Jucys-Murphy elements of partition algebras for the rook monoid

Abstract

Kudryavtseva and Mazorchuk exhibited Schur-Weyl duality between the rook monoid algebra $\mathbb{C}{R}_n$ and the subalgebra $\mathbb{C}{I}_k$ of the partition algebra $\mathbb{C}{A}_k(n)$ acting on $({\mathbb{C}}^n{)}^{\otimes k}$. In this paper, we consider a subalgebra $\mathbb{C}{I}_{k+\frac{1}{2}}$ of $\mathbb{C}{I}_{k+1}$ such that there is Schur-Weyl duality between the actions of $\mathbb{C}{R}_{n-1}$ and $\mathbb{C}{I}_{k+\frac{1}{2}}$ on $({\mathbb{C}}^n{)}^{\otimes k}$. This paper studies the representation theory of partition algebras $\mathbb{C}{I}_k$ and $\mathbb{C}{I}_{k+\frac{1}{2}}$ for rook monoids inductively by considering the multiplicity free tower $$\mathbb{C}{I}_1\subset \mathbb{C}{I}_{\frac{3}{2}}\subset \mathbb{C}{I}_2\subset \cdots \subset \mathbb{C}{I}_k\subset \mathbb{C}{I}_{k+\frac{1}{2}}\subset \cdots .$$ Furthermore, this inductive approach is established as a spectral approach by describing the Jucys-Murphy elements and their actions on the canonical Gelfand-Tsetlin bases, determined by the aforementioned multiplicity free tower, of irreducible representations of $\mathbb{C}{I}_k$ and $\mathbb{C}{I}_{k+\frac{1}{2}}$. Also, we describe the Jucys-Murphy elements of $\mathbb{C}{R}_n$ which play a central role in the demonstration of the actions of Jucys-Murphy elements of $\mathbb{C}{I}_k$ and $\mathbb{C}{I}_{k+\frac{1}{2}}$.

Revised version