$(k+1)$-potent Matrices in triangular matrix Groups and Incidence Algebras of Finite Posets

TL;DR

This paper characterizes all (k+1)-potent matrices within upper triangular matrix groups over fields with certain characteristics and provides formulas for counting such matrices in finite cases and incidence algebras of finite posets.

Contribution

It offers a complete description of (k+1)-potent matrices in triangular groups and derives formulas for counting these matrices in finite fields and incidence algebras.

Findings

Explicit description of (k+1)-potent matrices in upper triangular groups.

Formulas for counting (k+1)-potent matrices over finite fields.

Counting (k+1)-potent elements in incidence algebras of finite posets.

Abstract

Let $\mathbb{K}$ be a field such that $char(\mathbb{K})\nmid k$ and $char(\mathbb{K})\nmid k+1$. We describe all $(k+1)$-potent matrices over the group of upper triangular matrix. In the case that $\mathbb{K}$ is a finite field we show how to compute the number of these elements in triangular matrix groups and use this formula to compute the number of $(k+1)$-potent elements in the Incidence Algebra $\mathcal{I}(X,\mathbb{K})$ where $X$ is a finite poset.