The Multiset Partition Algebra

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1. Introduction

n ⊗k The symmetric group Sk acts on the k-fold tensor space (C ) by permuting n its tensor factors. The space C is the deﬁning representation of GLn(C), and n ⊗k GLn(C) acts on (C ) diagonally. These two actions commute and they generate the centralizers of each other. Further, as a (GLn(C),Sk)-bimodule

n ⊗k (C ) =∼ Wµ ⊗ Vµ, µ M where µ is a partition of k with at most n parts, Wµ is an irreducible polynomial representation of GLn(C) and Vµ is a Specht module of Sn. This phenomenon, discovered by Schur [22] and later popularized by Weyl [24], is called the classical Schur–Weyl duality and is a cornerstone of representation theory. n ⊗k Brauer [4] restricted the action of GLn(C) on(C ) to the orthogonal group On(C) and showed Schur–Weyl duality between the actions of Brauer algebra and n ⊗k On(C)on(C ) . Jones [9] and Martin [15], independently, further restricted the action of On(C) to the symmetric group Sn. This led to the definition of partition algebra Pk(ξ) over the polynomial ring F [ξ], where F is a field of characteristic 0. The algebra Pk(ξ) has a basis given by the partitions of a set of cardinality 2k. A partition of such a set can be pictorially represented as a graph, called the partition diagram. In particular, Pk(ξ) is a diagram algebra as described by Martin [16]. Martin and Saluer [17] showed that Pk(ξ) is semisimple over F unless ξ is an integer such that ξ < 2k − 1. Jones [9] showed that, when ξ is evaluated at a positive integer n, then Pk(n) n ⊗k maps onto the commutant of Sn acting on (F ) . Moreover, when n ≥ 2k, ∼ n ⊗k (1) Pk(n) = EndSn ((F ) ). In particular, this isomorphism gives a diagrammatic interpretation of the central- n ⊗k izer algebra EndSn ((F ) ). Martin [15] classified the irreducible representations of Pk(ξ) and a combinatorial description of the same was given by Halverson and Ram [7]. For n ≥ 2k, Benkart and Halverson [2] constructed a basis for the irreducible representations of Pk(n) in terms of the set-partition tableaux. Orellana and Zabrocki [19] stud- ied the combinatorics of the multiset tableaux, which generalizes the set-partition tableaux. The multiset tableaux play a central role in the representation theory of the algebra we define in this paper. Inspired by the Kazdhan–Lusztig basis of Hecke algebras, Graham and Lehrer [6] introduced the notion of a cellular algebra in terms of the existence of a special kind of basis, called cellular basis. An important application of cellular basis is that it allows one to construct the irreducible representations of the cellular algebra. König and Xi [11] gave a basis-free definition of a cellular algebra. Prominent examples of cellular algebras include Ariki–Koike Hecke algebras, Brauer algebras, and also partition algebras. k n The k-th symmetric power Sym (F ) is a polynomial representation of GLn(F ). k n In spirit of the isomorphism (1), we study the centralizer algebra EndSn (Sym (F )) by constructing a new diagram algebra. We define a unital associative algebra MPk(ξ) over F [ξ] in terms of a basis indexed by the multiset partitions of the multiset {1k, 1′k}. Pictorially these multiset partitions can be represented by certain bipartite multigraphs (Equation 9). The structure constants of MPk(ξ) with THE MULTISET PARTITION ALGEBRA 3 respect to this basis are polynomials in ξ (Equation 11) such that these polynomials give integer values when evaluated to a positive integer. In particular, the algebra MPk(n) obtained by evaluating ξ to any positive integer n may be defined over the ring of integers. Following Harman [8, p. 22], we call MPk(ξ) the multiset partition algebra. k n The Schur–Weyl duality between the actions of MPk(n) and Sn on Sym (F ) is demonstrated in Theorem 3.10. This provides a diagrammatic interpretation of k n EndSn (Sym (F )), thereby answering a question in [8, p. 22]. For a vector λ = (λ1,...,λs) of non-negative integers, the representation Symλ(F n) = Symλ1 (F n) ⊗···⊗ Symλs (F n) along with the tensor product of exterior powers of F n are the building blocks for constructing the irreducible polynomial representations of GLn(F ). By the λ n λ restriction Sym (F ) is also a representation of Sn. For a partition ν of n, let aν denote the multiplicity of the irreducible representation of Sn corresponding to ν in Symλ(F n). These multiplicities are interpreted as a count of certain multiset tableaux in [19] and in [8, Proposition 3.11]. In Theorem 4.2 we obtain the ordinary λ generating function for {aν }λ as the plethysm sν [1 + h1 + h2 + ··· ]. As a result, λ n we explicitly describe the irreducible representations of Sn occurring in Sym (F ) when λ = (k) in Theorem 4.3. Due to Schur–Weyl duality between MPk(n) and Sn, this is also an indexing set of the irreducible representations of MPk(n). Another application (Theorem 4.4) of this generating function yields a generating function for the restriction coefficients, i.e., the multiplicities of the irreducible representation of Sn in the restriction of an irreducible polynomial representation of GLn(F ) to Sn. A bijection between the set of bipartite multigraphs and certain equivalence classes of partition diagrams is described in Theorem 5.2. By utilizing this bijection we construct an embedding of MPk(ξ) in Pk(ξ) (Theorem 5.5). Moreover, the image of the embedding is ePk(ξ)e for an idempotent e ∈ Pk(ξ). As a result of this, over F , we obtain MPk(ξ) is a cellular algebra, and MPk(ξ) is semisimple when ξ is not an integer or ξ is integer such that ξ ≥ 2k − 1. In Section 6, we generalize the construction of MPk(ξ) to any vector λ of non-negative integers. The algebra so obtained has a basis indexed by the multiset partitions of {1λ1 ,...,sλs , 1′λ1 ,...,s′λs }, which we pictorially represent by certain bipartite multigraphs. The Schur–Weyl duality exists between the actions of λ n MPλ(n) and Sn on Sym (F ). In particular, for n ≥ 2|λ|, MPλ(n) is isomorphic λ n to the centralizer algebra EndSn (Sym (F )). We also call MPλ(ξ) the multiset partition algebra. We define an embedding of MPλ(ξ) inside P|λ|(ξ). The image of the embedding is eP|λ|(ξ)e for an idempotent e ∈ P|λ|(ξ). As an application of this embedding, we obtain that MPλ(ξ) over F is cellular, and semismiple when ξ is not an integer or ξ is an integer such that ξ ≥ 2|λ|− 1. The aforementioned embedding is an isomorphism when λ = (1k), so that ∼ Pk(ξ) = MP(1k)(ξ). Thus the partition algebra is a special case of the multiset partition algebra. n ⊗k In [8, Proposition 4.1], the action of GLn(F ) on (F ) is restricted to the subgroup consisting of monomial matrices. This prompts the definition of balanced partition algebra which is a subalgebra of Pk(ξ). In Section 6.3, we define balanced bal mulitset partition algebra MPλ whose structure constants do not depend on ξ. 4 NARAYANAN, PAUL, AND SRIVASTAVA

bal Theorem 6.14 gives Schur–Weyl duality between the actions of MPλ and the group of monomial matrices on Symλ(F n). In [5, Section 5], Colmenarejo, Orellana, Saliola, Schilling, and Zabrocki mod- ified the Robinson–Schensted–Knuth correspondence to give a bijection between ordered multiset partitions and pairs of tableaux, where the insertion tableau is a standard Young tableau, and the recording tableau is a semistandard multiset tableau. We provide a representation-theoretic interpretation of this bijection in Equation (53). In Theorem 7.1, we give a bijection between multiset partitions and pairs of semistandard multiset tableaux which corresponds to the decomposition of MPλ(n) as MPλ(n) × MPλ(n)-module. Throughout this paper, F denotes a field of characteristic 0. In order to avoid an excessive amount of notation, most of the results of this paper are proved for λ = (k). The proofs of the analogous results in the general case follow by minor modifications that we outline in Section 6 where required.

2. Preliminaries In this section, we give an overview of the multiset tableaux, partition algebras, and centralizer algebras of permutation representations of a ﬁnite group, which are relevant in this paper.

2.1. Multisets and tableaux. In this section, we define and introduce notation for various terms related to multisets, partitions of multisets, and tableaux. Multiset tableaux occur in the work of Colmenarejo, Orellana, Saliola, Schilling, and Zabrocki [5]. Here we include a self-contained recapitulation of these concepts. A multiset is a collection of possibly repeated objects. For example, {1, 1, 1, 2, 2} is a multiset with three occurrences of the element 1 and two occurrences of the element 2. More precisely: Definition 2.1. A multiset is an ordered pair (S,f) where S is a set, and f is a non-negative integer-valued function on S, which we call the multiplicity function. Given S = {1, 2,...,n} and a multiplicity function f, we shall denote (S,f)= {1f(1), 2f(2),...,nf(n)}. Here f(i) indicates the number of times i appears in the multiset. Thus the aforementioned multiset would be denoted {13, 22}. Given a set S, denote the set of all multisets with elements drawn from S by AS . Definition 2.2. A multiset partition of a given multiset M is a multiset of multisets whose disjoint union equals M. For example, π = {{1, 2}, {2}, {12}} is a multiset partition of {13, 22}. Given an ordered alphabet S, to define the notion of semistandard tableaux with entries as multisets in AS , we give a total order on AS . Definition 2.3. Let λ be a partition, and S be an ordered alphabet. Given a total order on multisets in AS , a semistandard multiset tableau (SSMT) of shape λ is a filling of the cells of Young diagram of λ with the entries from AS such that: • the entries increase strictly along each column, • the entries increase weakly along each row. THE MULTISET PARTITION ALGEBRA 5

The content of a SSMT P is the multiset obtained by the disjoint union of entries of P . Let SSMT(λ, C) denote the set of all semistandard multiset tableau of shape λ and content C. Unless speciﬁed otherwise, we assume the order on multisets to be the graded lexicographic order (see [5, Section 2.4]). Given multisets M1 = {a1,...,ar} with a1 ≤···≤ ar, and M2 = {b1,...,bs} with b1 ≤···≤ bs, we say M1 ≤L M2 in this order if (1) r ≤ s, (2) if r = s, then there exists a positive integer i ≤ r such that aj = bj for all 1 ≤ j

1′ 2′ 3′ 4′ 5′

′ ′ ′ Let Ak denote the set of all partition diagrams on {1, 2, . . . , k, 1 , 2 ,...,k }. Let Pk(ξ) denote the free module over the polynomial ring F [ξ] with basis Ak. Given d1, d2 ∈ Ak, let d1 ◦ d2 denote the concatenation of d1 with d2, i.e., place d2 on the top of d1, and then identify the bottom vertices of d2 with the top vertices of d1 and remove any connected components that lie completely in the middle row. (Borrowing analogy from the composition of maps, in the composition d1 ◦ d2, we apply d2 followed by d1, this diﬀers from the standard convention but it has been adapted, for example, in Bloss [3, p. 694].) For the basis elements in Ak of Pk(ξ), deﬁne the multiplication as follows: l (2) d1d2 := ξ d1 ◦ d2 where l is the number of connected components that lie entirely in the middle row while computing d1 ◦ d2. Example 2.5. The multiplication of ′ ′ ′ ′ ′ d1 = {{1, 1 }, {2}, {3}, {4}, {2 , 3 }, {5, 4 , 5 }} with ′ ′ ′ ′ ′ d2 = {{1, 2, 1 }, {3, 5}, {2 , 3 }, {4 }, {4, 5 }} 6 NARAYANAN, PAUL, AND SRIVASTAVA is illustrated below: 1 2 3 4 5

d2 =

1′ 2′ 3′ 4′ 5′

1 2 3 4 5

d1 =

1′ 2′ 3′ 4′ 5′ In the above, there are exactly two connected components which lie entirely in the middle, so 1 2 3 4 5

2 d1d2 = ξ

′ ′ ′ ′ ′ 1 2 3 4 5 .

