1 Introduction↩︎

The main goal of this paper is to prove characteristic-free filtration results for the orbit harmonics quotients of three important finite monoids, the rook monoid \(\mathop{\mathrm{\mathcal{IS}}}_n\), the partial transformation monoid \(\mathop{\mathrm{\mathcal{PT}}}_n\), and the full transformation monoid \(\mathop{\mathrm{\mathcal{T}}}_n\). These results are motivated by a theorem of Rhoades [1] who determined the \(\mathfrak{S}_n \times \mathfrak{S}_n\)-module structure of the orbit harmonics quotient of the symmetric group \(\mathfrak{S}_n\) over the complex numbers. The lack of semisimplicity in positive characteristic (and even over \(\mathbb{C}\) for \(\mathop{\mathrm{\mathcal{PT}}}_n\) and \(\mathop{\mathrm{\mathcal{T}}}_n\)) necessitates the use of suitable filtrations.

Outline of the paper↩︎

This paper concerns submonoids \(\mathop{\mathrm{\mathcal{M}}}(n)\) of \(\mathop{\mathrm{\mathcal{PT}}}_n\) that contain \(\mathfrak{S}_n\) with particular emphasis on \(\mathop{\mathrm{\mathcal{IS}}}_n, \mathop{\mathrm{\mathcal{PT}}}_n\) and \(\mathop{\mathrm{\mathcal{T}}}_n\). Our main contributions are summarized in items (i)-(v) below. By \(P_{\Bbbk}(n,r)\) we denote the category of finite dimensional rational representations of degree \(r\) of the monoid \(\mathop{\mathrm{\mathsf{M}}}_n(\Bbbk)\) of \(n \times n\) matrices over an infinite field \(\Bbbk\) of arbitrary characteristic.

(i) For any monoid \(\mathop{\mathrm{\mathcal{M}}}(n)\) as above, we define a functor \(\mathop{\mathrm{\mathcal{G}}}_{\mathop{\mathrm{\mathcal{M}}}(n)}: P_{\Bbbk}(n,r) \to \mathop{\mathrm{\mathsf{mod}}}\mathop{\mathrm{\mathcal{M}}}(n),\) the symmetrized Schur functor (see Definition 2). We examine how this is related to the classical Schur functor (see Corollary 1). We also show that \(\mathop{\mathrm{\mathcal{G}}}_{\mathop{\mathrm{\mathcal{M}}}(n)}(M)\) is an induced module from the symmetric group \(\mathfrak{S}_n\) (see Theorem 9).

(ii) Utilizing the symmetrized Schur functor, we define \(\mathop{\mathrm{\mathcal{M}}}(n)\)-modules \(\mathop{\mathrm{\mathcal{R}}}(n)^{\lambda/ \mu}\) for any submonoid \(\mathop{\mathrm{\mathcal{M}}}(n)\) of \(\mathop{\mathrm{\mathcal{PT}}}_n\) and any skew partition \(\lambda/ \mu\) (see Definition 5). These modules were motivated by the constructions of Grood [2] and Steinberg [3] of the irreducible representations of the rook monoid and the full transformation monoid, respectively, over the complex numbers. Our construction is characteristic-free and uniform. From the basis theorem for skew Schur modules of \(\mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\) [4] we obtain an analogous result for the modules \(\mathop{\mathrm{\mathcal{R}}}(n)^{\lambda/ \mu}\) (see Proposition 12). We prove two branching rules for the \(\mathop{\mathrm{\mathcal{M}}}(n)\)-modules \(\mathop{\mathrm{\mathcal{R}}}(n)^{\lambda/ \mu}\). These generalize Solomon’s branching rule for the rook monoid \(\mathop{\mathrm{\mathcal{IS}}}_n\) over the complex numbers [5] and branching rule for Specht modules of James and Peel [6] (see Theorem 17).

(iii) Let \(\boldsymbol{x}_{m,n}\) be an \(m \times n\) matrix of commuting variables \((x_{i,j})\), where \(1 \le i \le m\) and \(1 \le j \le n\). The group \(\mathop{\mathrm{\mathsf{GL}}}_m({\Bbbk}) \times \mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\) acts on the polynomial ring \(\Bbbk[\boldsymbol{x}_{m \times n}]\) in a natural way and when the characteristic of \(\Bbbk\) is zero, the irreducible decomposition of the degree \(r\) homogeneous component \(\Bbbk[\boldsymbol{x}_{m \times n}]_r\) is \(\bigoplus_{\lambda\vdash r} L_\lambda(V_m) \otimes L_\lambda(V_n)\), see [7]. Here, \(L_{\lambda'}\) denotes the Schur functor corresponding to a partition \(\lambda\) and \(V_m\) is the natural \(\mathop{\mathrm{\mathsf{GL}}}_m({\Bbbk})\)-module. In arbitrary characteristic, the previous decomposition holds up to filtration [8]. This is referred to as the Cauchy decomposition of \(\Bbbk[\boldsymbol{x}_{m \times n}]\) since the corresponding identity of formal characters is known as the Cauchy identity [9]. Related results appear in various contexts, for example invariant theory [10]–[12], highest weight categories [13] and cellular algebras [14]. In Section 5 we define natural homogeneous ideals \(J_{m,n}(Z)\) of \(\Bbbk[\boldsymbol{x}_{m \times n}]\) associated to the monoids \(\mathop{\mathrm{\mathcal{IS}}}_n\), \(\mathop{\mathrm{\mathcal{PT}}}_n\), \(\mathop{\mathrm{\mathcal{T}}}_n\) and prove the following result on the graded structure of the quotient rings (see Theorem 20).

Thus we obtain analogs of the Cauchy decomposition of the polynomial ring \(\Bbbk[\boldsymbol{x}_{m \times n}]\) and the basis theorem of Desarmenien, Kung and Rota [10].

(iv) We continue with the setup of (iii) and assume in addition that \(m=n\). Consider the monoids \(\mathop{\mathrm{\mathcal{IS}}}_n, \mathop{\mathrm{\mathcal{PT}}}_n\) and \(\mathop{\mathrm{\mathcal{T}}}_n\) as affine varieties in \(\mathop{\mathrm{\mathsf{M}}}_{n}(\Bbbk)\) and let \(\mathop{\mathrm{\mathbf{I}}}(Z)\) be the vanishing ideal of the variety \(Z\). Using the method of orbit harmonics, we prove that the above ideals \(J_{n,n}(Z)\) are the associated graded ideals \(\mathop{\mathrm{\mathsf{gr}}}(\mathop{\mathrm{\mathbf{I}}}(Z))\) of the corresponding vanishing ideals, where \(Z\) is any of \(\mathop{\mathrm{\mathcal{IS}}}_n, \mathop{\mathrm{\mathcal{PT}}}_n, \mathop{\mathrm{\mathcal{T}}}_n\). Thus from Theorem 1 we obtain results on the graded module structure of the orbit harmonics quotients \(\Bbbk[\boldsymbol{x}_{n \times n}]/ \mathop{\mathrm{\mathsf{gr}}}(\mathop{\mathrm{\mathbf{I}}}(Z))\) (see Corollary 5). We regard this and Theorem 1 as the main results of the paper.

Throughout the paper we take the point of view of utilizing the polynomial representation theory of the general linear group \(\mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\) [4], [8], [15] and related combinatorics. This is done using rational modules of the monoid \(\mathop{\mathrm{\mathsf{M}}}_n(\Bbbk)\) and the symmetrized Schur functor. Alternatively, one could use induced modules from the symmetric group based on work of Clifford [16], Munn [17], [18] and Ponizovskii [19] and its modern recast in terms of Green’s idempotent theory [15] by Ganyushkin, Mazorchuk and Steinberg [20]. However, our approach seems very suitable for the purposes of Theorem 1 and Corollary 5. For an approach to the representations of the rook monoid \(\mathop{\mathrm{\mathcal{IS}}}_n\) using Schur algebras, we refer to Andre and Martins [21].

