A combinatorial model for the decomposition of multivariate polynomials rings as an $S_n$-module

We consider the symmetric group $S_n$ module of the polynomial ring with $m$ sets of $n$ commuting variables and $m'$ sets of $n$ anti-commuting variables and calculate that the multiplicity of an irreducible indexed by the partition $\lambda$ (a partition of $n$) is the number of multiset tableaux of shape $\lambda$ satisfying certain column and row strict conditions. We also present a finite generating set for the ring of $S_n$ invariant polynomials of this ring.


Introduction
Let m and n be positive integers, then the multivariate polynomial ring of m sets of n commutative variables is a GL n × GL m module that is familiar in the combinatorial representation theory literature. Denote this module by then it is well known (e.g. [GoodWall] Theorem 5.6.7) that the space decomposes as where the direct sum is over all partitions with length less than or equal to min(m, n) and W λ n is a polynomial irreducible GL n module indexed by the partition λ. More precisely, as a GL n module, the multiplicity of the irreducible module W λ n is equal to the dimension of W λ m . This dimension is equal to the number of column strict tableaux of shape λ and content in the entries {1, 2, . . . , m}. The actions of GL n and GL m commute with each other and this decomposition is a consequence of the double centralizer theorem.
Let (a 1 , a 2 , . . . , a m ) be a sequence of non-negative integers that sum to k, then the homogeneous component of degree a i in the variables x 1i , x 2i , . . . , x ni is a GL n submodule of C[X n×m ] and has character equal to h a 1 [X n ]h a 2 [X n ] · · · h am [X n ] where the h r [X n ] are the complete homogenous symmetric functions.
The symmetric group S n is a subgroup of permutation matrices of GL n and so this subspace is also an S n module and the multiplicity of the irreducible symmetric group module indexed by the partition λ can be expressed in terms of plethysm [Lit, ST] of symmetric functions, (1) h a 1 h a 2 · · · h am , s λ [1 + h 1 + h 2 + · · · ] .
While there are no general techniques for computing plethysm multiplicities, it is possible to give a combinatorial interpretation for this particular expression (e.g. [LR,Theorem 10], [LW]). We extend the module under consideration by looking at polynomial rings in m sets of commuting variables and m ′ sets of anticommuting variables. That is, let where the variables x ij commute and commute with the θ ij ′ variables and θ ij θ ab = −θ ab θ ij if either i = a or j = b, and θ 2 ij = 0. There is a GL n × GL m × GL m ′ action on this space, but in this paper we are interested in the subgroup of GL n , S n , acting on the first indices of the variables and how, in particular, the subspaces of fixed homogeneous degree in the polynomials decompose. It is again possible to give a combinatorial interpretation for the multiplicity of an irreducible in terms of plethysm, but in this case we are not aware of general techniques that make it possible to give a combinatorial interpretation for this expression. The main goal of this paper is to give a tableau interpretation for this multiplicity.
After a section introducing the notation used in this paper, there are two sections presenting the two main results: • A combinatorial interpretation for the multiplicity of an irreducible symmetric group representation in a homogeneous component of C[X n×m ; Θ n×m ′ ] in terms of certain multiset tableaux (see Theorem 3.1). • A finite set of algebraic generators for the S n invariants of C[X n×m ; Θ n×m ′ ] (see Theorem 4.5). An interesting consequence of the combinatorial interpretation is that it shows that the symmetric group submodule of fixed homogeneous degree is representation stable in the sense defined in [CF, CEF].

Notation
Let X (i) n represent a collection of commuting variables x 1i , x 2i , . . . , x ni on which the symmetric group S n acts by permutation of the first index. That is, σ(x ri ) = x σ(r)i for all σ ∈ S n . The notation Θ (i) n will be used to represent a collection of anti-commuting variables (Grassmannian variables) θ 1i , θ 2i , . . . , θ ni (again, on which the symmetric group acts on the first index). Now denote the polynomial ring in m sets of the commuting variables and m ′ sets of anti-commuting variables by where the product satisfies the relations θ ri θ sj = −θ sj θ ri if r = s or i = j and θ 2 ri = 0 x rk x sd = x sd x rk and A monomial in C[X n×m ; Θ n×m ′ ] is said to be of degree α = (α 1 , α 2 , . . . , α m ) in the commuting variables and β = (β 1 , β 2 , . . . , β m ′ ) in the Grassmannian variables if the total degree in the variables X (k) n is α k and the total degree of the monomial in the variables Θ (i) n is β i for 1 ≤ i ≤ m ′ and 1 ≤ k ≤ m. The homogeneous subspace spanned by all monomials of degree α = (α 1 , α 2 , . . . , α m ) in the commuting variables and β = (β 1 , β 2 , . . . , β m ′ ) in the Grassmannian variables is an S n submodule of C[X n×m ; Θ n×m ′ ].