By linearly extending the multiplication (2), Pk(ξ) becomes an associative unital algebra over F [ξ], and it is called the partition algebra. Now we deﬁne another basis of Pk(ξ), called the orbit basis, which is particu- ′ ′ larly essential for this paper. Given d, d ∈ Ak, we say d is coarser than d, denoted as d′ ≤ d, if whenever i and j are in the same block in d, then i and j are in the same block in d′. For a partition diagram d, deﬁne the element xd ∈Pk(ξ) by setting:

(3) d = xd′ . d′ d X≤ It can be easily seen that the transition matrix between {d} and {xd} is unitrian- gular. Thus {xd} is also a basis of the partition algebra Pk(ξ). The structure constants of Pk(ξ) with respect to the orbit basis {xd} is given in [2, Theorem 4.8]. To state this result, we need the following deﬁnitions. Definition 2.6. For a non-negative integer l and f(ξ) ∈ F [ξ], the falling factorial polynomial is (f(ξ))l = f(ξ)(f(ξ) − 1) ··· (f(ξ) − l + 1).

Definition 2.7. Given a partition diagram d = {B1,...,Bn} ∈ Ak, we deﬁne u l ′ ′ Bj := Bj ∩{1,...,k} and Bj := Bj ∩{1 ,...,k } for 1 ≤ j ≤ n. u l u l u u u Then we can rewrite d = {(B1 ,B1),..., (Bn,Bn)}. Also deﬁne d := {B1 ,...,Bn} l l l and d := {B1,...,Bn}. u For d1, d2 ∈ Ak, we say d1 ◦ d2 matches in the middle if the set partition d1 is l same as the set partition obtained from d2 after ignoring the primed numbers in l d2. Denote [d1 ◦ d2] by the number of connected components that lie entirely in the middle row while computing d1 ◦ d2. For a partition diagram d ∈ Ak, let |d| denote the number of blocks in d. From [18, Lemma 3.1], we recall, in the following theorem, the structure constants of Pk(ξ) with respect to the basis {xd}. THE MULTISET PARTITION ALGEBRA 7

Theorem 2.8. For d1, d2 ∈ Ak, we have

d(ξ − |d|)[d1◦d2]xd if d1 ◦ d2 matches in the middle, (4) xd xd = 1 2 0 otherwise, (P where the sum is over all the coarsenings d of d1◦d2 which are obtained by connecting a block of d2, which lie entirely in the top row of d2, with a block of d1, which lie entirely in the bottom row of d1. 2.2.1. Schur–Weyl duality. Let F n be the n-dimensional vector space over F with standard basis {e1,e2,...,en}. The group Sn acts on a basis element ei as:

σei := eσ(i), where σ ∈ Sn. n ⊗k So Sn acts on the k-fold tensor product (F ) diagonally. For d ∈ Ak, deﬁne

1 if ir = is if and only if r and s (d)i1 ,...,ik = are in the same block of d, i1′ ,...,ik′  0 otherwise. Deﬁne a map φ : P (n) → End ((F n)⊗k) as follows: k k  F i1 ,...,ik (5) φk(xd)(ei ⊗···⊗ ei )= (d) ei ′ ⊗···⊗ ei ′ . 1 k i1′ ,...,ik′ 1 k i ,...,i 1′X k′ The Schur–Weyl duality between the actions of the group algebra FSn and the n ⊗k partition algebra Pk(n) on (F ) is given in [9] and in [14]. We state, in the following theorem, the Schur–Weyl duality from [7, Theorem 3.6], which is more applicable in this paper. n ⊗k Theorem 2.9. (1) The image of map φk : Pk(n) → EndF ((F ) ) is n ⊗k EndSn ((F ) ). The kernel of φk is spanned by

{xd | d has more than n blocks}. n ⊗k In particular, when n ≥ 2k, Pk(n) is isomorphic to EndSn ((F ) ). n ⊗k (2) The group Sn generates the centralizer algebra EndPk(n)((F ) ). 2.3. Centralizer algebras. For a finite set X, let F [X] denote the space of F -valued functions on X. Bya G-set X, we mean a group G acts on X on the left. Then G also acts on F [X] as follows: (6) g · f(x)= f(g−1x), for x ∈ X, g ∈ G, and f ∈ F [X]. The space F [X] is called the permutation representation of G associated to the G- set X. Suppose that X and Y are finite G-sets. Given a function φ : X × Y → F , the integral operator ζφ : F [Y ] → F [X] associated to φ is defined as

(7) (ζφf)(x)= φ(x, y)f(y), for f ∈ F [Y ]. y Y X∈ If Z is another G-set and φ : X × Y → F and ψ : Y × Z → F are functions, then

ζφ ◦ ζψ = ζφ∗ψ, where φ ∗ ψ : X × Z → F is the convolution product (8) φ ∗ ψ(x, z)= φ(x, y)ψ(y,z). y Y X∈ 8 NARAYANAN, PAUL, AND SRIVASTAVA

Let (G \ X × Y ) denote the set of orbits of the diagonal action of G on X × Y . From [21, Theorem 2.4.4] we have: Theorem 2.10. Let X and Y be ﬁnite G-sets. Deﬁne 1 if (x, y) ∈ O, φO(x, y)= (0 otherwise.

Write ζO = ζφO . Then the set

{ζO | O ∈ (G \ X × Y )} is a basis for HomG(F [Y ], F [X]). Consequently,

dim HomG(F [Y ], F [X]) = |G \ (X × Y )|.

3. The multiset partition algebra For a positive integer k, each multiset partition of {1k, 1′k} can be represented by the equivalence class of certain graphs by a procedure described below (see Equa- tion (9)). We use these graphs to define the multiset partition algebra MPk(ξ). Let Γ be a bipartite multigraph with a bipartition of vertices into {0, 1,...,k}⊔{0, 1,...,k} and edge multiset EΓ. We arrange the vertices in two rows, with edges joining a vertex in the top row to one in the bottom row. We define a weight function Z2 w : EΓ → ≥0 by setting w(e) = (i, j) for every edge e from the vertex i in the top row to the vertex j in the bottom row. The weight of Γ is w(Γ) := w(e). e∈EΓ Let Bk be the set of all bipartite multigraphs Γ with the weight w(Γ) = (k, k). P Given an edge e of Γ ∈ Bk, we say e is non-zero weighted if w(e) 6= (0, 0). Two bipartite multigraphs are said to be equivalent if they have the same non-zero weighted edges. Denote by B˜k the set of all equivalence classes in Bk. Define the rank of Γ, denoted rank(Γ), to be the number of non-zero weighted edges of Γ. Since all graphs in the same equivalence class have the same rank, the rank of the equivalence class [Γ] is defined as rank([Γ]) = rank(Γ).

Example 3.1. The following two graphs in B5 have rank four, and they are equivalent. 0 1 2 3 4 5

0 1 2 3 4 5 0 1 2 3 4 5

0 1 2 3 4 5

The set B˜k is in bijection with the set of multiset partitions of the multiset k ′k {1 , 1 }. Explicitly, for a class [Γ] ∈ B˜k, let EΓ = {(a1,b1),..., (an,bn)} be the collection of non-zero edges of Γ then we have the following bijective correspondence: (9) [Γ] ↔ {{1ai , 1′bi } | i =1, 2,...,n}. THE MULTISET PARTITION ALGEBRA 9

Thus graphs in B˜k are a diagrammatic interpretation of multiset partitions of {1k, 1′k}. For example, the multiset partition associated with the class of graphs in Example 3.1 is {{1′}, {1′}, {12, 1′3}, {13}}. Let MPk(ξ) denote the free module over F [ξ] with basis B˜k and Uk be the subset of B˜k consisting of equivalence classes of graphs whose edges are of the form (i,i) for i ∈{0, 1,...,k}. The following is the main theorem of this section.

Theorem 3.2. For [Γ1], [Γ2] in B˜k we deﬁne the operation:

(10) [Γ ] ∗ [Γ ]= Φ[Γ] (ξ)[Γ] 1 2 [Γ1][Γ2] ˜ [Γ]X∈Bk where Φ[Γ] (ξ) ∈ F [ξ] is given in Equation (11). The linear extension of this [Γ1][Γ2] operation makes MPk(ξ) an associative unital algebra over F [ξ], with the identity element id = [Γ]. [Γ]∈Uk P Let [Γ1], [Γ2] ∈ B˜k and pick the respective representatives Γ1, Γ2 with n edges, where n ≥ max{rank(Γ1), rank(Γ2)}. Place the graph Γ2 above Γ1 and identify the vertices in the bottom row of Γ2 with those in the top row of Γ1. A path on this diagram is an ordered triple (a,b,c) such that (a,b) is an edge of Γ2 and (b,c) is an edge of Γ1.