Section 2 is devoted to preliminaries concerning polynomial representations of \(\mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\) and representations of \(\mathfrak{S}_n\) that will be used in the sequel. In Section 3 we define the symmetrized Schur functor and study its basic properties. In Section 4 we introduce analogs \(\mathop{\mathrm{\mathcal{R}}}(n)^{\lambda/ \mu}\) and \(\mathop{\mathrm{\mathcal{R}}}(n)_{\lambda/ \mu}\) of skew Specht modules \(S^{\lambda/ \mu}\) and dual skew Specht modules \(S_{\lambda/ \mu}\). We establish for these a basis theorem and two branching rules. As an application of the first branching rule we show that when the characteristic of \(\Bbbk\) is zero, the \(\mathop{\mathrm{\mathcal{M}}}(n)\)-modules \(\mathop{\mathrm{\mathcal{R}}}^\lambda\) are distinct in most cases as \(\lambda\) varies over the partitions of \(r\) with \(r \le n\). Section 5 contains Theorem 20 and its corollary on orbit harmonics quotients. In Appendix A we prove a skew symmetric version of Theorem 1.

2 Preliminaries↩︎

In this section, we establish notation and gather preliminaries that will be used in the sequel. Throughout this paper, \(\Bbbk\) is an infinite field of arbitrary characteristic unless specified otherwise.

2.1 Monoids↩︎

Let \(\mathop{\mathrm{\mathcal{PT}}}_n\) be the subset of \(\mathop{\mathrm{\mathsf{M}}}_n(\Bbbk)\) of matrices with entries from \(\{0,1\}\) with at most one entry equal to 1 in each column. Under matrix multiplication, \(\mathop{\mathrm{\mathcal{PT}}}_n\) is a monoid known as the partial transformation monoid. The submonoid \(\mathop{\mathrm{\mathcal{IS}}}_n\) of \(\mathop{\mathrm{\mathcal{PT}}}_n\) consisting of the matrices with at most one entry equal to 1 in every column and every row is called the symmetric inverse monoid. This is also known as the rook monoid because of the correspondence between matrices in \(\mathop{\mathrm{\mathcal{IS}}}_n\) and placements of non-attacking rooks on an \(n \times n\) chessboard. The submonoid \(\mathop{\mathrm{\mathcal{T}}}_n\) of \(\mathop{\mathrm{\mathcal{PT}}}_n\) consisting of matrices that have exactly one entry equal to 1 in each column is called the full transformation monoid. The symmetric group \(\mathfrak{S}_n\) may be realized as the group of permutation matrices in \(\mathop{\mathrm{\mathsf{M}}}_n(\Bbbk)\).

We record the cardinalities.

A partial map \(p\) of \([n]\) is a map from a subset \(\mathop{\mathrm{\mathsf{dom}}}(p)\) of \([n]\), called the domain of \(p\), to \([n]\). If \(p_1 : [n] \to [n]\) and \(p_2 : [n] \to [n]\) are partial maps, we have the partial map given by their composition \(p_2 p_1 : [n] \to [n]\), \(p_2 p_1 (a)= p_2(p_1(a))\) for all \(a \in p_2^{-1}(\mathop{\mathrm{\mathsf{dom}}}(p_1))\). For \(A=(a_{ij}) \in \mathop{\mathrm{\mathcal{PT}}}_n\), we define the corresponding partial map \(p_A:[n] \to [n]\) by \(p_A(j)=i \Leftrightarrow a_{ij}=1\). Conversely, if \(p:[n]\to [n]\) is a partial map, we have the matrix \(A_p=(a_{ij}) \in \mathop{\mathrm{\mathcal{PT}}}_n\) defined by \(a_{ij}=1 \Leftrightarrow p(j)=i\). We see that the set of partial maps \([n] \to [n]\) is a monoid with respect to composition and is isomorphic to \(\mathop{\mathrm{\mathcal{PT}}}_n\).

2.2 Skew partitions and tableaux↩︎

A partition of an integer \(r\) is a finite sequence of weakly decreasing nonnegative integers that sum to \(r\). For a partition \(\lambda\), we denote by \(\ell(\lambda)\) the number of nonzero terms of the sequence. We identify two partitions if they differ by a string of zeros. For partitions \(\lambda=(\lambda_1, \dots, \lambda_q)\) and \(\mu=(\mu_1, \dots, \mu_q)\) we write \(\mu \subseteq \lambda\) if \(\mu_i \le \lambda_i\) for all \(i\). A skew partition \(\lambda/ \mu\) is a pair of partitions \(\lambda\) and \(\mu\) such that \(\mu \subseteq \lambda\). By a slight abuse of notation, we usually write \(\lambda/ \mu =(\lambda_1 - \mu_1, \dots, \lambda_q - \mu_q)\). If \(\mu=(0)\), we identify the skew partition \(\lambda/ (0)\) with the partition \(\lambda\). For a skew partition \(\lambda/ \mu\) we write \(|\lambda/ \mu|=r\) if \(|\lambda|-|\mu|=r\) and we say that \(\lambda/ \mu\) is a skew partition of \(r\).

For a skew partition \(\lambda/ \mu\), a tableau of shape \(\lambda/ \mu\) with entries from \([n]:=\{1,2,\dots, n\}\) is a filling of the diagram of \(\lambda/ \mu\) with elements of \([n]\). Let \(\mathop{\mathrm{Tab}}_{\lambda / \mu}([n])\) be the set of such tableaux. The weight \(\alpha=(\alpha_1,\dots, \alpha_n)\) of a tableau \(T \in \mathop{\mathrm{Tab}}_{\lambda / \mu}([n])\) is defined by letting \(\alpha_i\) be the number of appearances of the entry \(i\) in \(T\). We also say that the weight of \(i\) in \(T\) is equal to \(\alpha_i\). A tableau \(T \in \mathop{\mathrm{Tab}}_{\lambda / \mu}([n])\) is called semistandard (respectively co-semistandard) if the entries are strictly (respectively weakly) increasing across the rows from left to right and weakly (respectively strictly) increasing in the columns from top to bottom. (In [4], the term ‘standard’ is used for our semistandard, and ‘co-standard’ for our co-semistandard.) The subset of \(\mathop{\mathrm{Tab}}_{\lambda / \mu}([n])\) of semistandard (respectively co-semistandard) tableaux of shape \(\lambda/ \mu\) will be denoted by \(\mathrm{SST}_{\lambda/ \mu}([n])\) (respectively \(\mathrm{cSST}_{\lambda/ \mu}([n])\)).

2.3 Schur modules and Weyl modules↩︎

Suppose \(V\) is a finite dimensional vector space. Let \(\Lambda(V)=\bigoplus_{i \ge 0}\Lambda^i(V)\) be the exterior algebra of \(V\) with the usual grading and let \(Sym(V) = \bigoplus_{i\ge 0}Sym_i(V)\) be the symmetric algebra of \(V\) with usual grading. We also have the divided power algebra of \(D(V)=\bigoplus_{i\geq 0}D_i(V)\) ([4]), where \(D_i(V)\) is defined as the dual of \(Sym_i(V^*)\) and \(V^*\) denotes the dual of \(V\). If \(\alpha=(\alpha_1,\dots, \alpha_q)\) is a sequence of nonnegative integers, we use the notation \(\Lambda^{\alpha}(V):=\Lambda^{\alpha_1}(V) \otimes \dots \otimes \Lambda^{\alpha_q}(V)\) and likewise \(Sym_{\alpha}(V):=Sym_{\alpha_1}(V) \otimes \dots \otimes Sym_{\alpha_q}(V)\) and \(D_{\alpha}(V):=D_{\alpha_1}(V) \otimes \dots \otimes D_{\alpha_q}(V)\).

Suppose \(\lambda/ \mu\) is a skew partition. The Schur module \(L_{\lambda / \mu}(V)\) and the Weyl module \(K_{\lambda / \mu}(V)\) are defined in [4] as the images of particular \(GL(V)\)-maps\[\begin{align} d_{\lambda / \mu} (V):\Lambda^{\lambda / \mu}(V) \to Sym_{\lambda' / \mu'}(V), \; \;d'_{\lambda / \mu}(V): D_{\lambda / \mu}(V) \to \Lambda^{\lambda' / \mu'} (V). \end{align}\] For example, if \(\lambda=(r)\), then \(L_{\lambda}(V)=\Lambda^r(V)\) and \(K_{\lambda}(V)=D_r(V)\). If \(\lambda=(1^r)\), then \(L_{\lambda}(V)=Sym_r(V)\) and \(K_{\lambda}(V)=\Lambda^r(V)\).