There is other notation for this symmetric group module that is worth mentioning in terms of the symmetric tensor S r (V ) and antisymmetric tensor r ′ (V ). If V r is a vector space of dimension r with a basis {v 1 , v 2 , . . . , v r }. We note that as S n modules, where the symmetric group S n acts on the vector space V n in this expression.
A partition of an integer n is a sequence of positive weakly decreasing integers whose values sum to n. The notation λ ⊢ n is shorthand for the fact that λ is a partition of n and let ℓ(λ) denote the number of entries. The cells of a partition λ is the set of pairs {(i, j) : 1 ≤ i ≤ ℓ(λ), 1 ≤ j ≤ λ i }. In this paper, the cells will be graphically represented by displaying them in the first quadrant using French notation with the largest row of the partition on the bottom.
A multiset is a collection of objects where the entries are allowed to repeat. Multisets will be indicated by enclosing the collection of elements with { {, } } to indicate that the structure keeps the multiplicity of the elements. When the multiset has entries which are integers between 1 and m, the content vector of the multiset will be a vector (a 1 , a 2 , . . . , a m ) where a i ≥ 0 is the number of times that i appears in the multiset.
A multiset partition is a multiset of multisets of integers. That is, π = { {S 1 , S 2 , . . . , S r } } where each of the S i are mulitsets of integers. The entries in π are referred to as the parts of π and the length of π is the number of (non-empty) parts of π. The content of a multiset partition π is the disjoint union of the the entries of π, that is, The multiset partitions that appear in this application will be in two different alphabets. Fix two nonnegative integers m and m ′ , then the multiset partitions that are considered here will have entries in [m] ∪ [m ′ ] := {1, 2, . . . , m} ∪ {1, 2, . . . , m ′ } where barred entries are allowed to occur at most once in each part of the multiset partition (but may occur in several parts of a given multiset partition). The notation π ⊢ ⊢S will be used to indicate that that π is a multiset partition with content S.
Multisets are, by definition, an unordered structure, but it will be necessary to specify an order on multisets for constructing multiset tableaux. For this application it does not matter which order is chosen, but there is a step in the proof of Theorem 3.1 which determines the order in which a product is taken in a commutative monomial. The order on multisets will determine the order of how the terms in the monomial are taken. For the purposes of presenting examples of tableaux in this paper, the multisets will be ordered lexicographically, reading the entries in increasing order where the alphabet is totally ordered by 1 < 2 < . . . < m < 1 < 2 < . . . < m ′ (see Examples 2.1 and 3.9 for an illustration of the use of this multiset order). The empty multiset is considered to be the smallest multiset in this order.
Let λ be a partition and S be a multiset of entries from [m] ∪ [m ′ ]. A multiset tableau of shape λ and content S is a map, T , from the cells of λ to multisets of entries of [m] ∪ [m ′ ] satisfying the following conditions.
(1) For each c ∈ λ, T (c) is a multiset (which could be empty) in barred and unbarred entries, [m] ∪ [m ′ ] and barred entries may appear at most once in a cell.