Definition 3.3 (Conﬁguration of paths). A conﬁguration of paths with respect to an ordered pair (Γ1, Γ2) ∈ Bk ×Bk is a multiset P = {p1,...,pn} of n paths, pi = (ai,bi,ci), such that the covering condition (C) holds:

E = {(a ,b ) | 1 ≤ i ≤ n}, and (C) Γ2 i i EΓ1 = {(bi,ci) | 1 ≤ i ≤ n}.

n Let Supp (Γ1, Γ2) denote the set of all conﬁgurations of paths with respect to (Γ1, Γ2).

n For P ∈ Supp (Γ1, Γ2), let ΓP denote the multigraph with edge multiset EΓP = {(a1,c1),..., (an,cn)}. Let D = {(s1,t1),..., (sj ,tj )} be the set of distinct edges of ΓP , labelled in lexicographically increasing order. For (s,t) ∈ D, deﬁne the multiset Pst = {b | (s,b,t) ∈ P }. Let pst denote the cardinality of Pst, and for r = 0, 1,...,k denote the multiplicity of the path (s,r,t) occuring in P by pst(r). ˜ n Let [Γ] ∈ Bk with n ≥ rank(Γ). Let SuppΓ(Γ1, Γ2) be the set of path conﬁgurations n P in Supp (Γ1, Γ2) such that ΓP = Γ. The structure constant of [Γ] in the product [Γ1] ∗ [Γ2] is the following:

[Γ] (11) Φ (ξ)= KP · (ξ − rank(Γ)) Γ1 ◦Γ2 , where [Γ1][Γ2] [P ] P ∈Supp2k(Γ ,Γ ) XΓ 1 2 1 pst KP = , p00(1)! ··· p00(k)! pst(0),...,pst(k) (s,t)∈DY\{(0,0)} 

Γ1◦Γ2 k and [P ]= i=1 p00(i), the number of paths (0, i, 0) in P for all i ≥ 1. Example 3.4P. Let k = 2. Consider the following graph: 10 NARAYANAN, PAUL, AND SRIVASTAVA

0 1

Γ1 =

0 1

Let Γ2 = Γ1. We wish to ﬁnd [Γ1] ∗ [Γ2], so we pick representatives Γ1 and Γ2 of the classes with n = 4 edges each. We do so by adding the requisite number (n − 3) of extra (0, 0) edges. Then we identify the bottom row of Γ2 with the top row of Γ1 to obtain: 0 1 • ❆ • ❆❆ ⑥⑥ (n−3) ❆⑥❆⑥ ⑥⑥ ❆❆ ⑥⑥ ❆ •0 •1

0 1 • ❆ • ❆❆ ⑥⑥ (n−3) ❆⑥❆⑥ ⑥⑥ ❆❆ ⑥⑥ ❆ •0 •1

2k Thus Supp (Γ1, Γ2) comprises four elements {P1, P2, P3, P4} where

n−4 P1 = {(0, 0, 0) , (0, 1, 0), (0, 0, 1), (1, 0, 0), (1, 1, 1)}, n−3 P2 = {(0, 0, 0) , (0, 1, 1), (1, 1, 0), (1, 0, 1)}, n−3 P3 = {(0, 0, 0) , (0, 1, 0), (1, 0, 1), (1, 1, 1)}, n−4 P4 = {(0, 0, 0) , (0, 0, 1), (0, 1, 1), (1, 1, 0), (1, 0, 0)}.

The graphs ΓP1 , ΓP3 and ΓP4 , corresponding to conﬁgurations P1, P3 and P4 respectively, represented by:

•0 •1 •0 •1 •0 •1 ❆❆ ⑥ ❆❆❆❆ ⑥⑥ ❆❆ ⑥⑥ ❆❆❆⑥⑥⑥ (n−3) ⑥❆ (n−2) (n−4) ❆⑥❆⑥❆⑥ ⑥⑥ ❆❆ ⑥⑥⑥⑥ ❆❆❆❆ ⑥⑥ ❆ ⑥⑥⑥⑥ ❆❆ •0 •1 •0 •1 •0 •1

2k Observe that ΓP2 = ΓP1 = Γ1 and hence the set SuppΓ1 (Γ1, Γ2) consists of Γ1◦Γ2 Γ ◦Γ P1, P2. By Equation (11), KP1 · (ξ − rank(Γ1))[P1 1 2 ] = ξ − 3 as [P1 ] = 1 and [Γ1] KP = 1. Again, KP · (ξ − rank(Γ )) Γ1◦Γ2 =1. Therefore, Φ = ξ − 2. 1 2 1 [P2 ] [Γ1][Γ2] [Γ ] [Γ ] A similar computation for P , P yields Φ P3 = 2(ξ − 2) and Φ P4 = 4 3 4 [Γ1][Γ2] [Γ1][Γ2] Thus ﬁnally [Γ1] ∗ [Γ2] is equal to:

•0 •1 •0 •1 •0 •1 ❆❆ ⑥ ❆❆❆❆ ⑥⑥ ❆❆ ⑥⑥ ❆❆❆⑥⑥⑥ (ξ−2) ⑥❆ + 2(ξ−2) + 4 ❆⑥❆⑥❆⑥ ⑥⑥ ❆❆ ⑥⑥⑥⑥❆❆❆❆ ⑥⑥ ❆ ⑥⑥⑥⑥ ❆❆ •0 •1 •0 •1 •0 •1 THE MULTISET PARTITION ALGEBRA 11

Remark 1. Let n ≥ max{rank([Γ]), rank([Γ1]), rank(Γ2)}. Assume n ≤ 2k. Then the assignment n 2k (12) β : SuppΓ(Γ1, Γ2) → SuppΓ (Γ1, Γ2) P 7→ P ⊔{(0, 0, 0)2k−n}

Γ1◦Γ2 Γ1◦Γ2 is a bijection. Moreover, KP = Kβ(P ) and [P ] = [β(P ) ]. The case when n ≥ 2k can be done likewise by adding {(0, 0, 0)n−2k} to every conﬁguration in 2k SuppΓ (Γ1, Γ2). The polynomial deﬁned in Equation (11) is integer valued when evaluated at integers, and MPk(n) is an algebra over Z. Moreover, we have the following theorem:

Theorem 3.5. Let [Γ1], [Γ2], [Γ] ∈ B˜k and the respective representatives Γ1, Γ2, Γ have n edges. Deﬁne Γ n (13) CΓ1Γ2 (n) := (b1,...,bn) |{(a1,b1,c1),..., (an,bn,cn)} ∈ SuppΓ(Γ1, Γ2) . Then Φ[Γ] (n)= |CΓ (n)|. [Γ1][Γ2] Γ1Γ2 n Proof. Note that n ≥ max{rank(Γ), rank(Γ1), rank(Γ2)}. If SuppΓ(Γ1, Γ2) is Γ 2k empty, then CΓ1Γ2 (n)= ∅ and by Remark 1, SuppΓ (Γ1, Γ2)= ∅. We have Φ[Γ] (n)= |CΓ (n)| =0. [Γ1][Γ2] Γ1Γ2 n n When SuppΓ(Γ1, Γ2) 6= ∅, for P ∈ SuppΓ(Γ1, Γ2), let D and Pst be as deﬁned on page 9. Let Perm(Pst) be the set of distinct permutations of the elements of Pst. Γ Then Perm(P ) := Perm(Ps1t1 ) ×···× Perm(Psj tj ) is a subset of CΓ1Γ2 (n). From Equation (13) Γ CΓ1Γ2 (n)= Perm(P ). P ∈Suppn(Γ Γ ) GΓ 1 2 pst pst! Note that, |Perm(Pst)| = = . Hence pst(0),...,pst(k) pst(0)!pst(1)!···pst(k)!

pst |Perm(P )| = . pst(0),...,pst(k) (s,tY)∈D Γ1◦Γ2 Also, p00 = n − rank(Γ) = p00(0) + [P ], thus p p p − p (0) 00 = 00 00 00 p (0),...,p (k) p (0) p (1),...,p (k) 00 00 00 00 00 (n − rank(Γ)) P Γ1 ◦Γ2 p − p (0) = [ ] 00 00 [P Γ1◦Γ2 ]! p (1),...,p (k) 00 00 (n − rank(Γ)) P Γ1 ◦Γ2 = [ ] . p00(1)! ··· p00(k)!

Recall KP from Equation (11), then we have

3.1. Centralizer algebra and Schur–Weyl duality. The deﬁning action n k n of GLn(F ) on F extends uniquely to the kth symmetric power Sym (F ). So the k n subgroup Sn of GLn(F ) acts on Sym (F ) by the restriction. Let n

M(n, k)= {a := (a1,...,an) | ai = k, ai ∈ Z≥0}. i=1 X a a1 an k n The set {e = e1 ··· en ∈ Sym (F ) | a = (a1,...,an) ∈ M(n, k)} is a basis of k n Sym (F ). For w ∈ Sn and a ∈ M(n, k) deﬁne an action of Sn on M(n, k) by

w.a = (aw−1(1),...,aw−1(n)). a The Sn-linear map e 7→ 1a, where 1a is the indicator function of a ∈ M(n, k), determines the isomorphism: (14) Symk(F n) =∼ F [M(n, k)]. k n Thus Sym (F ) is a permutation representation of Sn. We have the following algebra isomorphism k n ∼ (15) EndSn (Sym (F )) = EndSn (F [M(n, k)]). a a ... a We express an element (a,b) ∈ M(n, k)×M(n, k) as the biword: 1 2 n . b b ... bn 1 2 The action of Sn on (a,b) corresponds to permuting the columns of its biword. De- ﬁne B˜k,n := {[Γ] ∈ B˜k | rank([Γ]) ≤ n}. Consider the map

(16) Sn\(M(n, k) × M(n, k)) → B˜k,n

a1 a2 ... an defined by 7→ [Γa,b] b b ... bn 1 2 where Γa,b is the graph with multiset of edges {(ai,bi) | i =1, 2,...,n}. The map is well-defined because two biwords, which are the same up to a permutation of their columns, determine the same graph. For [Γ] ∈ B˜k,n, one can choose the representative Γ′ with n edges. The multiset of edges of Γ′ uniquely determines an Sn-orbit of M(n, k) × M(n, k). Hence the map (16) is a bijection. Following Theorem 2.10 we define the integral operator corresponding to an Sn-orbit of M(n, k) × M(n, k). ˜ Definition 3.6. For each [Γ] ∈ Bk,n, define T[Γ] ∈ EndSn (F [M(n, k)]) by

T[Γ]1a = 1b, where a,b ∈ M(n, k). {b∈M(n,kX)|[Γa,b]=[Γ]} Example 3.7. For k = 2 and n = 3, consider the following graph: 0 1 2

Γ=

0 1 2

We compute the action of the integral operator T[Γ] on some basis elements of F [M(3, 2)]. For a = (2, 0, 0), we must choose tuples b such that the pairs (ai,bi) exhaust all the edges of the graph Γ. There are two such tuples: (1, 1, 0) and (1, 0, 1) THE MULTISET PARTITION ALGEBRA 13 which can easily be veriﬁed to yield the multiset {(0, 0), (0, 1), (2, 1)} of edges. So T[Γ](1(2,0,0))=1(1,1,0) +1(1,0,1). For a = (1, 1, 0), there are no admissible tuples b that yield the multiset of edges as above. Hence T[Γ](1(1,1,0)) = 0. ˜ k n Theorem 3.8. The set {T[Γ] | [Γ] ∈ Bk,n} is a basis of EndSn (Sym (F )). Proof. The proof follows from the isomorphism (15), the bijection (16), and Theorem 2.10.