Suppose \(\{e_1, \dots, e_n\}\) is a basis of the vector space \(V\) and \(T \in \mathrm{Tab}_{\lambda/ \mu}([n])\) is a tableau. We write \(T(i,j)\) for the entry of \(T\) in position \((i,j)\). Define elements \(X_T \in \Lambda^{\lambda / \mu}(V)\) and \(Y_T \in D_{\lambda/ \mu}(V)\) by \[\begin{align} X_T:=e_{T(1,1)} \cdots e_{T(1,\lambda_1 - \mu_1)}\otimes \cdots\otimes e_{T(q,1)} \dots e_{T(q,\lambda_q - \mu_q)}. \end{align}\] where \(\lambda= (\lambda_1, \dots, \lambda_q)\) and \(\mu = (\mu_1, \dots, \mu_q)\), and \[\begin{align} Y_T:=e_1^{(a_{11})} \cdots e_n^{(a_{1n})} \otimes \cdots \otimes e_1^{(a_{q1})} \cdots e_n^{(a_{qn})}, \end{align}\] where \(a_{ij}\) is equal to the number of entries in row \(i\) of \(T\) that are equal to \(j\).

We will need the following proposition. The proof of this follows from the proof of Theorem 3 given in [4].

2.4 Polynomial representations of \(\mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\) and rational representations of \(\mathop{\mathrm{\mathsf{M}}}_n(\Bbbk)\)↩︎

By \(M_{\Bbbk}(n,r)\) we denote the category of homogeneous polynomial representations of \(\mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\) of degree \(r\) and by \(P_{\Bbbk}(n,r)\) we denote the category of rational representations of \(\mathop{\mathrm{\mathsf{M}}}_n(\Bbbk)\) of degree \(r\) [15]. Now since \(\Bbbk\) is an infinite field, we may regard homogeneous polynomial representations of \(\mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\) of degree \(r\) as rational representations of \(\mathop{\mathrm{\mathsf{M}}}_n(\Bbbk)\) of degree \(r\) and conversely. The categories \(M_{\Bbbk}(n,r)\) and \(P_{\Bbbk}(n,r)\) are equivalent, see [15] or [22]. In this paper, we will often use results for the category \(M_{\Bbbk}(n,r)\) (from [4], [15] and [8]) and state them in their \(P_{\Bbbk}(n,r)\) equivalent form.

2.5 Schur functor and Specht modules↩︎

Let \(\Lambda(n,r)\) be the set of sequences \(\alpha=(\alpha_1, \dots, \alpha_n)\) of nonnegative integers such that \(\alpha_1+\cdots+\alpha_n=r\). We call the elements of \(\Lambda(n,r)\) weights. We denote the subset of diagonal matrices in \(\mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\) by \(T_n{(\Bbbk)}\). For a module \(M \in M_{\Bbbk}(n,r)\) and a weight \(\alpha\in \Lambda(n,r)\), define the \(\alpha\)- weight space of \(M\) by \[M^{\alpha}:= \{ v \in M: \mathop{\mathrm{\mathsf{diag}}}(t_1,\dots,t_n)v = t^{\alpha_1}_1\cdots t^{\alpha_n}_n, \mathrm{\;for \;all} \mathop{\mathrm{\mathsf{diag}}}(t_1, \dots, t_n) \in T_n(\Bbbk) \}.\] Since \(\Bbbk\) is an infinite field, we have the weight space decomposition of \(M\) \[\label{ws}M=\bigoplus_{\alpha\in \Lambda(n,r)}M^{\alpha}.\tag{1}\] If \(h:M \to N\) is a morphism in \(P_{\Bbbk}(n,r)\), then for every \(\alpha\in \Lambda(n,r)\) \[\label{wpres} h(M^\alpha) \subseteq N^\alpha.\tag{2}\]

We have the action of \(\mathfrak{S}_n\) on \(\Lambda(n,r)\) given by \[\label{sigmaa}\sigma\alpha = (a_{\sigma^{-1}(1)}, \dots, a_{\sigma^{-1}(n)}).\tag{3}\]

Suppose \(n \ge r\). Recall from [15] that the Schur functor is a functor \(f\) from the category \(M_{\Bbbk}(n,r)\) of homogeneous polynomial representations of \(\mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\) of degree \(r\) to the category of \(\mathfrak{S}_r\)-modules. For \(M\) an object in the first category, \(f(M)\) is the weight subspace \(M^\alpha\) of \(M\), where \(\alpha = (1^r, 0^{n-r})\) and for \(\theta :M \to N\) a morphism in the first category, \(f(\theta)\) is the restriction \(M^\alpha \to N^\alpha\) of \(\theta\).

For a skew partition \(\lambda/ \mu\) let us define the skew Specht module \(S^{\lambda / \mu }\) and the dual Specht module \(S_{\lambda / \mu }\) module by \[\label{fdef}S^{\lambda / \mu }:=f(L_{\lambda' / \mu'}(V_n)), \;\;S_{\lambda / \mu }:=f(K_{{\lambda} / \mu}(V_n)),\tag{4}\] where \(V_n\) is the natural \(\mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\)-module of column vectors. These are \(\mathfrak{S}_r\)-modules for \(r= |\lambda/ \mu|\). For example, \(S^{(r)}\) is the trivial representation and \(S^{(1^r)}\) is the sign representation.

Let \(f^{\lambda/ \mu} := \dim(S^{\lambda/ \mu}) \;\text{and} \;f_{\lambda/ \mu} := \dim(S_{\lambda/ \mu}).\) Recall that a standard tableau \(T \in \mathop{\mathrm{Tab}}_{\lambda / \mu}([r])\) is a semistandard tableau that has distinct entries. Then both \(f^{\lambda / \mu}\) and \(f_{\lambda / \mu}\) are equal to the number of standard tableaux \(T \in \mathop{\mathrm{SST}}_{\lambda' / \mu'}([r])\).

2.6 Table of frequently used notation↩︎

For the reader’s convenience we gather here notation used frequently in the paper.

3 Symmetrized Schur functor↩︎

In Section 3.2 we define for any submonoid \(\mathop{\mathrm{\mathcal{M}}}(n)\) of the partial transformation monoid \(\mathop{\mathrm{\mathcal{PT}}}_n\) a functor \(\mathop{\mathrm{\mathcal{G}}}_{\mathop{\mathrm{\mathcal{M}}}(n)}:P_{\Bbbk}(n,r) \to \mathop{\mathrm{\mathsf{mod}}}\mathop{\mathrm{\mathcal{M}}}(n)\) that attaches to a rational representation of the monoid \(\mathop{\mathrm{\mathsf{M}}}_n(\Bbbk)\) a finite dimensional representation of \(\mathop{\mathrm{\mathcal{M}}}(n)\). First we need some preparation concerning weights of \(P_{\Bbbk}(n,r)\)-modules. This is done in Section 3.1. In Section 3.4 we study properties of the functor \(\mathop{\mathrm{\mathcal{G}}}_{\mathop{\mathrm{\mathcal{M}}}(n)}\).

3.1 Weight spaces and submonoids of \(\mathop{\mathrm{\mathcal{PT}}}_n\)↩︎

We define here certain weights of a \(P_{\Bbbk}(n,r)\)-module \(M\) that are central for our purpose and we show a relevant lemma.

The set \(\Lambda(n,r)''\) is the complement of \(\Lambda(n,r)'\) in \(\Lambda(n,r)\) and thus by (1 ) \[\label{sum}M = M' \bigoplus M''.\tag{5}\]

3.2 The symmetrized Schur functor \(\mathop{\mathrm{\mathcal{G}}}_{\mathop{\mathrm{\mathcal{M}}}(n)}\)↩︎

Consider the decomposition (5 ) for \(M \in P_\Bbbk(n,r)\). If \(\eta :M \to N\) is a morphism in the category \(P_{\Bbbk}(n,r)\), then the image under \(\eta\) of the subspace \(M''\) of \(M\) is contained in the subspace \(N''\) of \(N\) according to (2 ). Hence we have the induced map on quotients \({M}/M'' \to {N}/N''\). By Lemma 1(2), this induced map is a homomorphism of \(\mathop{\mathrm{\mathcal{M}}}(n)\)-modules when \(M\) is an \(\mathop{\mathrm{\mathcal{M}}}(n)\)-module, for any submonoid \(\mathop{\mathrm{\mathcal{M}}}(n)\) of \(\mathop{\mathrm{\mathcal{PT}}}_n\). Thus we may give the following definition.

