A Proof of Solomon's Rule

TL;DR

This paper presents a matrix-based proof of Solomon's rule for the descent algebra of the symmetric group, utilizing graph representations derived from subsets of transpositions.

Contribution

It provides a novel matrix and graph-theoretic proof of Solomon's rule, enhancing understanding of the descent algebra's structure.

Findings

Matrix proof of Solomon's rule established

Graph representations from transposition subsets used

New insights into descent algebra structure gained

Abstract

We put forward a proof of Solomon's rule, in terms of matrices, for multiplication in the descent algebra of the symmetric group. Our proof exploits the graphs that we can obtain from all the subsets of the set of transpositions, $\{(i,i+1)\}_{i=1}^{n-1}$.