Given [Γ1], [Γ2] ∈ B˜k,n, we can express their product as follows: T T = α[Γ] T , [Γ1] [Γ2] [Γ1][Γ2] [Γ] ˜ [Γ]X∈Bk,n where α[Γ] is the structure constant. [Γ1][Γ2]

Theorem 3.9. Given graph [Γ1], [Γ2], [Γ] ∈ B˜k,n, the structure constant α[Γ] = |CΓ (n)|, [Γ1][Γ2] Γ1Γ2 where Γ, Γ1, Γ2 are the representative of the respective classes with n edges and Γ is deﬁned in . CΓ1Γ2 (n) (13)

Proof. Let (a, c) ∈ M(n, k) × M(n, k) such that [Γa,c] = [Γ]. Then

T[Γ1]T[Γ2](1a)= T[Γ1] 1b {b|[ΓaX,b]=[Γ2]} 

= T[Γ1](1b) {b|[ΓaX,b]=[Γ2]} = 1z. {b|[ΓaX,b]=[Γ2]} {z|[ΓbX,z ]=[Γ1]} [Γ] So α = |{b ∈ M(n, k) | [Γa,b] = [Γ2], [Γb,c] = [Γ1]}| which is same as [Γ1][Γ2] Γ |CΓ1Γ2 (n)|. In the seminal paper [9], Jones proved the Schur–Weyl duality (Theorem 2.9) n ⊗k between the actions of Pk(n) and Sn on (F ) . In this section we introduce an k n analog of the above scenario for the actions of MPk(n) and Sn on Sym (F ). Theorem 3.10 (Schur–Weyl duality for the multiset partition algebra). Deﬁne a map k n (17) φ : MPk(n) → EndSn (Sym (F )) T if rank[Γ] ≤ n, φ([Γ]) = [Γ] (0 otherwise. The map φ is a surjective algebra homomorphism with the kernel

ker(φ)= F -span{[Γ] ∈ B˜k | rank([Γ]) >n}. ∼ k n In particular, when n ≥ 2k, MPk(n) = EndSn (Sym (F )). Moreover, Sn gener- k n ates EndMPk(n)(Sym (F )). 14 NARAYANAN, PAUL, AND SRIVASTAVA

˜ k n Proof. From Theorem 3.8, {T[Γ] | [Γ] ∈ Bk,n} is a basis of EndSn (Sym (F )), so by the deﬁnition of φ it is a surjective linear map. Let [Γ1] and [Γ2] ∈ B˜k. If [Γ1], [Γ2] ∈ B˜k,n,

φ([Γ ] ∗ [Γ ]) = φ Φ[Γ] (n)[Γ] by Theorem 3.2 1 2 [Γ1][Γ2] ˜ [Γ]X∈Bk = Φ[Γ] (n)T by Deﬁnition 17 [Γ1][Γ2] [Γ] ˜ [Γ]X∈Bk,n

= T[Γ1]T[Γ2] from Theorem 3.9 and Theorem 3.5.

If rank(Γ1) > n or rank(Γ2) > n and [Γ] ∈ B˜k with rank([Γ]) ≤ n appears in 2k [Γ1] ∗ [Γ2], then we claim that (n − rank(Γ))[P Γ1 ◦Γ2 ] = 0 for all P ∈ SuppΓ (Γ1, Γ2). 2k Assume that the rank(Γ1) >n, then for any P ∈ SuppΓ (Γ1, Γ2), rewrite

(n − rank(Γ))[P Γ1◦Γ2 ] = (rank(Γ1) − rank(Γ) − (rank(Γ1) − n))[P Γ1◦Γ2 ].

Note that rank(Γ1) − rank(Γ) is the set of edges of Γ1 of the form (i, 0) for i ≥ 1 Γ ◦Γ that become (0, 0) edges in ΓP . This quantity is [P 1 2 ]. Thus the falling factorial is 0 since rank(Γ) − n> 0. A similar argument works when rank(Γ2) >n. Thus

φ([Γ1] ∗ [Γ2]) = 0 = φ([Γ1]) ◦ φ([Γ2]).

Note that [Γ] ∈ B˜k has rank at most 2k, so when n ≥ 2k, ker(φ) is trivial and thus φ is an isomorphism.

Corollary 3.11 (Associativity). For [Γ1], [Γ2], and [Γ3] ∈ B˜k,

([Γ]1 ∗ [Γ2]) ∗ [Γ3] = [Γ1] ∗ ([Γ2] ∗ [Γ3]).

Proof. For [Γ] ∈ B˜k, the coeﬃcients of [Γ] both in ([Γ]1 ∗ [Γ2]) ∗ [Γ3] and [Γ1] ∗ ([Γ2] ∗ [Γ3]) are polynomials in ξ. From Theorem 3.10, these polynomials are equal when evaluated to any positive integer n ≥ 2k. Since characteristic of F is 0, these polynomials must be the same.

4. Representation theory of the multiset partition algebra In this section, we give an indexing set of the irreducible representations of MPk(n) when n ≥ 2k. It is well known that the partitions of n index the irreducible representations of Sn. Given a partition ν = (ν1 ≥ ν2 ≥ · · · ) of n, the Specht module Vν has a combinatorial basis, known as Young’s seminormal basis [26], consisting of standard Young tableaux of shape ν. In this spirit, we observe that the dimension of an irreducible representation of MPk(n) is obtained by counting certain semistandard multiset tableaux. We ﬁnd a generating function for semistandard multiset tableaux in terms of the Schur functions. As an application of this identity, we obtain a generating function for the multiplicity of a Specht module in the restriction of an irreducible polynomial representation of GLn(F ) to Sn. k We know by Theorem 3.10 that the irreducible representations Mν are indexed by a subset Λk,n of partitions ν of n such that Vν occurs with positive multiplicity in the decomposition of Symk(F n). THE MULTISET PARTITION ALGEBRA 15

For every vector λ = (λ1, λ2,...,λs) of non-negative integers, consider the following representation of Sn: (18) Symλ(F n) := Symλ1 (F n) ⊗ Symλ2 (F n) ⊗···⊗ Symλs (F n).

λ λ n Let aν denote the multiplicity of Vν in Sym (F ). In the following proposition, we λ paraphrase Proposition 3.11 of [8] to express aν in the context of multiset tableaux. λ Proposition 4.1. The multiplicity aν is the number of semistandard multiset tableaux of shape ν and content {1λ1 , 2λ2 ,...,sλs }. Recall that the Schur function corresponding to ν is deﬁned as

T sν (x1, x2,...)= x , T X where the sum is over all the semistandard Young tableau (SSYT) T of shape ν, and ci(T ) is deﬁned to be the number of occurences of a positive integer i in T , T c1(T ) c2(T ) and x := x1 x2 ··· . Littlewood [12] introduced the concept of plethysm of two symmetric functions (for the deﬁnition also see [13, Section I.8]). Given symmetric functions f and g, let f[g] denote the plethystic substitution of g into f.

Theorem 4.2. Let hi be the complete homogeneous symmetric function of de- λ λ1 λ2 λs gree i in the variables q1, q2,..., and for λ = (λ1, λ2,...,λs), q = q1 q2 ··· qs . Then, λ λ sν [1 + h1 + h2 + ··· ]= aν q , λ X where λ runs over all vectors of non-negative integers.

Proof. Each monomial in the Schur function sν (q1, q2,...) corresponds to an SSYT T , and the degree of qi in this monomial is ci(T ). By definition of plethystic substitution, sν [1 + h1 + h2 + ··· ] can be regarded as the bijective substitution of the monomials of 1 + h1 + h2 + ··· into the variables q1, q2,... of sν . The monomials of the expression 1 + h1 + h2 + ··· are indexed by multisets of finite size (in particular the monomials of hk are indexed by multisets of size k), and the variables of sν are indexed by integers. The aforementioned correspondence is a bijection, denoted φ, from the positive integers to the multisets of finite size. Given a positive integer i, let φ(i)= {1φ(i)1 , 2φ(i)2 ,...}. Under the total order induced on multisets by this bijection, replacing each integer in the filling of a SSYT by its image yields a semistandard multiset tableau. This association is reversible. Thus

φ(1)1 φ(1)2 c1(T ) φ(2)1 φ(2)2 c2(T ) sν [1 + h1 + h2 + ··· ]= (q1 q2 ··· ) (q1 q2 ··· ) ··· , T X where the sum is over all the semistandard Young tableau (SSYT) T of shape ν. Thus sν [1 + h1 + h2 + ··· ] is the generating function for the multiset semistandard tableaux of shape ν by their content. The coeﬃcient of qλ in this function is thus the number of semistandard multiset tableaux of shape ν and content {1λ1 ,...,sλs } λ which is precisely aν . 16 NARAYANAN, PAUL, AND SRIVASTAVA

Remark 2. The above result might be restated for a ﬁnite number of variables i1 is q1,...,qs. Let I(s) = {q1 ··· qs |i1,...,is ∈ Z≥0} denote the set of monomials of 1+ h1 + h2 + ··· in the variables q1,...,qs. Then we have:

λ λ sν (I(s)) = aν q . λ X where sν (I(s)) is the Schur function in the set of variables I(s) and the sum is over vectors of non-negative integers with at most s parts.

(k) The following theorem describes the set Λk,n = {ν ⊢ n | aν 6=0}.

k Theorem 4.3. Let Mν denote the irrreducible representation of MPk(n) corresponding to the partition ν of n. For n ≥ 2k, as Sn × MPk(n)-bimodule, we have k n ∼ k Sym (F ) = Vν ⊗ Mν ν ∈MΛk,n n 1 where Λk,n := {ν ⊢ n | i=1(i − 1)νi ≤ k} .

k n Proof. Note thatP the decomposition of Sym (F ) as Sn ×MPk(n)-bimodule follows from the Schur–Weyl duality between Sn and MPk(n) when n ≥ 2k. From [23, Corollary 7.21.3], we have the identity:

b(ν) 2 q (19) sν (1,q,q ,...)= h(u) , u∈ν (1 − q ) n Q where b(ν)= i=1(i − 1)νi and, for a cell u in the Young diagram of ν, h(u) is the hook-length of u. For λ = (kP), by Remark 2, we have that the multiplicity of the Specht module b ν k n k q ( ) h(u) Vν in Sym (F ) is the coeﬃcient of q in h(u) . Each term (1 − q ) in Qu∈ν (1−q ) the denominator may be expanded as 1 + qh(u) + (qh(u))2 + ··· . Thus the lowest qb(ν) k exponent of q in h(u) is b(ν). If b(ν) > k then the coeﬃcient of q is zero, Qu∈ν (1−q ) and thus ν∈ / Λk,n. Conversely if b(ν) ≤ k, pick a corner cell u with hook-length 1. Then 1 = 0 1−qh(u0 ) 1+ q + q2 + ··· , and

qb(ν) = qb(ν)(1 + q + q2 + ··· ) (1 + qh(u) + (qh(u))2 + ··· ). (1 − qh(u)) u∈ν u u Y6= 0 Q Choosing the term 1 from each of the infinite sums corresponding to cells u 6= u0 k−b(ν) and the term q from the infinite sum corresponding to the cell u0 yields an instance of qk. Since the expression above is a positive sum, demonstrating a single k (k) instance is sufficient to prove the positivity of the coefficient of q . Thus aν 6= 0 if and only if b(ν) ≤ k.