(2) The cells are weakly increasing in both the rows and columns with respect to the chosen order on multisets. That is, T (c 1 , c 2 ) ≤ T (c 1 + 1, c 2 ) and T (c 1 , c 2 ) ≤ T (c 1 , c 2 + 1) whenever both cells are in the partition. (3) If a multiset label contains an even number of barred entries, then no two cells that are labelled with that multiset may occur in the same column (i.e. the cells with the same multiset label that have an even number of barred entries form a horizontal strip). That is, if S i is a multiset with an even number of barred entries and (c 1 , c 2 ), (c ′ 1 , c ′ 2 ) are cells of λ such that T (c 1 , c 2 ) = T (c ′ 1 , c ′ 2 ) = S i , then either (c 1 , c 2 ) = (c ′ 1 , c ′ 2 ) or c 2 = c ′ 2 . (4) If a multiset contains an odd number of barred entries, then no two cells that are labelled with that multiset may occur in the same row (i.e. the cells with the same multiset label that have an odd number of barred entries form a vertical strip). That is, if S i is a multiset with an odd number of barred entries and (c 1 , The content of a tableau is the multiset union of the content of the cells of the tableau. Example 2.1. Let λ = (7, 3, 2, 2, 1), then the following is an example of a multiset tableau 2 11 2 11 2 11 2 12 2 2 12 12 2 The proof of the main result will require the use of well known identities and notation in symmetric functions. We will mainly follow the notation which is common to references in this area [Mac, Sag, Sta] with a single addition that we describe below. The ring of symmetric functions is the polynomial algebra in generators p i (the power sum generators) for i ≥ 1 where the degree p i is i. That is, The elementary {e i } i≥1 and homogeneous generators {h i } i≥1 are related to the power sum generators by the equations To allow for simpler notation, let h 0 = e 0 = p 0 = 1 and h −r = e −r = p −r = 0 for r > 0. For an integer vector α = (α 1 , α 2 , . . . , α ℓ(α) ) products of of the generators will be represented by the shorthand For a partition λ of n, denote the the irreducible character of the representation of the symmetric group S n indexed by the partition λ by χ λ and the value of this character at a permutation of cycle type µ by χ λ (µ). The Schur symmetric functions are defined as is the number of times that i occurs in the partition µ.
The Hall scalar product on symmetric functions is defined for the power sum basis as There is a combinatorial rule for multiplying an elementary or homogeneous generator and a Schur function that is known as the Pieri rule. It says where the sum on the left is over partitions µ such that λ i ≤ µ i and for all cells (c 1 , c 2 ) and The sum on the right is over partitions γ such that λ i ≤ γ i and for all cells (c 1 , c 2 ) and (c ′ 1 , c ′ 2 ) in γ which are not also in λ, either (c 1 , c 2 ) = (c ′ 1 , c ′ 2 ) or c 1 = c ′ 1 . Symmetric functions play a role both as generating functions for characters of the symmetric group, but also as polynomial characters of GL n representations. The character of a GL n representation is the trace of the representation when it is evaluated at a diagonal matrix with eigenvalues x 1 , x 2 , . . . , x n . It will always be the case that this character, as a function of the eigenvalues, will be equal to a symmetric function f ∈ Λ where f is expanded in the power sum generators and p k is replaced by The symmetric group may be realized as the subgroup of permutation matrices inside of GL n . For each permutation σ, let A σ represent the corresponding permutation matrix. To compute the value of the character of an S n representation with character equal to f [X n ], the variables of f [X n ] are replaced by the eigenvalues of A σ . Let µ be a partition of n and σ a permutation of cycle structure µ. Up to reordering, the eigenvalues of A σ are dependent only on the cycle structure of the permutation σ (that is, on the partition µ). We will denote the evaluation of f [X n ] at the eigenvalues of a permutation matrix of cycle structure µ by f [Ξ µ ] and this is a character value of the S n representation.
is the character of the symmetric group representation corresponding to the module S 2 (V 3 ). To compute the value of the character at the permutations of cycle type µ = (1, 1, 1) with eigenvalues {1, 1, 1}, cycle type (2, 1) with eigenvalues {1, −1, 1} and cycle type (3) with eigenvalues {1, e 2πi/3 , e 4πi/3 }. The character values are the following evaluations The reason that the computation of the character is important for this problem is that the symmetric group character characterizes an S n representation up to isomorphism. That is, let X be a GL n representation with character f [X n ] as a function of the eigenvalues of a permutation matrix. The Frobenius image of the character is the generating function p µ z µ and the multiplicity of an irreducible S n representaton indexed by the partition λ in X is equal to the coefficient of One of the main results of this paper is the following combinatorial model for the decomposition of the multivariate polynomial ring as an S n module.
Theorem 3.1. The multiplicity of the symmetric group irreducible indexed by the partition λ ⊢ n in the subspace of degree α = (α 1 , α 2 , . . . , α m ) in the commuting variables and degree β = (β 1 , β 2 , . . . , β m ′ ) in the Grassmannian variables is equal to the number of multiset tableaux (see the definition in Section 2) of content and of shape λ.