3.3 The functor \(\mathop{\mathrm{\mathcal{F}}}\)↩︎

We now define a functor needed in the next subsection.

3.4 Some properties of the symmetrized Schur functor↩︎

For the proof of Theorem 8 below, we will need the following lemma on monomials in symmetric powers and divided powers. First some notation. Recall we have the canonical basis \(\{e_1, \dots, e_n\}\) of the natural \(\mathop{\mathrm{\mathsf{M}}}_{n}(\Bbbk)\)-module \(V\). For \(\alpha=(\alpha_1, \dots, \alpha_n) \in \Lambda(n,r)\) we denote by \(e_\alpha \in Sym_r(V_{n})\) and \(e_{(\alpha)} \in D_r(V_{n})\) the monomials \[e_\alpha:=e_{1}^{\alpha_1} \cdots e_{n}^{\alpha_n}, \; e_{(\alpha)}:=e_{1}^{(\alpha_1)} \cdots e_{n}^{(\alpha_n)}.\] We recall that the action of the symmetric group \(\mathfrak{S}_n\) on \(e_\alpha\) and \(e_{(\alpha)}\) is given by \[\label{sigmaea}\sigma e_\alpha = {e_{\sigma(1)}}^{\alpha_1} \cdots {e_{\sigma(n)}}^{\alpha_n}, \;\sigma e_{(\alpha)} = {e_{\sigma(1)}}^{(\alpha_1)} \cdots {e_{\sigma(n)}}^{(\alpha_n)}\tag{7}\] and the action of \(\mathfrak{S}_n\) on \(\Lambda(n,r)\) is given by (3 ).

Suppose we have modules \(N_1 \in P_{\Bbbk}(n,r_1)\) and \(N_2 \in P_{\Bbbk}(n,r_2)\) with \(r_1+r_2 \le n\). The monoid \(\mathop{\mathrm{\mathsf{M}}}_n(\Bbbk)\) acts on the tensor product \(N_1 \otimes N_2\) by \(A(u \otimes v)=Au \otimes Av\), where \(A \in \mathop{\mathrm{\mathsf{M}}}_n(\Bbbk)\) and \(v_1 \in N_1, v_2 \in N_2\). Then we have \(N_1 \otimes N_2 \in P_{\Bbbk}(n,r_1+r_2)\) and we may consider the image \(f(N_1\otimes N_2)\) of \(N_1\otimes N_2\) under the Schur functor \(f:M_{\Bbbk}(n,r_1+r_2) \to \mathop{\mathrm{\mathsf{mod}}}\mathfrak{S}_{r_1+r_2}\).

The main result of the present section is the next theorem and its corollary.

3.5 Induced modules and \(\mathop{\mathrm{\mathcal{G}}}_{\mathop{\mathrm{\mathcal{M}}}(n)}(M)\)↩︎

In the Clifford-Munn-Ponizovskii theory on the classification of simple modules of a finite monoid, the induction functor from maximal subgroups of the monoid is a central object, see [20] or [3]. We show next that \(\mathop{\mathrm{\mathcal{G}}}_{\mathop{\mathrm{\mathcal{M}}}(n)}(M)\) is an induced module from \(\mathfrak{S}_n\) to \(\mathop{\mathrm{\mathcal{M}}}(n)\). This result will be used in Theorem 13 below.

Following [3] we have that \(\Bbbk I_{n,r}\) is a right \(\mathfrak{S}_r\)-module: for \(h \in I_{n,r}\) and \(A \in \mathfrak{S}_r\), define \(hA\) to be the composition \([r] \xrightarrow{\sigma} [r] \xrightarrow{h} [n]\) where \(\sigma\) is the permutation corresponding to the permutation matrix \(A\). One verifies that \(\Bbbk I_{n,r}\) is a free \(\mathfrak{S}_r\)-module with basis the order preserving injective maps \([r] \to [n]\). Hence the rank of \(\Bbbk I_{n,r}\) is equal to \(\binom{n}{r}\).

We also have that \(\Bbbk I_{n,r}\) is a left \(\mathop{\mathrm{\mathcal{PT}}}_n\)-module: for \(h \in I_{n,r}\) and \(A \in \mathop{\mathrm{\mathcal{PT}}}_n\), define \[A h = \begin{cases} p_A h, &\text{if} \;p_A h \in I_{n,r},\\ 0, &\text{else}, \end{cases}\] where \(p_A: [n] \to [n]\) denotes the partial map corresponding to \(A \in \mathop{\mathrm{\mathcal{PT}}}_n\) and \(p_A h\) denotes the composition \([r] \xrightarrow{h} [n] \xrightarrow{p_A} [n]\) of the map \(h\) and the partial map \(p_A\). In fact, \(\Bbbk I_{n,r}\) is a \((\mathop{\mathrm{\mathcal{PT}}}_n,\mathfrak{S}_r)\)-bimodule.

4 Basis theorem and branching rules↩︎

For a submonoid \(\mathop{\mathrm{\mathcal{M}}}(n)\) of the partial transformation monoid \(\mathop{\mathrm{\mathcal{PT}}}_n\) we define \(\mathop{\mathrm{\mathcal{M}}}(n)\)-modules \(\mathop{\mathrm{\mathcal{R}}}(n)^{\lambda/ \mu }\) and \(\mathop{\mathrm{\mathcal{R}}}(n)_{\lambda/ \mu }\). We establish for these a basis theorem and two branching rules. In Section 4.6 we give an application of the first branching rule when \(\Bbbk\) is a field of characteristic zero.

From the present section, we only need Definition 5 and Proposition 12 for Section 5.

4.1 Definition of \(\mathop{\mathrm{\mathcal{R}}}(n)^{\lambda/ \mu}\) and \(\mathop{\mathrm{\mathcal{R}}}(n)_{\lambda/ \mu}\)↩︎

4.2 Basis theorem↩︎

For a partition \(\lambda\) of \(r\), Grood [2] considered semistandard tableaux of shape \(\lambda\) that have distinct entries. Similarly we give the following definition.

For the next theorem we adopt the notation of Section 2.3.

4.3 Irreducibility in characteristic zero↩︎

4.4 Filtrations↩︎

Let us recall the following definitions. For \(N \in P_{\Bbbk}(n,r)\), a Schur filtration (respectively, Weyl filtration) of \(N\) is a sequence of submodules in \(P_{\Bbbk}(n,r)\) \[\label{Sfil}0=N_s \subseteq N_{s-1} \subseteq \cdots \subseteq N_0=N\tag{11}\] such that for \(1 \le i \le s-1\) the quotient \(N_i / N_{i+1}\) is zero or isomorphic to a Schur module \(L_\lambda (V_n)\) (respectively, Weyl module \(K_\lambda (V_n)\)) for some partition \(\lambda\) of \(r\) [23]. Likewise, for \(Q \in \mathop{\mathrm{\mathsf{mod}}}\mathfrak{S}_r\), a Specht filtration (respectively, dual Specht filtration) is a sequence of \(\mathfrak{S}_r\)-submodules \[0=Q_t \subseteq Q_{t-1} \subseteq \cdots \subseteq Q_0=Q\] such that for \(1 \le i \le t-1\) the quotient \(Q_i / Q_{i+1}\) is zero or isomorphic to a Specht module \(S^\lambda\) (respectively, dual Specht module \(S^\lambda\)) for some partition \(\lambda\) of \(r\).

In the sequel, we will apply the following lemma.

This follows since the functor \(f\) is exact (see [15]) and satisfies eqs. (4 ).

4.5 First branching rule↩︎

We consider the next theorem as the main result of this section.

4.6 An application of the first branching rule↩︎

As an application of Theorem 14, we can show that when the characteristic of \(\Bbbk\) is zero, the \(\mathop{\mathrm{\mathcal{M}}}(n)\)-modules \(\mathop{\mathrm{\mathcal{R}}}^\lambda\) and \(\mathop{\mathrm{\mathcal{R}}}^\mu\) are not isomorphic for almost all distinct partitions \(\lambda\) and \(\mu\). To be precise we have the following result.