1 The description of Λk,n does not require the condition n ≥ 2k. THE MULTISET PARTITION ALGEBRA 17

4.1. Restriction from general linear group to symmetric group. The irreducible polynomial representations Wλ of GLn(F ) of degree d are indexed by the partitions λ of d with at most n parts. Consider the decomposition of the restriction of Wλ to Sn into Specht modules:

rλ ResGLn(F ) W = ⊕ ν Sn λ Vν . ν n M⊢ By Littlewood’s formula [12]:

λ rν = hsν [1 + h1 + h2 + ··· ],sλi, where h−, −i denotes the usual inner product on the ring of symmetric functions. As an application of Theorem 4.2 we deduce a generating function for the integers λ rν .

Theorem 4.4. For every partition λ = (λ1, λ2,...,λn) of d with at most n λ λ λ1 λn parts and every partition ν of n, rν is the coeﬃcient of q := q1 ...qn in

i1 i2 in sν ({q1 q2 ··· qn }) (1 − qj /qi). i

µ µ i1 i2 in aν q = sν ({q1 q2 ··· qn }) where µ is a composition of n. µ X The Jacobi–Trudi identity can be expressed at the level of the Grothendieck ring of the category of polynomial representations of GLn(F ) (see [1]) as:

λi+j−i n (20) Wλ = det(Sym (F )), so

λ w·λ rν = sgn(w)aν w S X∈ n w·λ i1 i2 in = sgn(w)[q ]sν ({q1 q2 ··· qn }) w∈Sn X n λ i1 i2 in i−w(i) = [q ] sgn(w)sν ({q1 q2 ··· qn }) qi w∈Sn i=1 X Yn λ i1 i2 in i−w(i) = [q ]sν ({q1 q2 ··· qn }) sgn(w) qi w S i=1 X∈ n Y λ i1 i2 in i−j = [q ]sν ({q1 q2 ··· qn }) det(qi ).

λ where w · λ is the tuple whose ith coordinate is λi + w(i) − i, and [q ]f(q1,...,qn) λ is the coeﬃcient of q in the function f(q1,...,qn). Recognizing the determinant on the last line as a Vandermonde determinant −1 −1 that evaluates to i

5. Structural properties of the multiset partition algebra Xi [25] proved that partition algebras are cellular with respect to an involution. The goal of this section is to prove the multiset partition algebras are also cellular. We achieve this by exhibiting that MPk(ξ) is isomorphic to ePk(ξ)e for an idempotent e ∈ Pk(ξ) which is ﬁxed by the involution. We also obtain that MPk(ξ) is semisimple when ξ is not an integer or when ξ is an integer such that ξ ≥ 2k − 1.

5.1. An embedding of MPk(ξ) into Pk(ξ). We ﬁrst give a bijection between B˜k and certain class of partition diagrams in Ak. We begin with describing a partition diagram associated to given element in B˜k. Given Γ ∈Bk with n edges, associate to it the canonical biword

a1 a2 ... an b b ... bn 1 2 by arranging the edges of Γ in weakly increasing lexicographic order. For 1 ≤ i ≤ ′ ′ ′ n, we associate the edge (ai,bi) to a subset of {1, 2, . . . , k, 1 , 2 ,...,k } with ai ′ ′ ′ elements from {1, 2,...,k} and bi elements from {1 , 2 ,...,k } as follows: i−1 i i−1 i ′ ′ (21) Bi = ar +1,..., ar, br +1 ,..., br . ( r=1 r=1 r=1 r=1 ) X X X X Note that the set Bi is empty if and only if the pair (ai,bi)=(0, 0). Then

(22) dΓ := {B1,...,Bn} gives a set partition of {1, 2, . . . , k, 1′, 2′,...,k′}. Note that the partition diagram dΓ is independent of choice of representatives of [Γ] ∈ B˜k, henceforth, we call dΓ the canonical partition diagram associated to [Γ]. Example 5.1. Let Γ be the following graph: 0 1 2

0 1 2 0 0 2 Then the corresponding canonical biword of Γ is . By Equation (21), we 0 1 1 ′ ′ have B1 = ∅,B2 = {1 }, B3 = {1, 2, 2 }, and so dΓ is as follows: 1 2

1′ 2′

If we consider the biword of edges of Γ ∈ Bk in any other order and perform the above procedure then the partition diagram so obtained is in the orbit of dΓ under the following action of Sk × Sk on Ak. For (σ, τ) ∈ Sk × Sk and d ∈ Ak, the permutations σ and τ act by the value permutation of the vertices of the top row and the bottom row of d respectively. Note that this action preserves the number of blocks of d, and also preserves the number of elements of the top row and bottom ˜ row occurring in each block. Thus the orbit OdΓ uniquely determines the [Γ] ∈ Bk, and so we obtain the following bijection. THE MULTISET PARTITION ALGEBRA 19

Theorem 5.2. Let (Sk × Sk) \ Ak denote the set of orbits of Sk × Sk on Ak. ˜ For [Γ] ∈ Bk, let OdΓ denote the Sk × Sk-orbit of dΓ. Then the following map is a bijection,

(23) ψ : B˜k → (Sk × Sk) \ Ak

ψ([Γ]) = OdΓ . Example 5.3. Under the above bijection, the graph in Example 5.1 is mapped to the orbit consisting of the following partition diagrams: 1 2 1 2

and

1′ 2′ 1′ 2′

l l Definition 5.4. Given d ∈ Ak, let d = {B1,...,Bn} (for the deﬁnition of d see Deﬁnition 2.7), and |Bi| = bi for i =1,...,n. Then k! αd = . b1! ··· bn!

Now utilizing the bijection in Theorem 5.2 we give an embedding of MPk(ξ) inside Pk(ξ) in the following theorem. Theorem 5.5. The following map is an injective algebra homomorphism

(24) η˜k : MPk(ξ) →Pk(ξ) 1 η˜k([Γ]) = xd. αd Γ d∈O XdΓ Before we oﬀer a proof, we need to set up some notation. n ⊗k k n The canonical projection map πk : (F ) → Sym (F ) has a section τk : Symk(F n) → (F n)⊗k, which is deﬁned as 1 (25) τ (e ··· e )= (e ⊗···⊗ e ) for 1 ≤ j ≤···≤ j ≤ n. k j1 jk k! jσ(1) jσ(k) 1 k σ S X∈ k We have,

(26) πk ◦ τk = idSymk(F n). Recall that both (F n)⊗k and Symk(F n) are permutation modules, and using the following deﬁnitions we interpret the maps πk and τk at the level of permutation modules.

n ⊗k Definition 5.6. We represent an element ej1 ⊗···⊗ ejk in (F ) as the indicator functor corresponding to an ordered set partition of {1,...,k}. That is,

ej1 ⊗···⊗ ejk ↔ 1(A1,...,An) n where Am = {l | ejl = em, 1 ≤ l ≤ k} and ⊔m=1Am = {1,...,k}.

Definition 5.7. For a = (a1,...,an) ∈ M(n, k), deﬁne n D(a)= {S := (S1,...,Sn) | ⊔i=1Si = {1,...,k}, |Si| = ai}. 20 NARAYANAN, PAUL, AND SRIVASTAVA

4 ⊗3 Example 5.8. The element e2⊗e1⊗e1 in (F ) is represented by the indicator function corresponding to the tuple ({2, 3}, {1}, ∅, ∅). For a = (2, 1) ∈ M(2, 3), D(a)= {({1, 2}, {3}), ({1, 3}, {2}), ({2, 3}, {1})}.

Using Deﬁnitions 5.6 and 5.4, the maps πk and τk are described as follows:

(27) πk(1(A1,...,An))=1(|A1|,...,|An|), 1 (28) τ (1 )= 1 . k a |D(a)| S S∈XD(a)

Definition 5.9. For a set X and a tuple r = (r1,...,rn) with ri ∈ X, deﬁne

(29) set(r) := {r1,...,rn}.

Let A = (A1,...,An) and C = (C1,...,Cn) then using Deﬁnition 2.7, the map (5) can be written as

1 if set(A)= du, {C|{(A1,C1),...,(An,Cn)}=d} C (30) xd(1A)= (P0 otherwise. Now we are ready to give a proof of Theorem 5.5.

Proof. The structure constants for both MPk(ξ) and Pk(ξ) are polynomials in ξ therefore it is suﬃcient to proveη ˜k is an algebra homomorphism when ξ is evaluated at any positive integer. For this deﬁne the following map, which is an algebra homomorphism,

k n n ⊗k ηk : EndSn (Sym (F )) → EndSn ((F ) )

ηk(f)= τk ◦ f ◦ πk.

k n for f ∈ EndSn (Sym (F )), and consider the following diagram:

 η˜k / MPk(n) Pk(n) ,

η k n k / n ⊗k EndSn (Sym (F )) EndSn ((F ) ) where the leftmost and the rightmost vertical maps are as in Theorem 3.10 and Theorem 2.9 respectively. By Schur–Weyl duality the vertical maps are algebra isomorphisms for n ≥ 2k. Therefore it is enough to prove

1 ˜ (31) ηk(T[Γ])= xd, for [Γ] ∈ Bk,n, αd Γ d∈O XdΓ where αdΓ is given in Deﬁnition 5.4. Equation (31) as this ensures that the above diagram commutes. a a ... a Let Γ be the graph whose canonical biword is 1 2 n . Fix a = b b ... bn 1 2 (a1,...,an), b = (b1,...,bn). Let A = (A1,...,An) be a set partition of {1,...,k}. THE MULTISET PARTITION ALGEBRA 21

The right hand side of Equation (31) is

τk ◦ T[Γ] ◦ πk(1A)= τk(T[Γ](1(|A1|,...,|An|)))

= τk 1c

 {c|[Γ(|A |,...,|A |),c]=[Γ]} 1 Xn 1 = 1 |D(c)| S {c|[Γ ]=[Γ]} S∈D(c) (|A1|,...,X|An|),c X 1 (32) = 1S. αd Γ {c|[Γ ]=[Γ]} S∈D(c) (|A1|,...,X|An|),c X

For the last equality recall that for Γ = Γ(|A1|,...,|An|),c, we have αdΓ = |D(c)|. If set((|A1|,..., |An|)) 6= set(a), then for every c ∈ M(n, k) we have

[Γ(|A1|,...,|An|),c] 6= [Γ].