We present an example of this theorem in Example 3.10 at the end of this section. All of the hard combinatorial effort for proving this theorem appears in two recent references of the authors [OZ,OZ2]. By referring the reader to the combinatorial interpretations in those papers we can present a relatively short proof of this result, but there is a part which is admittedly not completely self contained.
Remark 3.2. The case of m = 1 and m ′ = 0 or m = 0 and m ′ = 1 is a well known result in the theory of symmetric functions due to A. C. Aitkin [Ait1,Ait2] (see [Sta] p.474-5 exercises 7.72 and 7.73). The case of m > 0 and m ′ = 0 follows from a result of Littlewood [Lit, ST] and known techniques for calculating plethysm coefficients. The multivariate version that we present here is a repeated tensor of Aitkin's results. What we hope to convey is the surprising fact that the decomposition of this symmetric group module has a simple description in terms of 'multiset tableaux' and these combinatorial objects specialize to several well known special cases.
The following lemmas and propositions involve finding combinatorial interpretations between algebraic expressions and combinatorial objects to explain the coefficients that we see in the ring of symmetric functions. When we began extending our combinatorial results to explain the decomposition of expressions that are no longer bases of the symmetric functions, the multiset tableaux that appear in Theorem 3.1 were a consequence.
Before we prove the theorem we state the following lemma which is a typical calculation of a computation of a GL n character.
Lemma 3.3. The GL n character of the subspace of degree α = (α 1 , α 2 , . . . , α m ) in the x ij variables and degree β = (β 1 , β 2 , . . . , β m ′ ) in the θ ik Grassmannian variables is equal to The proof will also require a combinatorial interpretation for the evaluation of this character at eigenvalues of a permutation matrix of cycle structure µ because we will explicitly compute the Frobenius character. For this we need the following combinatorial definitions.
Definition 3.4. (Definition 33 of [OZ]) Let T α,µ be the set of fillings of some of the cells of the partition µ with multisets such that the total content of the filling is { {1 α 1 , 2 α 2 , . . . , ℓ(α) α ℓ(α) } } such that any number of labels can go into the same cell but all cells in the same row must have the same multiset of labels.
Definition 3.5. (Definition 5.13 of [OZ2]) For a non-negative integer vector β and a partition µ let T β,µ be the fillings of some of the cells of the diagram of the partition µ with subsets of {1, 2, . . . , ℓ(β)} such that the total content of the filling is { {1 β 1 , 2 β 2 , . . . , ℓ(β) β ℓ(β) } } and such that all cells in the same row have the same subset of entries. For F ∈ T β,µ , we define the weight of F , wt(F ), to be −1 to the power of the number of cells plus the number of rows occupied by the sets of odd size.
These two combinatorial definitions are used to describe the following expressions for symmetric group characters whose characters are given as homogeneous and elementary symmetric functions.
Proposition 3.6. (Proposition 27 and Theorem 37 of [OZ]; Proposition 5.6 and Lemma 5.15 of [OZ2]) For non-negative integer vectors α and β and any partition µ, Now a product of these expressions will have terms indexed by elements in T α,µ × T β,µ and combining the objects into multiset fillings of µ establishes that is equal to the sum over all fillings of the diagram for µ with multisets of content where barred entries are not allowed to be repeated more than once and there is a weight of the filling equal to −1 to the power of the number of cells plus the number of rows occupied by the multisets with an odd number of barred entries. Our theorem follows from one final result that we pull from [OZ2] and state here without proof.
Proposition 3.7. (Proposition 5.8 [OZ2]) For partitions λ, τ and µ, let F µ λ,τ be the fillings of the diagram for the partition µ with λ i labels i and τ j labels j ′ such that all cells in a row are filled with the same label. For F ∈ F µ λ,τ , the weight of the filling, wt(F ) is equal to −1 raised to the number of cells filled with primed labels plus the number of rows occupied by the primed labels. Then This last proposition indicates that we should assign a 'type' to each filling described above and group the fillings with the same type together.
Definition 3.8. For a filling F of the diagram for µ with multisets such that the multiset union of all of labels is of content where barred entries are not allowed to be repeated more than once in a cell. We associate the filling to a multiset partition, MSP(F ), which is equal to the multiset collection of the non-empty labels of the cells. For a given multiset partition, definem e (π) to be the partition whose entries are the multiplicities of the parts of π that have an even number of barred entries andm o (π) be the partition whose entries are the multiplicities of the parts of π that have an odd number of barred entries.