4.7 Second branching rule↩︎

Let \(\mathop{\mathrm{\mathcal{M}}}(n)\) be one of the monoids \(\mathop{\mathrm{\mathcal{IS}}}_n, \mathop{\mathrm{\mathcal{PT}}}_n\) or \(\mathop{\mathrm{\mathcal{T}}}_n\). For \(1 \le s < n\) consider the monomorphism of monoids \(\mathop{\mathrm{\mathcal{M}}}(s) \times \mathop{\mathrm{\mathcal{M}}}(n-s) \to \mathop{\mathrm{\mathcal{M}}}(n)\) \[(A,B) \mapsto \left( \begin{array}{c|c} A & \boldsymbol{0} \\ \hline \boldsymbol{0}&B \end{array} \right),\] where the matrix \(A\) (respectively, \(B\)) is located in the upper left \(s \times s\) corner (respectively, lower right \((n-s) \times (n-s)\) corner). Thus every \(\mathop{\mathrm{\mathcal{M}}}(n)\)-module can be considered an \(\mathop{\mathrm{\mathcal{M}}}(s) \times \mathop{\mathrm{\mathcal{M}}}(n-s)\)-module and we have the following branching rule.

4.8 \(GL_n\) branching rule↩︎

For the proof of Theorem 17 we will apply the branching rule for the general linear group \(\mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\) which we now recall. We follow closely [4].

For sequences \(\alpha=(\alpha_1, \dots, \alpha_q), \beta = (\beta_1, \dots, \beta_q)\) of nonnegative integers, we write \(\alpha\ge \beta\) if there is an index \(i\) such that \(\alpha_1=\beta_1, \dots, \alpha_{i-1}=\beta_{i-1}\) and \(\alpha_i >\beta_i\).

Let \(\lambda / \mu\) be a skew partition of \(r\), where \(r \le n\). Consider the direct sum decomposition of the natural \(\mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\)-module \(V_n\) as vector spaces \(V_n= F \bigoplus G\), where \(F\) and \(G\) are the subspaces of \(V_n\) with basis \(e_1, \dots, e_r\) and \(e_{r+1}, \dots, e_n\) respectively. If \(T \in \mathop{\mathrm{Tab}}_{\lambda' / \mu'}\) has entries from \([n]\), we define the sequence \(\eta(T)\) of nonnegative integers whose \(i\)th coordinate is equal to \(\mu_i\) + the number of entries in the \(i\)th row of \(T\) that are in \([r]\). For a partition \(\nu\) satisfying \(\mu' \subseteq \nu \subseteq \lambda'\), let \(M_\nu\) be the subspace of the Schur module \(L_{\lambda' / \mu'}(V_n)\) spanned by the elements

(With the notation of [4], our \(M_\nu\) is equal to \(M_\nu (\Lambda_{\lambda' / \mu'}(F\bigoplus G))\)). Also let \(\dot{M}_\nu\) be the subspace \(\sum_{\theta} M_\theta\) of \(M_\nu\) given by \(\dot{M}_\nu:= \sum M_\xi\), where \(\xi\) ranges over all partitions such that \(\mu' \subseteq \xi \subseteq \lambda'\) and \(\xi > \nu\). According to [4] we have a filtration \[\label{fil3}0 \subseteq M_{\lambda'} \subseteq \cdots \subseteq M_{\nu} \subseteq \cdots \subseteq M_{\mu'}= L_{\lambda' / \mu'}(V_n)\tag{14}\] of \(L_{\lambda' / \mu'}(V_n)\) by \(\mathop{\mathrm{\mathsf{GL}}}_r \times \mathop{\mathrm{\mathsf{GL}}}_{n-r}\)-submodules such that \[\label{iso3}L_{\nu / \mu'}(F) \otimes L_{\lambda' / \nu}(G) \cong M_\nu / \dot{M}_{\nu}.\tag{15}\] Moreover the isomorphism in (15 ) sends \(d_{\nu / \mu'}(F)(X_{T_1}) \otimes d_{\lambda' / \nu}(G)(X_{T_2})\) to \(d_{\lambda' / \mu'}(V_n)(X_{T}) + \dot{M}_{\nu}\), where the \(q\)th row of \(T\) is

if the \(q\)th rows of \(T_1\) and \(T_2\) are respectively

for \(q=1, \dots, \ell(\lambda' / \mu')\), where \(u=\nu_q - \mu'_q\) and \(v=\lambda'_q - \nu_q\).

4.9 Proof of Theorem 17↩︎

With the previous preparation, we may now prove the branching rule associated with \(\mathop{\mathrm{\mathcal{M}}}(s) \times \mathop{\mathrm{\mathcal{M}}}(n-s) \to \mathop{\mathrm{\mathcal{M}}}(n)\).

For \(r=n-1\) in Theorem 17 we obtain the following corollary.

5 Cauchy decompositions and orbit harmonics quotients↩︎

As mentioned in items (iv) and (v) of the Introduction, the purpose of this section is to establish ‘Cauchy decompositions’ (a) for certain algebras associated to the monoids \(\mathop{\mathrm{\mathcal{IS}}}_n, \mathop{\mathrm{\mathcal{PT}}}_n, \mathop{\mathrm{\mathcal{T}}}_n\) (see Theorem 20) and (b) for the orbit harmonics quotients of these monoids (see Corollary 5).

From Section 4 we will need only Proposition 12.

In Section 5.1 we introduce a family of modules that will be used later. In Section 5.2 we compute dimensions of the graded components of the quotient rings whose module structure is studied in Section 5.3. In Section 5.4 we study orbit harmonic quotients.

5.1 The modules \(L_\lambda(S^{(m-1,1)})\)↩︎

The goal of this subsection is to introduce a family of modules \(L_\lambda(S^{(m-1,1)})\) for the symmetric group \(\mathfrak{S}_m\) that feature in a Cauchy decomposition associated to the full transformation monoid \(\mathop{\mathrm{\mathcal{T}}}_m\), see Theorem 20(3). We will use the presentation of these modules given in Lemma 4 below.

Recall from Section 2.3 that for any finite dimensional \(\Bbbk\)-vector space \(V\) and for any partition \(\lambda\) we have the Schur module \(L_\lambda(V)\). Here, we will take \(V\) to be the standard \(\mathfrak{S}_m\)-module which we denote by \(U_m\). To be precise, let \(V_m\) be the natural \(\mathfrak{S}_m\)-module. We have a basis \(\{e_1, \dots, e_m\}\) of \(V_m\) and the action of \(\mathfrak{S}_m\) is defined by \(\sigma e_i = e_{\sigma (i)}\), where \(\sigma \in \mathfrak{S}_m\). The subspace of \(V_m\) spanned by the sum \(e_1+\dots + e_m\) is an \(\mathfrak{S}_m\)-submodule of \(V_m\). The quotient module \[U_m:=V_m / \langle e_1+\cdots+e_m \rangle\] is called the standard module of \(\mathfrak{S}_m\). It is well known that \(U_m\) is isomorphic to the Specht module \(S^{(m-1,1)}\) corresponding to the partition \((m-1,1)\) of \(m\). We will not need this remark.

For every \(k \ge 0\), the natural surjection \(\pi: V_m \to U_m\) gives a surjection of \(\mathfrak{S}_m\)-modules \[\label{pik}\pi_k: \Lambda^k(V_m) \to \Lambda^k(U_m), v_1 \cdots v_k \mapsto \pi(v_1) \cdots \pi(v_k), \;v_i \in V_m.\tag{20}\] Define \(\boldsymbol{e}:=e_1 +\cdots+ e_m.\) Since \(\boldsymbol{e} \in \ker\pi\), we have \[\label{kerpi}v_1 \cdots v_k \in \ker \pi_k\tag{21}\] if \(v_j=\boldsymbol{e}\) for some \(j \in \{1,\dots, k\}\).

