Jucys-Murphy elements of partition algebra for the rook monoid

Ashish Mishra, Shraddha Srivastava

Published 2019-12-23Version 1

Kudryavtseva and Mazorchuk exhibited Schur-Weyl duality between the rook monoid algebra $\mathbb{C}R_n$ and a subalgebra, denoted by $\mathbb{C}I_k$, 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 $(\mathbb{C}^n)^{\otimes k}$. We call $\mathbb{C}I_k$ and $\mathbb{C}I_{k+\frac{1}{2}}$ totally propagating partition algebras. This paper studies the representation theory of $\mathbb{C}I_k$ and $\mathbb{C}I_{k+\frac{1}{2}}$ inductively by considering the tower ($\mathbb{C}I_1\subset \mathbb{C}I_{\frac{3}{2}}\subset \mathbb{C}I_{2}\subset \cdots$) whose Bratteli diagram turns out to be a simple graph. 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 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 $\mathbb{C}I_{k+\frac{1}{2}}$. In addition, we compute the Kronecker product of $\mathbb{C}^n$ and an irreducible representation of $\mathbb{C}R_n$ which is further used to specify the decomposition of $(\mathbb{C}^n)^{\otimes k}$ inductively.