On Tensor Spaces for Rook Monoid Algebras

TL;DR

This paper constructs a specific quasi-idempotent in the rook monoid algebra's annihilator of a tensor space, revealing the structure of this annihilator in relation to the rook monoid algebra.

Contribution

It introduces a novel quasi-idempotent element in the rook monoid algebra's annihilator, clarifying its structure and generating the entire annihilator.

Findings

Identified a quasi-idempotent in the annihilator of $U^{ ensor n}$

Proved the generated ideal equals the entire annihilator

Enhanced understanding of rook monoid algebra representations

Abstract

Let $m,n\in \mathbb{N}$, and $V$ be a $m$-dimensional vector space over a field $F$ of characteristic $0$. Let $U=F\oplus V$ and $R_n$ be the rook monoid. In this paper, we construct a certain quasi-idempotent in the annihilator of $U^{\otimes n}$ in $FR_n$, which comes from some one-dimensional two-sided ideal of rook monoid algebra. We show that the two-sided ideal generated by this element is indeed the whole annihilator of $U^{\otimes n}$ in $FR_n$.