The Multiset Partition Algebra

Sridhar Narayanan, Digjoy Paul, Shraddha Srivastava

Published 2019-03-26Version 1

For each partition $\lambda$, we introduce the multiset partition algebra $\mathcal{MP}_\lambda(\xi)$ over $F[\xi]$, where $F$ is a field of characteristic $0$. Upon specializing $\xi$ to a positive integer $n$, the resulting algebra $\mathcal{MP}_\lambda(n)$ is in Schur-Weyl duality with the action of the symmetric group $S_n$ on $\text{Sym}^\lambda(F^n)$. The generating function of the dimensions of irreducible representations of $\mathcal{MP}_{\lambda}(n)$ is given in terms of Schur functions. When $\lambda$ is a partition of $k$, $\mathcal{MP}_\lambda(\xi)$ is isomorphic to a subalgebra of the partition algebra $\mathcal{P}_k(\xi)$ of Jones and Martin. When $\lambda=(1^k)$, $\mathcal{MP}_\lambda(\xi)$ is isomorphic to $\mathcal{P}_k(\xi)$. We identify a subalgebra of $\mathcal{MP}_{\lambda}(\xi)$ called the balanced multiset partition algebra whose structure constants do not depend on $\xi$. This algebra is in Schur-Weyl duality with the group of monomial matrices acting on $\text{Sym}^\lambda(F^n)$.