Jucys-Murphy elements of partition algebras for the rook monoid

TL;DR

This paper develops a spectral approach to the representation theory of partition algebras related to rook monoids, using Jucys-Murphy elements to analyze their irreducible representations and establish Schur-Weyl duality.

Contribution

It introduces a new inductive and spectral method for studying partition algebras of rook monoids via Jucys-Murphy elements and Gelfand-Tsetlin bases.

Findings

Established a multiplicity free tower of algebras involving $\, ext{I}_k$ and $ ext{I}_{k+1/2}$

Described the actions of Jucys-Murphy elements on irreducible representations

Connected Jucys-Murphy elements of rook monoids with those of partition algebras

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 $\mathbb{C}I_{k+\frac{1}{2}}$.