Example 3.9. Let n = 24 and µ = (5, 5, 3, 2, 2, 2, 2, 1, 1, 1) and consider the filling F , where (while a multiset is by definition an unordered structure) we have listed the entries of by increasing reading word to be consistent with the order that we will use on tableaux. There are 6 cells which have labels with an odd number of barred entries and they occupy 3 rows so the weight of this filling is −1. If we let π = M SP (F ), theñ m e (π) = (5, 2, 1, 1) andm o (π) = (4, 2).
Proof. (of Theorem 3.1) The GL n character of the subspace of degree α = (α 1 , α 2 , . . . , α m ) in the x ij variables and degree β = (β 1 , β 2 , . . . , β m ′ ) in the θ ik Grassmannian variables is equal to h α [X n ]e β [X n ] by Lemma 3.3. We will use the evaluation of this character at elements of the symmetric group as a subset of elements of GL n . With the GL n character, we can compute the S n character by evaluating h α [X n ]e β [X n ] at the eigenvalues of a permutation matrix. Fix a partition µ of n and we will calculate, using the combinatorial gadgets, the character of the subspace at a permutation matrix of cycle structure µ. Proposition 3.6 implies that h α [Ξ µ ]e β [Ξ µ ] is equal to a sum over fillings of the diagram for µ with multisets of content cont α,β where barred entries are not allowed to be repeated more than once. We then group all fillings by the associated multiset partition to the filling, MSP(F ), hence where the outer sum is over all multiset partitions π of such that π ⊢ ⊢{ {1 α 1 , 2 α 2 , . . . , m αm , 1 β 1 , 2 β 2 , . . . , m ′ β m ′ } } where barred entries are not allowed to repeat in the same part.
We next apply Proposition 3.7 and consider the labels of the filling of multisets where the 'primed' entries of the filling are those with an odd number of barred entries and the 'unprimed' entries of the filling are those with an even number of barred entries. This where again the sum is over all multiset partitions π of content cont α,β where barred entries are not allowed to repeat in the parts.
Since we have computed the character of the subspace for each permutation of of cycle structure µ, it follows that we can compute the Frobenius image of the character of the subspace as where the sum is over all multiset partitions π of content cont α,β where barred entries are not allowed to repeat in the same part.
To conclude the proof of the theorem we need only to establish that the multiplicity of a Schur function indexed by a partition λ in this expression agrees with the description stated in the theorem. Here is where the order of the multisets in the tableau plays a roll in determining the combinatorial interpretation. Each of the generators in the product h n−|me(π)|−|mo(π)| hm e(π) em o (π) will represent the cells in the multiset tableau which are labeled by a fixed multiset. The generator h n−|me(π)|−|mo(π)| are the blank cells, and those with an even number of barred entries are represented by the generators in the product hm e(π) , while those with an odd number of barred entries are represented by the generators in the product em o(π) . Since the multiplication in the ring of symmetric functions is commutative, we can choose to order these terms with respect to the total order that we have placed on these multisets.
To determine the multiplicity of the Schur function s λ in this expression, we repeatedly apply the Pieri rule and to keep track of the terms in the Schur expansion of h n−|me(π)|−|mo(π)| hm e (π) em o(π) .
Tableaux are used where the labels of the tableaux are the multisets represented by the h or e-generators. The Pieri rule implies we will record i cells in a horizontal strip for each product of an h i generator, while we will record i cells in a vertical strip for each e i generator.
We provide an example below to ensure that it is clear that the the coefficient of a Schur function s λ in h n−|me(π)|−|mo(π)| hm e (π) em o(π) is equal to the number of multiset tableaux whose entries are the multisets of π. By equation (5) the coefficient of s λ in the Frobenius image of the subspace has multiplicity equal to the total number of multiset tableaux of shape λ and content cont α,β .