So τk ◦ T[Γ] ◦ πk(1A) =0. For d ∈ OdΓ , the cardinality of a given block of d is equal 1 to ai for some 1 ≤ i ≤ n. So by Equation (30), we get α d∈O xd(1A) = 0. dΓ dΓ Assume set((|A1|,..., |An|)) = set(a). Following the notationP of Deﬁnition 2.7: X = (X1,...,Xn) ∈ D(a), O = d = {(X , Y ),..., (X , Y )} Y = (Y ,...,Y ) ∈ D(b), . dΓ  1 1 n n 1 n  Γ =Γ  |X|,|Y |  u If d ∈ OdΓ and d 6= set(A) then from Equation (30), xd(1A)=0  u  If d ∈ OdΓ and d = set(A) then d = {(X1, Y1),..., (Xn, Yn)}, where

{X1,...,Xn} = set(A),

{|Y1|,..., |Yn|} = set(b) such that {(|X1|, |Y1|),..., (|Xn|, |Yn|)} is the biword associated to a graph that is in the class [Γ]. From Equation (30) we know that:

xd(1A)= 1S. {S=(S1,...,Sn)|{(AX1,S1),...,(An,Sn)}=d} Thus we have: 1 1 (33) xd(1A)= 1S. αdΓ αdΓ d∈Od {c|[Γ ]=[Γ]} S∈D(c) XΓ (|A1|,...,X|An|),c X Comparing Equation (33) and Equation (32) we see that Equation (31) is true.

5.2. Cellularity of MPk(ξ). The following proposition [11, Proposition 4.3] allows one to realize another cellular algebra from given a cellular algebra and an idempotent in it. Proposition 5.10. Let A be a cellular algebra with respect to an involution i. Let e ∈ A be an idempotent such that i(e) = e. Then the algebra eAe is also a cellular algebra with respect to the involution i restricted to eAe. u l u l The involution i is deﬁned for d = {(B1 ,B1),..., (Bn,Bn)} ∈ Ak by: l u l u (34) i(d)= {(B1,B1 ),..., (Bn,Bn)} 22 NARAYANAN, PAUL, AND SRIVASTAVA which may be visualized as interchanging each primed element j′ with unprimed element j and vice versa. This map is extended linearly to Pk(ξ). Xi [25] showed that partition algebras are cellular with respect to i.

Definition 5.11. Let Yk be the subset of Ak consisting of all partition diagram d = {B1,B2,...,Bn} satisfying the following condition: u l (35) |Bj | = |Bj | for 1 ≤ j ≤ n.

Example 5.12. For k = 2, the identity element of MPk(ξ) is 0 1 2 0 1 2

id = +

0 1 2 0 1 2 . 1 The element e = 2 (xd1 + xd2 )+ xd3 is an idempotent in MPk(ξ) where 1 2 1 2 1 2

d1 = , d2 = , d3 =

1′ 2′ 1′ 2′ 1′ 2′

1 Lemma 5.13. Let e = xd. Then e is an idempotent and the embedding d∈Yk αd η˜ induces an isomorphism MP (ξ) ≃ eP (ξ)e. k P k k Proof. We showη ˜k(id)= e, whereη ˜k is the embedding deﬁned in Theorem 5.5 and id = [Γ] is the identity element of MP (n) and U consists of elements [Γ]∈Uk k k ˜ in Bk whoseP edges are of the form (a,a) for a ∈ M(n, k). From the deﬁnition of embedding (24) 1 η˜k(id)= xd. αdΓ [Γ]∈U d∈Od X k XΓ

For [Γ] ∈ Uk, by construction of dΓ we have dΓ ∈ Yk, and so the orbit OdΓ ⊂ Yk. ′ Conversely, given d ∈ Yk, then under the bijection (23), [Γ ] ∈ B˜k corresponding to the orbit O belongs to U . So we obtain O = Y . Now the result follows d k [Γ]∈Uk dΓ k by observing that for d , d in the same orbit, α = α . 1 2 F d1 d2 Sinceη ˜k is an embedding, it is enough to show that image ofη ˜k is ePk(ξ)e. The mapη ˜k is an algebra homomorphism andη ˜k(id) = e therefore e is an idempotent and also the image ofη ˜k is contained in ePk(ξ)e. For d0 ∈ Ak, we show that exd0 e u u is in the image ofη ˜k. For d = {B1,...,Bn} ∈ Ak deﬁne U(d) := (B1 ,...,Bn) and l l L(d) := (B1,...,Bn). We have 1 exd0 e = xd1 xd0 xd2 αd αd d ,d 1 2 1 X2∈Yk 1 = xd1 xd0◦d2 αd1 αd2 d1,d2∈Yk ,LX(d2)=U(d0) 1 (36) = xd1◦d0◦d2 αd1αd2 d1,d2∈Yk ,L(d2)=XU(d0),L(d0)=U(d1) THE MULTISET PARTITION ALGEBRA 23

Since d1 ∈ Yk and L(d0) = U(d1), we have αd0 = αd1 . Since L(d2) = U(d0) therefore αd2 is the cardinality βd0 of Sk-orbit of set(U(d1)). For d1, d2 ∈ Yk,L(d2) = U(d0),L(d0) = U(d1), d0 and d1 ◦ d0 ◦ d2 are in the same orbit. Conversely, if d in the orbit of d0 then there exist d1, d2 ∈ Yk such that d = d1 ◦ d0 ◦ d2. u u If d0 has exactly i a,b blocks Bm ,...,Bm such that |B | = |B | = a ( ) 1 i(a,b) ms mt l l and |Bms | = |Bmt | = b for 1 ≤ s,t ≤ i(a,b) then there are θd0 = (a,b)∈N×N(i(a,b)!) partition diagrams d and d in Y such that d ◦ d ◦ d = d. Then the sum (36) 1 2 k 1 0 2 Q simpliﬁes to

θd0 θd0 1 θd0 (37) xd = xd = η˜k([Γ]) αd βd βd αd βd d∈O 0 0 0 0 d∈O 0 Xd0 Xd0 ˜ where [Γ] ∈ Bk corresponds to Od0 under the bijection (23).

Theorem 5.14. The algebra MPk(ξ) over F is cellular. Furthermore, MPk(ξ) over F is semisimple when ξ is not an integer or when ξ is an integer such that ξ ≥ 2k − 1.

Proof. From Lemma 5.13 we have MPk(ξ) ≃ ePk(ξ)e where 1 e = xd. αd d X∈Yk We show that ePk(ξ)e is a cellular algebra with respect to the involution i. The restriction of map i to Yk is a bijection on Yk. Moreover, for d ∈ Yk, 1 we have i(xd) = x . So i(e) = x . By observing that for d = i(d) d∈Yk αd i(d) u l u l u u {(B1 ,B1),..., (Bn,Bn)} ∈ Yk, the cardinality of Sk-orbit of {B1 ,...,Bn} is the P l l same as the cardinality of Sk-orbit of {B1,...,Bn}, i.e, αd = αi(d) we conclude that i(e)= e. By Proposition 5.10 we have ePk(ξ)e is a cellular algebra. Suppose ξ is not an integer or ξ is an integer such that ξ ≥ 2k − 1. In this case, Pk(ξ) is semisimple, so rad(Pk(ξ)) = {0}. Since e is an idempotent, rad(ePk(ξ)e)= e(rad(Pk(ξ)))e. By Lemma 5.13 we have rad(MPk(ξ)) = {0} and thus MPk(ξ) is semisimple.

6. The generalized multiset partition algebra

For every vector λ = (λ1,...,λs) of non-negative integers, we deﬁne a generalized multiset partition algebra MPλ(ξ) over F [ξ] with a basis indexed by the multiset partitions of {1λ1 ,...,sλs , 1′λ1 ,...,s′λs }. We associate such a multiset partition to a class of bipartite multigraphs by a process analogous to Section 3 by generalizing the deﬁnition of Bk for λ. The multiplication rule in MPλ(ξ) is completely analogous to the case of MPk(ξ). So here, we will omit the proofs. Let Γ be a bipartite multigraph on Vλ ⊔Vλ where s (38) Vλ = {I = (i1,i2,...,is) ∈ Z≥0 | ij ≤ λj , 1 ≤ j ≤ s}.

For an edge e of Γ joining (i1,i2,...,is) in the top row with (j1, j2,...,js) in the s s bottom row, the weight w(e) of e is ((i1,i2,...,is), (j1, j2,...,js)) ∈ Z≥0 ×Z≥0. Let Bλ denote the set of bipartite multigraphs Γ on Vλ ⊔Vλ such that the total weight s of edges of Γ is (λ, λ) := ((λ1,...,λs), (λ1,...,λs)). For 0 = (0,..., 0) ∈ Z≥0, an edge e of Γ is non-zero if w(e) 6= (0, 0). The rank(Γ) is the number of non-zero edges of Γ. 24 NARAYANAN, PAUL, AND SRIVASTAVA

Two bipartite multigraphs in Bλ are said to be equivalent if they have the same non-zero weighted edges. Let B˜λ denote the set of all equivalence classes in Bλ. We deﬁne rank([Γ]) := rank(Γ). Let (I,J) := ((i1,i2,...,is), (j1, j2,...,js)) be a non-zero weighted edge of a graph in Bλ. Then the following correspondence

(39) (I,J) ↔{1i1 ,...,sis , 1′j1 ,...,s′js } extends to a bijection between the set B˜λ and the set of multiset partitions of λ λs ′λ ′λs {1 1 ,...,s , 1 1 ,...,s }. Thus the elements of B˜λ are the diagrammatic inter- pretations of the multiset partitions of {1λ1 ,...,sλs , 1′λ1 ,...,s′λs }.

Example 6.1. Consider the following graph in B(2,1). (0, 0) (0, 1) (1, 0) (1, 1) (2, 0) (2, 1)

(0, 0) (0, 1) (1, 0) (1, 1) (2, 0) (2, 1)

The multiset partition of {12, 2, 1′2, 2′} associated to this graph is {{1′}, {2, 1′, 2′}, {12}}.

Deﬁne MPλ(ξ) to be the free module over F [ξ] with basis B˜λ. Let Uλ be the subset of B˜λ consisting of equivalence classes of graphs whose edges are of the form (I, I) for I ∈Vλ. The following theorem gives the multiplication in MPλ(ξ).

