Combinatorics of Centers of 0-Hecke Algebras in Type A

TL;DR

This paper provides a combinatorial description of the basis of the center of the 0-Hecke algebra for the symmetric group, linking it to maximal compositions and hooks, thereby clarifying its structure and enumeration.

Contribution

It explicitly characterizes the equivalence classes of the center basis in type A 0-Hecke algebras using maximal compositions and hooks, building on prior work.

Findings

Maximal compositions of n index the basis of the center.

Explicit combinatorial description for classes parametrized by hooks.

Complete set of representatives for the equivalence classes obtained.

Abstract

A basis of the center of the 0-Hecke algebra of an arbitrary finite Coxeter group was described by He in 2015. This basis is indexed by certain equivalence classes of the Coxeter group whose explicit description is rather complicated. Even their number is not obvious. We consider case of the symmetric group $\mathfrak{S}_n$. Building on work of Geck, Kim and Pfeiffer, we obtain a complete set of representatives of the equivalence classes. This set is naturally parametrized by certain compositions of $n$ called maximal. It follows that the maximal compositions of $n$ index the basis of the center of the 0-Hecke algebra of $\mathfrak{S}_n$. We then develop an explicit combinatorial description for the equivalence classes that are parametrized by the maximal compositions whose odd parts form a hook.