Example 3.10. In Example 3.9, our filling F was just one of the terms which contributed to the Frobenius image h 24−9−6 h (5,2,1,1) e (4,2) and if we order the generators with respect to the multiset partition π = MSP(F ), the product h 9 h 1 h 5 e 2 h 2 h 1 e 4 will agree with the order on multisets mentioned in Section 2, The number of multiset tableaux with these entries will be the coefficient of the Schur function in the symmetric function h (9,5,2,1,1) e (4,2) . We choose λ = (10, 8, 5, 1) to illustrate the computation of the tableaux. They are: Since the multiplicity of an irreducible representation indexed by a partition λ in the S n module of C[X n×m ; Θ n×m ′ ] is equal to the number of multiset tableaux of shape λ, then the the multiplicity of S n invariants (the irreducible indexed by (n)) is equal to the number of single row multiset tableaux. Single row multiset tableaux are in bijection with multiset partitions as defined in Section 2 with one additional condition imposed by the construction on tableau.
Corollary 4.1. The ring of S n invariants of C[X n×m ; Θ n×m ′ ] is indexed by multiset partitions whose entries are in [m] ∪ [m ′ ] of length less than or equal to n where barred entries may occur at most once in a multiset and multisets with an odd number of barred entries may appear at most once in the multiset partition.
The following special cases of these rings of invariants are examples that we are aware of that are considered in the algebraic combinatorics literature.
• If m = 1 and m ′ = 0, then the ring of invariants of C[X n ] are known as symmetric polynomials and are equal to the span of the polynomials Sym n := k≥0 {p λ [X n ] : λ ⊢ k} . A basis for the ring of invariants in this case is indexed by partitions which have length of λ ≤ n.
A well known result [C] states that as an S n module, where I is the ideal p k [X n ] : 1 ≤ k ≤ n and this is equal to the ideal generated by symmetric polynomials with non-constant term. The quotient C[X n ]/I are often referred to as the coinvariants and the inverse system corresponding to that quotient are the harmonics. • If m = 2 and m ′ = 0, the quotient of C[X n×2 ] by the ideal generated by the invariants of the ring was defined by Haiman and is known as the ring of diagonal coinvariants [Hai94]. A combinatorial formula for the monomial expansion of the Frobenius characteristic of this S n module was known as the shuffle conjecture [HHLRU05,CM15]. • For m > 2 and m ′ = 0, F. Bergeron and L.-F. Préville-Ratelle [Ber, BPR] considered quotients and harmonics in multivariate polynomial spaces and their general linear group and symmetric group characters.
• If m = 2 and m ′ = 1, then the second author [Zab19] recently proposed the quotient C[X n×2 ; Θ n ] by the ideal generated by the invariants as a representation theoretic model for a generalization of the shuffle conjecture known as the delta conjecture [HRW]. • If m > 1 and m ′ = 0, then the ring of invariants of C[X n×m ] is known as MacMahon symmetric functions [Ges, Ros] (MacMahon [MacM] called these symmetric functions in several systems of parameters). MacMahon indexed the basis of this space of symmetric functions by vector partitions instead of multiset partitions. • If m = 1 and m ′ = 1, then the ring of invariants are known as symmetric functions in superspace and was studied by Desrosiers, Lapointe and Mathieu [DLM]. There the invariants are indexed by objects called superpartitions, which are pairs of the form (Λ a ; Λ s ), where Λ a is a strict partition and Λ s is a partition. • If m = 1 and m ′ ≥ 2, then there the ring of invariants was studied by Alarie-Vézina, Lapointe and Mathieu [ALM]. In that work, the invariants are indexed by generalizations of superpartitions. Borrowing the 'super' prefix from the references [DLM, ALM] mentioned above, we will say that π is a super multiset partition of [m] ∪ [m ′ ] if barred entries occur at most once in each part of π and the parts with an odd number of barred entries appear at most once in the multiset partition. The main result of the rest of this section is to establish a finite list of algebraic generators for the ring of S n invariants of C[X n×m ; Θ n×m ′ ] (the analogue of the power sums) in Theorem 4.5.
Analogues of the elementary and complete homogeneous generators exist and we will hint, but not explicitly state, how to define them in terms of generating functions. The generators of this ring are not 'free' because they will satisfy relations coming from the Grassmannian variables.
