Bruhat-Chevalley order on the rook monoid

TL;DR

This paper provides a combinatorial description of the Bruhat-Chevalley order on the rook monoid and introduces a formula for the length function, extending concepts from the symmetric group.

Contribution

It offers a new combinatorial framework for understanding the Bruhat-Chevalley order on the rook monoid and derives a length function formula.

Findings

Efficient combinatorial description of Bruhat-Chevalley order on R_n

New combinatorial formula for the length function on R_n

Extension of Bruhat order concepts from symmetric group to rook monoid

Abstract

The rook monoid $R_n$ is the finite monoid whose elements are the 0-1 matrices with at most one nonzero entry in each row and column. The group of invertible elements of $R_n$ is isomorphic to the symmetric group $S_n$. The natural extension to $R_n$ of the Bruhat-Chevalley ordering on the symmetric group is defined in \cite{Renner86}. In this paper, we find an efficient, combinatorial description of the Bruhat-Chevalley ordering on $R_n$. We also give a useful, combinatorial formula for the length function on $R_n$.