A representation on the labeled rooted forests

TL;DR

This paper studies the action of the symmetric group on partial functions, develops tools for character analysis, and applies findings to rook theory, revealing new insights into orbit structures and representations.

Contribution

It introduces a new framework for analyzing symmetric group actions on partial functions and connects these to rook theory applications.

Findings

Identifies nilpotent matrices with orbits affording the sign representation

Develops machinery for character formulas and multiplicities

Provides applications to rook theory

Abstract

We consider conjugation action of symmetric group on the semigroup of all partial functions and develop a machinery to investigate character formulas and multiplicities. In particular, we determine nilpotent matrices whose orbit under symmetric group afford the sign representation. Applications to rook theory are offered.