We have that the Schur module \(L_\lambda (V_m)\) is an \(\mathfrak{S}_m\)-module for any partition \(\lambda\). We now define an \(\mathfrak{S}_m\)-submodule of \(L_\lambda (V_m)\). We adopt the notation of Section 2.3 and in particular recall we have a map \(d_{\lambda} (V_m):\Lambda^{\lambda}(V_m) \to Sym_{\lambda' }(V_m)\) whose image is \(L_\lambda(V_m)\). In order to have more compact notation, for a tableau \(T \in \mathop{\mathrm{Tab}}_\lambda([m])\) let us denote the element \(d_\lambda(V_m)(X_T)\) of \(L_\lambda(V_m)\) by \([T]\).

It follows that \(W_\lambda\) is an \(\mathfrak{S}_m\)-submodule of \(L_\lambda (V_m)\).

5.2 The ideals \(J_{m,n}(Z)\) for \(Z \in \{ \mathop{\mathrm{\mathcal{IS}}}, \mathop{\mathrm{\mathcal{PT}}}, \mathop{\mathrm{\mathcal{T}}}\}\)↩︎

Let \(\boldsymbol{x}_{m,n}\) be an \(m \times n\) matrix of commuting variables \((x_{i,j})\), where \(1 \le i \le m\) and \(1 \le j \le n\), and consider the polynomial ring \(\Bbbk[\boldsymbol{x}_{m \times n}]\).

We have the grading \(\Bbbk[\boldsymbol{x}_{m \times n}] = \bigoplus_{r \ge0}\Bbbk[\boldsymbol{x}_{m \times n}]_r\), where \(\Bbbk[\boldsymbol{x}_{m \times n}]_r\) is the subspace of \(\Bbbk[\boldsymbol{x}_{m \times n}]\) generated by the homogeneous polynomials of degree \(r\). If \(I\) is a homogeneous ideal of \(\Bbbk[\boldsymbol{x}_{m \times n}]\) we obtain the grading \(\Bbbk[\boldsymbol{x}_{m \times n}]/ I= \bigoplus_{r\ge 0} \big(\Bbbk[\boldsymbol{x}_{m \times n}]/ I\big)_r\), where \[\big(\Bbbk[\boldsymbol{x}_{m \times n}]/ I\big)_r:=\Bbbk[\boldsymbol{x}_{m \times n}]_r/ (\Bbbk[\boldsymbol{x}_{m \times n}]_r \cap I).\] Since the ideals \(J_{m, n}(Z)\), where \(Z \in \{\mathop{\mathrm{\mathcal{PT}}}, \mathop{\mathrm{\mathcal{IS}}}, \mathop{\mathrm{\mathcal{T}}}\}\), of \(\Bbbk[\boldsymbol{x}_{m \times n}]\) are homogeneous, we have the corresponding grading \(\Bbbk[\boldsymbol{x}_{m \times n}]/ J_{m,n}(Z)= \bigoplus_{r\ge 0} \big(\Bbbk[\boldsymbol{x}_{m \times n}]/ J_{m,n}(Z)\big)_r\).

Recall that our convention for binomial coefficients is \(\binom{a}{b}=0\) if \(b >a\).

5.3 Cauchy decompositions↩︎

We consider the polynomial ring \(\Bbbk[\boldsymbol{x}_{m \times n}]\) as an \(\mathop{\mathrm{\mathsf{M}}}_m(\Bbbk) \times \mathop{\mathrm{\mathsf{M}}}_n(\Bbbk)\)-module by defining \[\label{sec:Iact}Ax_{i,j}:=\sum_{k=1}^m a_{k,i}x_{k,j}, \; Bx_{i,j}:=\sum_{k=1}^n b_{k,i}x_{i,k},\tag{31}\] for \(A=(a_{st}) \in \mathop{\mathrm{\mathsf{M}}}_m(\Bbbk)\) and \(B=(b_{st}) \in \mathop{\mathrm{\mathsf{M}}}_n(\Bbbk)\), and extending this action multiplicatively and linearly.

Recall from Section 2.1 that if \(A\) is a matrix in \(\mathop{\mathrm{\mathcal{PT}}}_m\), then we have the associated partial map \(p_A: [m] \to [m]\). The restriction of the above action to \(\mathop{\mathrm{\mathcal{PT}}}_m \times \mathop{\mathrm{\mathcal{PT}}}_n\) is given as follows. For \((A, B) \in \mathop{\mathrm{\mathcal{PT}}}_m \times \mathop{\mathrm{\mathcal{PT}}}_n\) and \(1 \le i \le m\), \(1 \le j \le n\) we have \[(A,B)x_{i,j} = \begin{cases}x_{p_A(i),p_B(j)}, &i \in \mathop{\mathrm{\mathsf{dom}}}(p_A), \;j \in \mathop{\mathrm{\mathsf{dom}}}(p_B ),\\ 0, & otherwise. \end{cases}\] For a monomial \(x_{i_1, j_1} \cdots x_{i_r, j_r}\) in \(\Bbbk[\boldsymbol{x}_{m \times n}]\), we have \[\label{act}(A,B)x_{i_1, j_1} \cdots x_{i_r, j_r}=x_{p_A(i_1), p_B(j_1)} \cdots x_{p_A(i_r), p_B(j_r)}.\tag{32}\]

It is straightforward to verify that, with respect to the above actions, the ideal

• \(J_{m,n}(\mathop{\mathrm{\mathcal{IS}}})\) is an \(\mathop{\mathrm{\mathcal{IS}}}_m \times \mathop{\mathrm{\mathcal{IS}}}_n\) -submodule of \(\Bbbk[\boldsymbol{x}_{m \times n}]\),

• \(J_{m,n}(\mathop{\mathrm{\mathcal{PT}}})\) is a \(\mathop{\mathrm{\mathsf{GL}}}_m(\Bbbk) \times \mathop{\mathrm{\mathcal{PT}}}_n\) -submodule of \(\Bbbk[\boldsymbol{x}_{m \times n}]\), and

• \(J_{m,n}(\mathop{\mathrm{\mathcal{T}}})\) is an \(\mathfrak{S}_m \times \mathop{\mathrm{\mathcal{T}}}_n\) -submodule of \(\Bbbk[\boldsymbol{x}_{m \times n}]\).

The graded structure of the corresponding quotient rings is given in Theorem 20 below, which we regard as the main result of the paper together with Corollary 5.

Cauchy decomposition of \(\Bbbk[\boldsymbol{x}_{m \times n}]\). For the proof of the above theorem we need to recall the characteristic-free Cauchy decomposition of the polynomial ring \(\Bbbk[\boldsymbol{x}_{m \times n}]\) as a \(\mathop{\mathrm{\mathsf{GL}}}_m(\Bbbk) \times \mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\)-module. There are several approaches due to Desarmenien, Kung and Rota [10], to DeConcini, Eisenbud and Procesi [11], to Akin, Buchsbaum and Weyman [4] and to Krause [32]. We follow closely [4].

If \(i_1, \dots, i_k \in [m]\) and \(j_1, \dots, j_k \in [n]\), where \(k \le \min\{m,n\}\), let us denote by \((i_1, \dots, i_k | j_1, \dots, j_k)\) the \(k \times k\) minor of the matrix \(\boldsymbol{x}_{m \times n}\) corresponding to rows \(i_1, \dots, i_k\) and columns \(j_1, \dots, j_k\), \[(i_1, \dots, i_k | j_1, \dots, j_k):=\det(x_{i_u, j_v}), \;1\le u, v \le k.\]

Next, we order the partitions of \(r\) lexicographically. For \(\lambda \vdash r\), let \(M_\lambda\) be the subspace of \(\Bbbk[\boldsymbol{x}_{m \times n}]\) spanned by the elements \((S|T)\) as \(S, T\) run over the tableaux of shape \(\ge \lambda\) and weight \(r\). We thus have a filtration of \(\Bbbk[\boldsymbol{x}_{m \times n}]_k\), \[\label{C1} 0 \subseteq M_{(r)} \subseteq M_{(r-1,1)} \subseteq \dots \subseteq M_{(1^r)}=\Bbbk[\boldsymbol{x}_{m \times n}]_r\tag{35}\] by \(\mathop{\mathrm{\mathsf{M}}}_m(\Bbbk) \times \mathop{\mathrm{\mathsf{M}}}_n(\Bbbk)\)-submodules. Finally, let \(\dot{M}_\lambda\) be the subspace of \(\Bbbk[\boldsymbol{x}_{m \times n}]_r\) defined by \(\dot{M}_\lambda :=\sum_{\gamma > \lambda}{M}_\gamma\), where the sum is over all partitions \(\gamma\) of \(r\) such that \(\gamma > \lambda\). It is shown in [4] that the map \[\begin{align} \label{beta} \beta_\lambda: L_\lambda(V_m) \otimes L_\lambda (V_n) &\to M_\lambda/\dot{M}_\lambda, \\\nonumber d_{\lambda}(V_m)(X_S) \otimes d_{\lambda}(V_n)(X_T) &\mapsto ( S| T ) + \dot{M}_\lambda \end{align}\tag{36}\] is an isomorphism of \(\mathop{\mathrm{\mathsf{M}}}_m(\Bbbk) \times \mathop{\mathrm{\mathsf{M}}}_n(\Bbbk)\)-modules.

