The Multiset Partition Algebra

Abstract

We introduce the multiset partition algebra ${\mathcal{M}\mathcal{P}}_k(\xi )$ over $F[\xi ]$, where $F$ is a field of characteristic $0$ and $k$ is a positive integer. When $\xi$ is specialized to a positive integer $n$, we establish the Schur-Weyl duality between the actions of resulting algebra ${\mathcal{M}\mathcal{P}}_k(n)$ and the symmetric group $S_n$ on ${\text{Sym}}^k(F^n)$. The construction of ${\mathcal{M}\mathcal{P}}_k(\xi )$ generalizes to any vector $\lambda$ of non-negative integers yielding the algebra ${\mathcal{M}\mathcal{P}}_{\lambda }(\xi )$ over $F[\xi ]$ so that there is Schur-Weyl duality between the actions of ${\mathcal{M}\mathcal{P}}_{\lambda }(n)$ and $S_n$ on ${\text{Sym}}^{\lambda }(F^n)$. We find the generating function for the multiplicity of each irreducible representation of $S_n$ in ${\text{Sym}}^{\lambda }(F^n)$, as $\lambda$ varies, in terms of a plethysm of Schur functions. As consequences we obtain an indexing set for the irreducible representations of ${\mathcal{M}\mathcal{P}}_k(n)$, and the generating function for the multiplicity of an irreducible polynomial representation of $G{L}_n(F)$ when restricted to $S_n$. We show that ${\mathcal{M}\mathcal{P}}_{\lambda }(\xi )$ embeds inside the partition algebra ${\mathcal{P}}_{|\lambda |}(\xi )$. Using this embedding, over $F$, we prove that ${\mathcal{M}\mathcal{P}}_{\lambda }(\xi )$ is a cellular algebra, and ${\mathcal{M}\mathcal{P}}_{\lambda }(\xi )$ is semisimple when $\xi$ is not an integer or $\xi$ is an integer such that $\xi \ge 2|\lambda |-1$. We give an insertion algorithm based on Robinson-Schensted-Knuth correspondence realizing the decomposition of ${\mathcal{M}\mathcal{P}}_{\lambda }(n)$ as ${\mathcal{M}\mathcal{P}}_{\lambda }(n)\times {\mathcal{M}\mathcal{P}}_{\lambda }(n)$-module.

arXiv commentMany changes, final version. To appear in Israel J. Math

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• An insertion algorithm on multiset partitions with applications to diagram algebras