Ribbon complexes for the 0-Hecke algebra

TL;DR

This paper constructs explicit tableau-level maps for the 0-Hecke algebra, forming complexes that prove Koszulness and connect to noncommutative symmetric functions, advancing understanding of algebraic structures in combinatorics.

Contribution

It introduces explicit tableau maps and complexes for the 0-Hecke algebra, demonstrating their acyclicity and applying them to prove Koszulness and relate to noncommutative symmetric functions.

Findings

Constructed tableau-level maps forming exact sequences.

Proved complexes are acyclic in positive degrees.

Established Koszulness of a graded algebra object.

Abstract

We construct explicit tableau-level maps between indecomposable projective modules for the type A 0-Hecke algebra that assemble into canonical split short exact sequences lifting the basic ribbon product rule in NSym via concatenation and near-concatenation. Iterating these maps yields cochain complexes indexed by generalized ribbons; we prove these complexes are acyclic in positive degrees and that their zeroth cohomology is the projective module indexed by full concatenation. We apply these complexes, together with VandeBogert's ribbon Schur module criterion, to prove Koszulness for a naturally defined internally graded algebra object built from the 0-Hecke tower. Finally, we define skew projective modules whose noncommutative Frobenius characteristics realize skewing by fundamental quasisymmetric functions on NSym.