A Combinatorial Formula for Orthogonal Idempotents in the $0$-Hecke Algebra of the Symmetric Group

TL;DR

This paper provides combinatorial formulas for decomposing the identity into orthogonal idempotents in the 0-Hecke algebra of the symmetric group, extending previous work and maintaining compatibility with group branching.

Contribution

It introduces explicit combinatorial formulas for maximal decompositions of the identity into orthogonal idempotents in the 0-Hecke algebra, building on Norton's work.

Findings

Formulas for two maximal decompositions of the identity

Compatibility with branching from S_{N-1} to S_N

Extension of Norton's previous results

Abstract

Building on the work of P.N. Norton, we give combinatorial formulae for two maximal decompositions of the identity into orthogonal idempotents in the $0$-Hecke algebra of the symmetric group, $\mathbb{C}H_0(S_N)$. This construction is compatible with the branching from $S_{N-1}$ to $S_{N}$.