Theorem 6.2. For [Γ1], [Γ2] in B˜λ, we deﬁne the following operation:

(40) [Γ ] ∗ [Γ ]= Φ[Γ] (ξ)[Γ], 1 2 [Γ1][Γ2] ˜ [Γ]X∈Bλ where Φ[Γ] (ξ) ∈ F [ξ] is given in Equation (41). The linear extension of the op- [Γ1][Γ2] eration (40) makes MPλ(ξ) an associative unital algebra over F [ξ] with the identity element id = [Γ]. [Γ]∈Uλ

In order toP define Φ[Γ] (ξ), as stated in Theorem 6.2, we need to give the [Γ1][Γ2] following set-up. Given Γ, Γ1, Γ2 ∈Bλ with n edges each, now by merely replacing the vertices by tuples of integers in the case of graphs in Bk, we have analogous def- n initions of the configuration of paths P = {p1,p2,...,pn}, the sets Supp (Γ1, Γ2), n Γ SuppΓ(Γ1, Γ2) and CΓ1Γ2 (n). For an edge of weight (I,J) = ((i1,i2,...,is), (j1, j2,...,js)) in ΓP with P ∈ n SuppΓ(Γ1, Γ2), and L ∈Vλ, define

pIJ (L) := cardinality of the multiset {pt ∈ P | pt = (I,L,J)}. Then p (L) is the multiplicity p of the edge (I,J) of Γ . Suppose L∈Vλ IJ IJ P |Vλ| = a and let v , v ,...,va be an enumeration of the elements of Vλ in the P 0 1 weakly increasing lexicographic order (so in particular v0 = 0). THE MULTISET PARTITION ALGEBRA 25

Let [Γ1], [Γ2], and [Γ] be in B˜λ. Deﬁne [Γ] (41) Φ (ξ)= KP · (ξ − rank(Γ)) Γ1◦Γ2 , [Γ1][Γ2] [P ] P ∈Supp2|λ|(Γ ,Γ ) XΓ 1 2

Γ1◦Γ2 a where [P ]= i=1 p00(vi), P 1 pIJ KP = 00 00 p (v1)! ··· p (va)! 0 0 pIJ (v0),...,pIJ (va) (I,J)∈DY\{( , )} and D denotes the set of all distinct edges of Γ. 6.1. Centralizer algebra and Schur–Weyl duality. In this section, we λ n give a basis of EndSn (Sym (F )), and also state the Schur–Weyl duality between λ n the actions of Sn and MPλ(n) on Sym (F ). Recall the deﬁnition of Symλ(F n) from (18). The choice of indexing set λj n M(n, λj ) for a basis of each Sym (F ) yields the following indexing set M(n, λ) for a basis of Symλ(F n)

a11 a12 ... a1n n . . . M(n, λ) := A =  ......  | aij ∈ Z≥0, aij = λi .  a a ... a j=1   s1 s2 sn X λ n  A  The space Sym (F) has a basis {e | A ∈ M(n, λ)}, where  A a11 a1n a21 a2n as1 asn e := e1 ...en ⊗ e1 ...en ⊗···⊗ e1 ...en .

The symmetric group Sn acts on an element A ∈ M(n, λ) by permuting columns of A. As previously (see Equation (14)), the following isomorphism makes Symλ(F n) a permutation representation of Sn (42) Symλ(F n) =∼ F [M(n, λ)].

As in Section 3 (see Equation (16)), the set of Sn-orbits of M(n, λ) × M(n, λ) is in bijection with the set B˜λ,n consisting of elements of rank at most n in B˜λ. In the following deﬁnition, we give the integral operator corresponding to an Sn-orbit of M(n, λ) × M(n, λ). ˜ Definition 6.3. For each [Γ] ∈ Bλ,n deﬁne T[Γ] ∈ EndSn (F [M(n, λ)]) by

T[Γ]1A = 1B, where A, B ∈ M(n, λ). {B∈M(n,λX)|[ΓA,B ]=[Γ]} ˜ Theorem 6.4. The set {T[Γ] | [Γ] ∈ Bλ,n} is a basis of EndSn (F [M(n, λ)]). s For vector λ = (λ1,...,λs) of non-negative integers, let |λ| = i=1 λi. Theorem 6.5. (1) Deﬁne a map P λ n (43) φ : MPλ(n) → EndSn (Sym (F )) by T if rank[Γ] ≤ n, φ([Γ]) = [Γ] (0 otherwise. Then map φ is a surjective algebra homomorphism with the kernel

ker(φ)= F -span{[Γ] ∈ B˜λ | rank([Γ]) >n}. 26 NARAYANAN, PAUL, AND SRIVASTAVA

∼ λ n In particular, when n ≥ 2|λ|, MPλ(n) = EndSn (Sym (F )). λ n (2) The group Sn generates EndMPλ(n)(Sym (F )).

6.2. Cellularity of MPλ(ξ). In this section, we give embedding of MPλ(ξ) inside P|λ|(ξ). We obtain this embedding by generalizing the bijection in Theo- rem 5.2. Let λ = (λ1, λ2,...,λs) be a vector of non-negative integers such that |λ| = k. Given Γ ∈ B˜λ with n edges, we have a pair (P,Q) of s × n matrices that uniquely determines the graph, where

p11 p12 ... p1n q11 q12 ... q1n p21 p22 ... p2n q21 q22 ... q2n P =  . . .  and Q =  . . .  ......     p p ... p  q q ... q   s1 s2 sn  s1 s2 sn     Each column of P is the label for a vertex in the top rowof Γ and each column of Q is the label for a vertex in the bottom row of Γ. A column of P is connected by an edge in Γ to the corresponding column of Q. The columns of P and Q are simultaneously permuted such that columns of P are in weakly increasing lexicographic order, and T T T T if (p1j ,...,psj ) = (p1j+1,...,psj+1) then (q1j ,...,qsj ) ≤ (q1j+1,...,qsj+1) . We deﬁne a set partition of {1, . . . , k, 1′,...,k′} corresponding to the pair of matrices (P,Q):

(44) dΓ = {B1,...,Bn}, where

Bj = B1j ∪···∪ Bsj and i−1 j−1 i−1 j (45) Bij := λr + pil +1,..., λr + pil , r=1 l r=1 l X X=1 X X=1 i−1 j−1 i−1 j ′ ′ λr + qil +1 ,..., λr + qil . r=1 l=1 r=1 l=1 X X X X The block of dΓ corresponding to edge (0, 0) is empty. It follows from the construction that if two graphs in Bλ are equivalent, then their corresponding partition diagrams deﬁned as above are the same. The diagram dΓ is called the canonical partition diagram associated to [Γ] ∈ B˜λ.

Example 6.6. Let λ = (2, 1). Here

V(2,1) = {(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1)}.

Let Γ be the following bipartite multigraph on the vertex set V(2,1) ⊔V(2,1). (0, 0) (0, 1) (1, 0) (1, 1) (2, 0) (2, 1)

(0, 0) (0, 1) (1, 0) (1, 1) (2, 0) (2, 1) THE MULTISET PARTITION ALGEBRA 27

Then corresponding pair of matrices are follows: 0 0 2 1 1 0 P = and Q = . 0 1 0 0 1 0 And we have: ′ ′ B11 = {1 }, B12 = {2 }, B13 = {1, 2}, ′ B21 = ∅, B22 = {3, 3 }, B23 = ∅. Following Equation (44), the blocks of canonical set partition associated to Γ are: ′ ′ ′ B1 = {1 },B2 = {3, 2 , 3 },B3 = {1, 2}.

This canonical partition diagram dΓ in A3 is depicted as follows: 1 2 3

1′ 2′ 3′

The subgroup Sλ × Sλ of Sk × Sk acts on Ak by the restriction. So analogously to Theorem 5.2 we have:

Theorem 6.7. Let (Sλ × Sλ) \ Ak be the set of orbits of the action of Sλ × Sλ on Ak. The following map is a bijection

(46) ψ : B˜λ → (Sλ × Sλ) \ Ak

ψ([Γ]) = OdΓ , ˜ where [Γ] ∈ Bλ and OdΓ is the Sλ × Sλ-orbit of dΓ. ˜ Corollary 6.8. The set B(1k) is in bijection with Ak. Proof. The results follows from Theorem 6.7, because for λ = (1k), the subgroup Sλ × Sλ is trivial. Utilizing the bijection in Theorem 6.7, we have the following theorem. Theorem 6.9. The following map is an injective algebra homomorphism

η˜λ : MPλ(ξ) →Pk(ξ) 1 (47) η˜λ([Γ]) = xd, αd Γ d∈O XdΓ k where [Γ] ∈ B˜λ. In particular, for λ = (1 ), MPλ(ξ) =∼ Pk(ξ). u l u l Theorem 6.10. Let Yλ be the set of d = {(B1 ,B1),..., (Bn,Bn)} in Ak satisfying the following property, for 1 ≤ i ≤ s and 1 ≤ j ≤ n, i−1 i u Bj ∩ λr +1,..., λr r=1 r=1 X X i−1 ′ i ′ l = Bj ∩ λr +1 ,..., λr . r=1 r=1 X X 1 Let e = x d. Then e is an idempotent in Pk(ξ) and MP λ(ξ) =∼ ePk(ξ)e. d∈Yλ αd P 28 NARAYANAN, PAUL, AND SRIVASTAVA

Theorem 6.11. The algebra MPλ(ξ) over F is cellular. Furthermore, MPλ(ξ) over F is semisimple when ξ is not an integer or when ξ is an integer such that ξ ≥ 2k − 1.

6.3. The balanced multiset partition algebra. In this section, we explore the balanced multiset partition algebra, a subalgebra of MPλ(ξ), motivated by a question in [8]. Our answer to his question is the Schur–Weyl duality between the actions of the balanced multiset partition algebra and the group of monomial matrices on Symλ(F n). We say that a multiset partition is balanced if each block B of the multiset partition satisﬁes:

(48) |B ∩{1λ1 ,...,sλs }| = |B ∩{1′λ1 ,...,s′λs }|.

˜bal ˜ Denote by Bλ the set of all graphs [Γ] in Bλ such that their associated multiset bal partition is balanced. Let MPλ (ξ) denote the submodule of MPλ(ξ) over F [ξ] ˜bal with basis Bλ .

bal Proposition 6.12. The submodule MPλ (ξ) is a subalgebra of MPλ(ξ) with the basis

˜bal ˜ Bλ = {[Γ] ∈ Bλ | |I| = |J| for all edges in Γof weight (I,J)} s where for I = (i1,...,is), |I| = l=1 il. P ˜bal Proof. Take an edge of weight (I,J) of a graph in Bλ . Using Equation (39) j it determines the multiset {1i1 ,...,sis , 1′ 1 ,...,s′js }. Then the balanced condition (48) gives |I| = |J|. bal Now we claim that the algebra structure on MPλ is inherited from the mul- ˜bal ˜ tiplication deﬁned in Equation (40). For [Γ1], [Γ2] ∈ Bλ , suppose [Γ] ∈ Bλ occurs in [Γ1] ∗ [Γ2]. For every edge in [Γ] of weight (I,K), there exists a vertex J such that an edge in [Γ2] of weight (I,J) is concatenated with an edge in [Γ1] of weight ˜bal (J, K). This implies |I| = |J| = |K|, thus [Γ] ∈ Bλ .

