}$-module, there are unique multiplicities $c_{\lambda}\geq 0$ so that $V\cong\bigoplus_{\lambda\vdash n}c_{\lambda}V^{\lambda}$. The Frobenius image of $V$ is the symmetric function

obtained by replacing each irreducible $V^{\lambda}$ with the corresponding Schur function $s_{\lambda}$. For example, the trivial representation ${\mathbf{1}}_{{\mathfrak{S}}_{n}}$ corresponds to the Schur function $s_{n}$ indexed by the one-part partition $(n)$. If $V=\bigoplus_{d\geq 0}V_{d}$ is a graded ${\mathfrak{S}}_{n}$-module, we define the graded Frobenius image by

More generally, if $V=\bigoplus_{d\geq 0}V_{d}$ is a graded vector space, the Hilbert series of $V$ is

Given $n,m\geq 0$, we have an embedding ${\mathfrak{S}}_{n}\times{\mathfrak{S}}_{m}\subseteq{\mathfrak{S}}_{n+m}$ by letting ${\mathfrak{S}}_{n}$ permute the first $n$ letters of $[n+m]$ and letting ${\mathfrak{S}}_{m}$ permute the last $m$ letters. If $V$ is an ${\mathfrak{S}}_{n}$-module an $W$ is an ${\mathfrak{S}}_{m}$-module, the tensor product $V\otimes W$ is naturally an ${\mathfrak{S}}_{n}\times{\mathfrak{S}}_{m}$-module. The induction product of $V$ and $W$ is

Induction product of representations corresponds to multiplication of symmetric functions; we have (see e.g. [Macdonald, Sagan])

For $n,m\geq 0$, the wreath product ${\mathfrak{S}}_{n}\wr{\mathfrak{S}}_{m}$ is the subgroup of ${\mathfrak{S}}_{nm}$ generated by $\dots$

• •

  the $n$-fold direct product ${\mathfrak{S}}_{m}\times{\mathfrak{S}}_{m}\times\cdots\times{\mathfrak{S}}_{m}$ permuting elements of the sets

  $$\{1,2,\dots,m\},\{m+1,m+2,\dots,2m\},\dots,\{nm-m+1,nm-m+2,\dots,nm\}$$

  independently, as well as

• •

  products of transpositions of the form

  $$(im-m+1,jm-m+1)(im-m+2,jm-m+2)\cdots(im,jm)$$

  for $1\leq i,j\leq n$ which interchange two of the above $n$ sets ‘wholesale’.

One way to visualize the group ${\mathfrak{S}}_{n}\wr{\mathfrak{S}}_{m}$ is as follows. Fill the entries of an $n\times m$ grid with the numbers $1,2,\dots,nm$ in English reading order. For example, if $n=3$ and $m=4$ we have the following figure.

The group ${\mathfrak{S}}_{n}\wr{\mathfrak{S}}_{m}$ is the subgroup of ${\mathfrak{S}}_{nm}$ generated by permutations which either permute entries within rows or interchange rows wholesale. We have embeddings ${\mathfrak{S}}_{m}^{n}\subseteq{\mathfrak{S}}_{n}\wr{\mathfrak{S}}_{m}$ and ${\mathfrak{S}}_{n}\subseteq{\mathfrak{S}}_{n}\wr{\mathfrak{S}}_{m}$; the images of these embeddings generate ${\mathfrak{S}}_{n}\wr{\mathfrak{S}}_{m}$.

If $V$ is an ${\mathfrak{S}}_{n}$-module and $W$ is an ${\mathfrak{S}}_{m}$-module, we have a $({\mathfrak{S}}_{n}\wr{\mathfrak{S}}_{m})$-module $V\wr W$ with underlying vector space

and group action determined by

Plethysm of symmetric functions relates to this construction via

See [Macdonald] for a textbook treatment of this fact. We will only use this construction when both $V$ and $W$ are trivial modules; we note the combinatorial interpretation of the relevant symmetric function $s_{n}[s_{m}]$.

In particular, if $n$ is even, Lemma 2.5 implies that the Frobenius image of the action of ${\mathfrak{S}}_{n}$ on the set ${\mathcal{PM}}_{n}$ of perfect matchings on $[n]$ is $s_{n/2}[s_{2}]$. It is an extremely difficult open problem to determine the $s$-expansion $s_{a}[s_{b}]$ for arbitrary $a,b$. However, if $n$ is even it is known that

is the multiplicity-free sum over all $s_{\lambda}$ where $\lambda\vdash n$ has all even parts.

We will need a standard result on symmetrizers acting on ${\mathfrak{S}}_{n}$-irreducibles. Let $j\leq n$ and consider the embedding ${\mathfrak{S}}_{j}\subseteq{\mathfrak{S}}_{n}$ where ${\mathfrak{S}}_{j}$ acts on the first $j$ letters. This induces an embedding of group algebras ${\mathbb{C}}[{\mathfrak{S}}_{j}]\subseteq{\mathbb{C}}[{\mathfrak{S}}_{n}]$. We let $\eta_{j}\in{\mathbb{C}}[{\mathfrak{S}}_{j}]\subseteq{\mathbb{C}}[{\mathfrak{S}}_{n}]$ be the element which symmetrizes with respect to ${\mathfrak{S}}_{j}$, i.e.

If $V$ is an ${\mathfrak{S}}_{n}$-module, the element $\eta_{j}$ acts as an operator on $V$. The following result characterizes when $\eta_{j}$ annihilates irreducible ${\mathfrak{S}}_{n}$-modules.

Lemma 2.6 is standard, but we include a proof for the convenience of the reader.

In order to study the ring $R({\mathcal{M}}_{n})$, it will be useful to have an explicit generating set of its defining ideal ${\mathrm{gr}}\,{\mathbf{I}}({\mathcal{M}}_{n})$. It will turn out that such a generating set is as follows.

We aim to show that ${\mathrm{gr}}\,{\mathbf{I}}({\mathcal{M}}_{n})=I^{\mathcal{M}}_{n}$ as ideals in ${\mathbb{C}}[{\mathbf{x}}_{n\times n}]$. One of these containments is straightforward.

As explained in Remark 2.1, the top degree components of a given generating set of ${\mathbf{I}}(\mathcal{Z})$ do not in general suffice to generate ${\mathrm{gr}}\,{\mathbf{I}}(\mathcal{Z})$. In order to show that we have generating in the context of Lemma 3.2, we show that the quotient ${\mathbb{C}}[{\mathbf{x}}_{n\times n}]/I^{\mathcal{M}}_{n}$ has vector space dimension at most $|{\mathcal{M}}_{n}|$. Recall from the introduction that if $w\in{\mathcal{M}}_{n}$ is a matching, the matching monomial ${\mathfrak{m}}(w)\in{\mathbb{C}}[{\mathbf{x}}_{n\times n}]$ is the product over all variables $x_{i,j}$ for which $i<w(i)=j$.

For example, if $n=4$ we have the matching monomial basis of $R({\mathcal{M}}_{4})$ given by