We adopt the notation of Definition 9.

5.4 Orbit harmonics quotients↩︎

Next we want to relate Theorem 20 when \(m=n\) to the coordinate rings of the affine varieties \(\mathop{\mathrm{\mathcal{IS}}}_n\), \(PT_n\) and \(T_n\) (see Corollary 5 below). This will be done using orbit harmonics and thus we need to recall some relevant background. For more details on orbit harmonics we refer to the paper by Rhoades [1] or the paper by Reiner and Rhoades [33].

If \(Z \subseteq \mathop{\mathrm{\mathsf{M}}}_{n}(\Bbbk)\) is a finite locus of points in affine \(n \times n\)-space, we have the vanishing ideal \[\mathop{\mathrm{\mathbf{I}}}(Z) := \{f \in \Bbbk[\boldsymbol{x}_{n \times n}]: f(\boldsymbol{z})=0 \;\mathrm{for all} \;\boldsymbol{z} \in Z\}\] and an identification of \(\Bbbk\)-algebras \(\Bbbk[Z] \cong \Bbbk[\boldsymbol{x}_{n \times n}/\mathop{\mathrm{\mathbf{I}}}(Z)\), where \(\Bbbk[Z]\) is the algebra of polynomial functions on \(Z\). Since \(Z\) is a finite, Lagrange interpolation yields that \(\Bbbk[Z]\) is the space of functions \(Z \to \Bbbk\). Hence we may regard \(\Bbbk[Z]\) as the \(\Bbbk\)-vector space of \(Z\), ie the \(\Bbbk\)-vector space that has basis the points of \(Z\).

If \(f \in \Bbbk[\boldsymbol{x}_{n \times n}]\) is a nonzero polynomial, we may write \(f=f_d+\cdots+f_1+f_0\) where \(f_i\) is a homogeneous polynomial of degree \(i\) and \(f_d \neq 0\). Define \(h(f):=f_d,\) the highest degree component of \(f\).

The result we need is the following.

For a proof see [1].

The coordinate rings of \(\mathop{\mathrm{\mathcal{IS}}}_n\), \(\mathop{\mathrm{\mathcal{PT}}}_n\) and \(\mathop{\mathrm{\mathcal{T}}}_n\) Recall we have the ideals \(J_{n}(Z)\) of Definition 8(4) for \(Z \in \{\mathop{\mathrm{\mathcal{IS}}}, \mathop{\mathrm{\mathcal{PT}}}, \mathop{\mathrm{\mathcal{T}}}\}\).

6 Skew symmetric Cauchy decompositions↩︎

Our purpose here is to obtain a skew symmetric analog of Theorem 20. Let us recall that the group \(\mathop{\mathrm{\mathsf{GL}}}_m({\Bbbk}) \times \mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\) acts on the exterior algebra \(\Lambda(V_m \otimes V_n)\) by extending the usual action on \(V_m \otimes V_n\). When the characteristic of \(\Bbbk\) is zero, the irreducible decomposition of the degree \(r\) homogeneous component \(\Lambda^r(V_m \otimes V_n)\) is \(\bigoplus_{\lambda\vdash r} L_\lambda(V_m) \otimes K_\lambda(V_n)\), see [34] or [8]. In arbitrary characteristic, the previous decomposition holds up to filtration [4].

6.1 Definition and statement↩︎

Let \(\boldsymbol{y}_{m,n}=(y_{i,j})\) be an \(m \times n\) matrix of skew commuting variables \(y_{i,j}\), where \(1 \le i \le m\) and \(1 \le j \le m\), and consider the corresponding skew polynomial ring \(\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle\). So we have \(y_{i,j}^2=0\) and \(y_{i,j}y_{k,l}=- y_{k,l}y_{i,j}\) for all \(1 \le i,k \le m\) and \(1 \le j, l \le n\).

From the definition we have \(J'_{m,n}(\mathop{\mathrm{\mathcal{PT}}}) \subseteq J'_{m,n}(\mathop{\mathrm{\mathcal{IS}}})\).

We have the grading \(\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle = \bigoplus_{r \ge0}\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle_r\), where \(\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle_r\) is the subspace of \(\Bbbk[\boldsymbol{y}_{m \times n}\) generated by the homogeneous polynomials of degree \(r\). Since the ideals \(J'_{m, n}(Z)\), where \(Z \in \{\mathop{\mathrm{\mathcal{PT}}}, \mathop{\mathrm{\mathcal{IS}}}, \mathop{\mathrm{\mathcal{T}}}\}\), of \(\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle\) are homogeneous, we have the corresponding grading \(\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle_r/ J'_{m,n}(Z)= \bigoplus_{r\ge 0} \big(\Bbbk\langle \boldsymbol{y}_{m \times n}\rangle / J'_{m,n}(Z)\big)_r\).

The formula in (31 ) for \(y_{i,j}\) in place of \(x_{i,j}\) define actions of \(\mathop{\mathrm{\mathcal{PT}}}_m \times \mathop{\mathrm{\mathcal{PT}}}_n\) and \(\mathop{\mathrm{\mathsf{GL}}}_m(\Bbbk)\) on \(\Bbbk\langle \boldsymbol{y}_{m \times n}\rangle\). It is straightforward to verify that the ideal

• \(J'_{m,n}(\mathop{\mathrm{\mathcal{IS}}})\) is an \(\mathop{\mathrm{\mathcal{IS}}}_m \times \mathop{\mathrm{\mathcal{IS}}}_n\) -submodule of \(\Bbbk\langle \boldsymbol{y}_{m \times n}\rangle\),

• \(J'_{m,n}(\mathop{\mathrm{\mathcal{PT}}})\) is a \(\mathop{\mathrm{\mathsf{GL}}}_m(\Bbbk) \times \mathop{\mathrm{\mathcal{PT}}}_n\) -submodule of \(\Bbbk\langle \boldsymbol{y}_{m \times n}\rangle\), and

• \(J'_{m,n}(\mathop{\mathrm{\mathcal{T}}})\) is an \(\mathfrak{S}_m \times \mathop{\mathrm{\mathcal{T}}}_n\) -submodule of \(\Bbbk\langle \boldsymbol{y}_{m \times n}\rangle\).

The main result of the Appendix is the following.

6.2 Cauchy decomposition of \(\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle\).↩︎

For the proof of the above theorem we need to recall the Cauchy decomposition of \(\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle\) as a \(\mathop{\mathrm{\mathsf{GL}}}_m(\Bbbk) \times \mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\)-module. We follow closely [4].

We remark that eq. (50 ) in the previous definition is not the same given in [4], but is easily seen to be equal to the definition given in the fourth line on p. 247 in [4].