Let π = { {S 1 , S 2 , . . . , S ℓ(π) } } be a multiset partition whose entries are in [m]∪[m ′ ] of length less than or equal to n where barred entries may occur at most once in a multiset. Furthermore let us assume that the parts of the multisets S i are ordered in weakly increasing order and S i = { {1 a i1 , 2 a i2 , . . . , m a im , s i1 , s i2 , . . . , s iℓ i } }. The monomial symmetric polynomial indexed by π is denoted m π and it is defined as the polynomial in C[X n×m ; Θ n×m ′ ] that the sum of the distinct S n orbits of (X; Θ) π := where S n acts on the first indices of the variables x ij and θ ij . If ℓ(π) > n then the monomial symmetric polynomial is 0. Looking carefully at this set of symmetric group invariants, we notice that if S i = S i+1 and ℓ i is odd, then σ(i, i + 1)(X; Θ) π = −σ(X; Θ) π . In this case m π = 0 under the condition that π contains two equal parts with an odd number of barred entries. Therefore the indexing set for the monomial basis are the super multiset partitions.
Let S be a multiset where barred elements are not allowed to repeat and let S = T ∪ T where T = { {1 a 1 , 2 a 2  x a 1 r1 x a 2 r2 · · · x am rm θ rs 1 θ rs 2 · · · θ rs k .
where the sum is over all sequences (a, b, c, d) of distinct entries with c < d (since θ c1 θ c2 θ d1 θ d2 = θ d1 θ d2 θ c1 θ c2 ). The following power sum generators are Some explicit expansion of the products of the power sum generators shows that The power symmetric polynomials {p π } span the ring of S n invariants of C[X n×m ; Θ n×m ′ ] where the π run over all super multiset partitions with ℓ(π) ≤ n.
Proof. The ring of S n invariants of C[X n×m ; Θ n×m ′ ] are clearly spanned by the monomial symmetric polynomials since they are the S n orbits of a single monomial in the polynomial ring. This basis is indexed by the super multiset partitions with length less than or equal to n.
Consider the power sum symmetric function indexed by a super multiset partition and order the monomials m < m ′ if the number of distinct first indices in m ′ is greater than the number of distinct first indices in m and if that quantity is equal, then order them with some other monomial order (it doesn't particularly matter which order is used). It follows that p π = c π m π plus terms which are smaller with respect to this order. Therefore the set {p π } with ℓ(π) ≤ n also spans the same space and is linearly independent.
Let q = q 1 , q 2 , . . . , q m be a commuting set of variables and z = z 1 , z 2 , . . . , z m ′ be an anticommuting set of variables. Define the following generating functions for the generators It is easily checked that these generating functions are related by E(q, z) = exp (−P (−q, −z)).
The following lemma involves a standard calculation on the generating function using the definitions in Equation (6) and the expansions of the expression in Equation (8). The variables z 2 j ′ = θ 2 ij ′ = 0 so no barred entries may be repeated in the multiset. There is no sign introduced in the expression from the θ ij ′ variables because it is cancelled by the sign from the z j ′ variables.
The coefficients of a monomial q a 1 1 q a 2 2 · · · q am m z s 1 z s 2 · · · z s k in Equation (9) is the expansion of the elementary generator in the power sum symmetric polynomial.
In the notation of the following theorem S = T ∪ T and S ′ = T ′ ∪ T ′ are two multisets with T, T ′ are both multisets of [m] and T , T ′ are both subsets of [m ′ ].
Lemma 4.3 establishes that products of the power sum generators will span the space of invariants. The next result states that we only need the power sum generators of degree less than or equal to n to generate the space of invariants.
Theorem 4.5. The set {p S }, running over all possible multisets S with no repeated barred entries and |S| ≤ n, is a generating set for the ring of invariants of C[X n×m ; Θ n×m ′ ].
Proof. Observe that the generating function expression in Equation (7) is a polynomial, so the coefficient from Equation (9) in E(q, z) is equal to 0 if a 1 + a 2 + · · · + a m + k > n. This implies that for all S = { {1 a 1 , 2 a 2 , . . . , m am , s 1 , . . . , s k } } with |S| > n, then (10) p S = − π ⊢ ⊢S π ={ {S} } (−1) |S|+ℓ(π) a π p π This implies that for all p π that contains a part S ∈ π with |S| > n can be expressed in terms of p π with all parts smaller than or equal to n by repeatedly applying this relation.
These generators are not free however and also satisfy the relation p S p S ′ = (−1) |T |·|T ′ | p S ′ p S if S and S ′ are not equal and p 2 S = 0 if |T | is odd.