˜bal Remark 3. For [Γ1], [Γ2] in Bλ and P ∈ Supp(Γ1, Γ2), note that the polyno- [Γ ] mial Φ P (ξ) deﬁned in Equation (41) is independent of ξ as [P Γ1◦Γ2 ] is 0. So [Γ1][Γ2] bal bal bal MPλ (ξ) is independent of ξ, and henceforth we denote MPλ (ξ) by MPλ .

Let D be the subgroup of the diagonal matrices in GLn(F ). Then D⋊Sn is the λ n subgroup of all the monomial matrices in GLn(F ). So D ⋊ Sn acts on Sym (F ) by the restriction.

Theorem 6.13 (Centralizer of monomial matrices). The set

˜bal {T[Γ] | [Γ] ∈ Bλ , rank([Γ]) ≤ n}

λ n is a basis of EndD⋊Sn (Sym (F )).

˜bal Proof. Let [Γ] ∈ Bλ such that rank([Γ]) ≤ n. Let diag(x1,...,xn) be a diagonal matrix in D and A = (I1,...,In) ∈ M(n, λ), where Ij is the jth column THE MULTISET PARTITION ALGEBRA 29 of A. Then

A |I1| |In| A T[Γ](diag(x1,...,xn)e )= T[Γ](x1 ··· xn e )

|I1| |In| A = x1 ··· xn T[Γ](e )

|I1| |In| B = x1 ··· xn e . [ΓA,BX]=[Γ] On the other hand, A B diag(x1,...,xn)T[Γ](e ) = diag(x1,...,xn) e [ΓA,BX]=[Γ] |J1| |Jn| B = x1 ··· xn e , if B = (J1,...,Jn). [ΓA,BX]=[Γ] ˜bal Since [Γ] ∈ Bλ and ΓA,B = Γ, this forces |I1| = |J1|,..., |In| = |Jn|. Thus λ n T[Γ] ∈ EndD⋊Sn (Sym (F )). λ n Let ψ ∈ EndD⋊Sn (Sym (F )). The symmetric group Sn is a subgroup of D ⋊ Sn, so we have an embedding of algebras λ n λ n (49) EndD⋊Sn (Sym (F )) ⊆ EndSn (Sym (F )). From Theorem 3.8,

(50) ψ = αΓT[Γ]. ˜ [Γ]X∈Bλ ˜bal In order to prove the result it is suﬃcient to show that in Equation (50) if [Γ] ∈/ Bλ then αΓ = 0. ′ ˜bal ′ ′ ′ ′ ′ So let [Γ ] ∈/ Bλ . Then Γ has an edge of weight (I ,J ) such that |I | 6= |J |. ′ ′ ′ Let A and B in M(n, λ) such that ΓA′,B′ =Γ . Note that there exists 1 ≤ s ≤ n such that I′ and J ′ are the sth columns of A′ and B′, respectively. Let X = diag(x1,...,xn) such that xs = x and xi = 1 for i 6= s, then ′ ′ ′ ψ(XeA )= ψ(x|I |eA ) ′ ′ = x|I |ψ(eA ) |I′| A′ = x αΓT[Γ](e ) ˜ [Γ]X∈Bλ |I′| B (51) = x αΓ e .

˜ [Γ ′ ]=[Γ] [Γ]X∈Bλ A X,B On the other hand, A′ B Xψ(e )= X αΓ e

˜ [Γ ′ ]=[Γ] [Γ]X∈Bλ A X,B |Js| B (52) = αΓ x e , where Js is the sth column of B.

˜ [Γ ′ ]=[Γ] [Γ]X∈Bλ A X,B ′ Now equating the coeﬃcient of eB in the Equations (51) and (52), we get |I′| |J′| x αΓ′ = x αΓ′ , for all x ∈ F. ′ ′ Since |I | 6= |J |, we get αΓ′ = 0. 30 NARAYANAN, PAUL, AND SRIVASTAVA

Theorem 6.14. (1) The map φ in Equation (43) restricts to a surjec- bal λ n tive algebra homomorphism from MPλ onto EndD⋊Sn (Sym (F )) with kernel ˜bal ker(φ)= F -span{[Γ] ∈ Bλ | rank([Γ]) >n}. bal ∼ λ n In particular, when n ≥ |λ|, MPλ = EndD⋊Sn (Sym (F )). λ n (2) The group D ⋊ Sn generates End bal (Sym (F )). MPλ Recall from [8, p. 21] a set partition is called balanced if it satisﬁes Equa- k bal tion (48) for λ = (1 ). Let Pk denote the balanced partition algebra, i.e., the subalgebra of Pk(ξ) with basis consisting of balanced set partitions. As a corollary of Theorem 5.5 we get: Corollary 6.15. For any vector λ of non-negative integers, the map (47) bal bal k ,( restricts to an embedding of algebras MPλ ֒→Pk . In particular, when λ = (1 bal ∼ bal MPλ = Pk . Remark 4. When λ = (1k), Theorem 6.14 recovers Schur–Weyl duality ([8, bal ⋊ n ⊗k Proposition 4.1]) between Pk and D Sn acting on (F ) .

7. RSK correspondence for the multiset partition algebra The Robinson–Schensted–Knuth (RSK) [10, Section 3] correspondence is a bijection from two-line arrays of positive integers onto pairs of semistandard Young tableaux of the same shape. This correspondence reﬂects a classical direct sum decomposition [20, Theorem 4.1]

d m n n m Sym (F ⊗ F )= Wλ ⊗ Wλ λ d M⊢ n m as (GLn(F ), GLm(F ))-bimodule. Here Wλ and Wλ are irreducible polynomial representations of GLn(F ) and GLm(F ) respectively. In [5, Section 5], the authors gave a variant of the RSK correspondence by applying Schensted’s insertion algorithm on two-line arrays consisting of multiset partitions instead of positive integers. In this section, we provide a representation theoretical interpretation for their correspondence. For n ≥ 2|λ|, by Theorem 6.5, we have, as Sn × MPλ(n)-bimodule, λ n ∼ λ (53) Sym (F ) = Vν ⊗ Mν ν n M⊢ λ where Mν , if non-zero, denotes the irrreducible representation of MPλ(n) corresponding to the partition ν of n. Observe from Proposition 4.1 that the dimension λ of Mν is given by the number of semistandard multiset tableaux of shape ν and con- λ1 λs tent {1 ,...,s } where λ = (λ1,...,λs). We obtain the following enumerative identity [5, Section 5.3] by counting dimension in both sides of the decomposition (53) s n + λi − 1 = |SYT (ν)| × |SSMT (ν, {1λ1 ,...,sλk })|, λi i=1 ν n Y X⊢ where SYT (ν) is the set of standard Young tableaux of shape ν. THE MULTISET PARTITION ALGEBRA 31

In [5, Theorem 6.3], a bijection between the partition diagrams and pairs of set partition tableaux is exhibited, that gives a combinatorial proof of the decomposition of Pk(n) as Pk(n) ×Pk(n)-module. In the following theorem, we extend their algorithm to give a combinatorial interpretation of the decomposition ∼ λ λ MPλ(n) = Mν ⊗ Mν ν n M⊢ as MPλ(n) × MPλ(n)-module.

Theorem 7.1. The multiset partitions of {1λ1 ,...,sλs , 1′λ1 ,...,s′λs } with at most n parts are in bijection with pairs (T,S) of semistandard multiset tableaux of same shape ν, where ν is a partition of n, and S has content {1λ1 ,...,sλs }, T has content {1′λ1 ,...,s′λs }.

Proof. Let d = {B1,...,Bn}, some Bi’s are allowed to be empty, be a multiset partition of {1λ1 ,...,sλs , 1′λ1 ,...,s′λs }. Given a block B of d, we deﬁne Bu := B ∩{1λ1 ,...,sλs } and Bl := B ∩{1′λ1 ,...,s′λs }. Consider the biword of multisets u u Bi1 ... Bin Bl ... Bl i1 in satisfying u u • Bij ≤L Bij+1 for all 1 ≤ j ≤ n, l l u u • Bij ≤L Bij+1 whenever Bij = Bij+1 . Observe that this biword uniquely determines the given multiset partition. By applying RSK insertion algorithm, we associate a pair of semistandard multiset tableaux (T,S) to the biword

Bu ... Bu RSK : i1 in → (T,S), Bl ... Bl i1 in where T is the insertion tableau and S is the recording tableau of same shape of size n. It is easy to see T , S have the required content. The inverse RSK algorithm completes the proof. Example 7.2. Let λ = (2, 2, 1) and n = 6. Consider the multiset partition {{1, 2}, {1′, 2′}, {1, 3, 1′}, {2, 2′}, {3′}} of {12, 22, 31, 1′2, 2′2, 3′1}. We add an empty block and then arrange all six blocks in the prescribed manner to obtain the following biword: ∅∅ ∅ {2}{1, 2}{1, 3} . ∅ {3′}{1′, 2′}{2′} ∅ {1′} By applying RSK insertion algorithm, we get

∅ ∅ {1′} ∅ ∅ ∅ T = , S = . {2′} {1′, 2′} {2} {1, 3}

{3′} {1, 2} 32 NARAYANAN, PAUL, AND SRIVASTAVA

Remark 5 (Symmetry and Schützenberger’s lemma). Given a block B of a t multiset partition d = {B1,...,Bn}, let B denote the block obtained from B by exchanging i by i′ for all i ∈ B. A block is called fixed block if B = Bt. t t t t Define d = {B1,...,Bn}. We call a multiset partition d symmetric if d = d . If d ↔ (T,S) via the RSK correspondence for multisets, then dt ↔ (S,T ). Moreover, if d is symmetric with l fixed blocks and T be the associated multiset tableau of shape ν, then ν has l columns of odd length.

Acknowledgements The authors thank Amritanshu Prasad for his consistence guidance and fruitful advice. The authors also thank Nate Harman and Mike Zabrocki for their valu- able suggestions on this manuscript. We thank the referee for pointing out a few crucial errors in the manuscript and giving us the opportunity to address them. SS was supported by a national postdoctoral fellowship (PDF/2017/000861) of the Department of Science & Technology, India.

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The Institute of Mathematical Sciences, HBNI, Chennai Email address: [email protected]

Tata Institute of Fundamental Research, Mumbai Email address: [email protected]