Next, we order the partitions of \(r\) lexicographically. For \(\lambda \vdash r\), let \(M_\lambda\) be the subspace of \(\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle\) spanned by the elements \((S|T)\) as \(S, T\) run over the tableaux of shape \(\ge \lambda\) and weight \(r\) with entries from \([m]\) and \([n]\) respectively. We thus have a filtration of \(\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle_r\), \[\label{skewC1} 0 \subseteq M_{(r)} \subseteq M_{(r-1,1)} \subseteq \dots \subseteq M_{(1^r)}=\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle_r\tag{52}\] by \(\mathop{\mathrm{\mathsf{GL}}}_m(\Bbbk) \times \mathop{\mathrm{\mathsf{GL}}}_n(\Bbbk)\)-submodules. Finally, let \(\dot{M}_\lambda\) be the subspace of \(\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle\) spanned by the elements \((S|T)\) as \(S, T\) run over the tableaux of shape \(> \lambda\) and weight \(r\). It is shown in [4] that the map \[\begin{align} \label{betaskew} \beta_\lambda: L_\lambda(V_m) \otimes K_\lambda (V_n) \to M_\lambda/\dot{M}_\lambda, \;\;d_{\lambda}(V_m)(X_S) \otimes d'_{\lambda}(V_n)(Y_T) \mapsto (S| T ) + \dot{M}_\lambda \end{align}\tag{53}\] is an isomorphism of \(\mathop{\mathrm{\mathsf{M}}}_m(\Bbbk) \times \mathop{\mathrm{\mathsf{M}}}_n(\Bbbk)\)-modules.

6.3 Two lemmas↩︎

For the proof of Theorem 22 we will need two lemmas concerning the filtration (52 ) and the ideals of Definition 11.

For the next lemma let us recall the following notation from Definition 7. Let \(S \in \mathop{\mathrm{Tab}}_\lambda([m])\) be a tableau and consider the box in position \((i,j)\), where \(1 \le i \le \ell (\lambda)\) and \(1 \le j \le \lambda_i\). For \(u \in [m]\), we denote by \(S_{i,j}(u) \in \mathop{\mathrm{Tab}}_\lambda([m])\) the tableau obtained from \(S\) by replacing the \((i,j)\) entry with \(u\).

6.4 Proof of Theorem 22↩︎

We may now prove Theorem 22. For \(Z \in \{\mathop{\mathrm{\mathcal{IS}}}, \mathop{\mathrm{\mathcal{PT}}}, \mathop{\mathrm{\mathcal{T}}}\}\) consider the ideal \(J'_{m,n}(Z)\) of \(\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle\). From the filtration (52 ) we obtain the following filtration \[\label{sec:skewC2} 0 \subseteq \frac{ M_{(r)}+J'_{m,n}(Z)}{J'_{m,n}(Z)} \subseteq \frac{ M_{(r-1,1)}+J'_{m,n}(Z)}{J'_{m,n}(Z)} \subseteq \dots \subseteq \frac{ M_{(1^r)}+J'_{m,n}(Z)}{J'_{m,n}(Z)}.\tag{55}\]

(1) We consider case (1) of Theorem 22, so let \(Z=\mathop{\mathrm{\mathcal{IS}}}\). We have \[\big(\frac{\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle}{J'_{m,n}(\mathop{\mathrm{\mathcal{IS}}})}\big)_r \cong \frac{ M_{(1^r)}+J'_{m,n}(\mathop{\mathrm{\mathcal{IS}}})}{J'_{m,n}(\mathop{\mathrm{\mathcal{IS}}})}\] as \(\mathop{\mathrm{\mathcal{IS}}}_m \times \mathop{\mathrm{\mathcal{IS}}}_n\)-modules.

If \(r > \min\{m,n\}\), then by Lemma 8(1) we have \(\big(\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle/ J'_{m,n}(\mathop{\mathrm{\mathcal{IS}}})\big)_r=0\). Let \(r \le \min\{m,n\}\). Suppose \(\lambda\vdash r\) and \(S \in \mathop{\mathrm{Tab}}_\lambda([m])\), \(T \in \mathop{\mathrm{Tab}}_\lambda([n]).\) If either of \(S,T\) has an entry of weight at least 2, then by Lemma 52 we have \((S|T) \in J'_{m,s}(\mathop{\mathrm{\mathcal{IS}}})\). Hence the map (53 ) induces a surjective map of \(\mathop{\mathrm{\mathcal{IS}}}_m \times \mathop{\mathrm{\mathcal{IS}}}_n\)-modules \[\label{betabarskew} \bar{\beta}_\lambda:\mathop{\mathrm{\mathcal{R}}}(m)^{\lambda'} \otimes \mathop{\mathrm{\mathcal{R}}}(n)_\lambda \to \frac{ M_{\lambda}+J'_{m,n}(\mathop{\mathrm{\mathcal{IS}}})}{\dot{M}_{\lambda} + J'_{m,n}(\mathop{\mathrm{\mathcal{IS}}})}.\tag{56}\] By summing with respect to all partitions of \(r\) we obtain a surjective map of vector spaces \[\label{surjskew1}\bigoplus_{\lambda \vdash r} \mathop{\mathrm{\mathcal{R}}}(m)^{\lambda'} \otimes \mathop{\mathrm{\mathcal{R}}}(n)_\lambda \to \frac{ M_{(1^r)}+J'_{m,n}(\mathop{\mathrm{\mathcal{IS}}})}{J'_{m,n}(\mathop{\mathrm{\mathcal{IS}}})}.\tag{57}\] Using Lemma 8(1) and a computation with dimensions identical to the proof of Theorem 5.6 (at the end of part (1)), yields that the domain and codomain of the linear map (57 ) have equal dimensions. Thus (57 ) is an isomorphism which implies that the map in (56 ) is an isomorphism.

(2) Let \(Z=\mathop{\mathrm{\mathcal{PT}}}\). The proof of this case is similar to (1).

(3) Let \(Z=\mathop{\mathrm{\mathcal{T}}}\). This too is similar to (1), but there is one difference. If \(r > n\), then by Lemma 8(1) we have \(\big(\Bbbk\langle\boldsymbol{y}_{m \times n}\rangle/ J'_{m,n}(\mathop{\mathrm{\mathcal{PT}}})\big)_r=0\). Let \(r \le n\). Suppose \(\lambda\vdash r\) and \(S \in \mathop{\mathrm{Tab}}_\lambda([m])\), \(T \in \mathop{\mathrm{Tab}}_\lambda([n]).\) If \(T\) has an entry of weight at least 2, then by Lemma 9(1) we have \((S|T) \in J'_{m,s}(\mathop{\mathrm{\mathcal{T}}})\). Hence the map (53 ) induces a surjective map of \(\mathop{\mathrm{\mathsf{GL}}}_m(\Bbbk) \times \mathop{\mathrm{\mathcal{T}}}_n\)-modules \[\label{betabarskew3} L_{\lambda}(V_m) \otimes \mathop{\mathrm{\mathcal{R}}}(n)_\lambda \to \frac{ M_{\lambda}+J'_{m,n}(\mathop{\mathrm{\mathcal{T}}})}{\dot{M}_{\lambda} + J'_{m,n}(\mathop{\mathrm{\mathcal{T}}})}.\tag{58}\] Consider the subspace \(W_\lambda\) of \(L_{\lambda}(V_m)\) defined in Definition 7(2). The kernel of the map (58 ) contains the subspace \(W_{\lambda} \otimes \mathop{\mathrm{\mathcal{R}}}(n)_\lambda\) according to Lemma 10. Now by Lemma 4 we conclude that the map (58 ) induces a surjective map of \(\mathfrak{S}_m \times \mathop{\mathrm{\mathcal{T}}}_n\)-modules \[\label{betabarskew4} L_{\lambda}(U_m) \otimes \mathop{\mathrm{\mathcal{R}}}(n)_\lambda \to \frac{ M_{\lambda}+J'_{m,n}(\mathop{\mathrm{\mathcal{T}}})}{\dot{M}_{\lambda} + J'_{m,n}(\mathop{\mathrm{\mathcal{T}}})}.\tag{59}\] By summing with respect to all partitions of \(r\) we obtain a surjective map of vector spaces \[\label{surjskew2}\bigoplus_{\lambda \vdash r} L_\lambda(U_m) \otimes \mathop{\mathrm{\mathcal{R}}}(n)_\lambda \to \frac{ M_{(1^r)}+J'_{m,n}(\mathop{\mathrm{\mathcal{IS}}})}{J'_{m,n}(\mathop{\mathrm{\mathcal{IS}}})}.\tag{60}\] Finally using Lemma 8(3) and a computation with dimensions identical to the proof of eq. (45 ) in the proof of Theorem 20(3), yields that the domain and codomain of the linear map (60 ) have equal dimensions. Thus (60 ) is an isomorphism which implies that the map in (59 ) is an isomorphism. This completes the proof of Theorem 22.

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