The Multiset Partition Algebra

Understand

We introduce the multiset partition algebra $\mathcal{MP}_k(\xi)$ 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 $\mathcal{MP}_k(n)$ and the symmetric group $S_n$ on $\text{Sym}^k(F^n)$.
• The construction of $\mathcal{MP}_k(\xi)$ generalizes to any vector $\lambda$ of non-negative integers yielding the algebra $\mathcal{MP}_{\lambda}(\xi)$ over $F[\xi]$ so that there is Schur-Weyl duality between the actions of $\mathcal{MP}_{\lambda}(n)$ 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.