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Ribbon complexes for the 0-Hecke algebra

Ayah Almousa, Bryan Lu

TL;DR

This work develops a homological framework for the type $A$ $0$-Hecke algebra by lifting NSym ribbon identities to explicit tableau-level projective maps. It constructs split short exact sequences encoding concatenation and near-concatenation, assembles them into acyclic ribbon complexes, and proves $H^0$ recovers the expected projective module, thereby categorifying the ribbon product identity. Via a ribbon-Schur module criterion, it establishes Koszulness of an internally graded algebra built from the $0$-Hecke tower and defines skew projectives with noncommutative Frobenius characteristics $L_\beta^\perp R_\alpha$, including explicit branching for $\beta=(1)$ and row/column skewing. The results connect the homological representation theory of $H_n(0)$ with NSym and QSym, yielding concrete branching rules and skewing operations that reflect noncommutative symmetric-function identities in a tableau-theoretic setting.

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.

Ribbon complexes for the 0-Hecke algebra

TL;DR

This work develops a homological framework for the type -Hecke algebra by lifting NSym ribbon identities to explicit tableau-level projective maps. It constructs split short exact sequences encoding concatenation and near-concatenation, assembles them into acyclic ribbon complexes, and proves recovers the expected projective module, thereby categorifying the ribbon product identity. Via a ribbon-Schur module criterion, it establishes Koszulness of an internally graded algebra built from the -Hecke tower and defines skew projectives with noncommutative Frobenius characteristics , including explicit branching for and row/column skewing. The results connect the homological representation theory of with NSym and QSym, yielding concrete branching rules and skewing operations that reflect noncommutative symmetric-function identities in a tableau-theoretic setting.

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.
Paper Structure (25 sections, 13 theorems, 119 equations)

This paper contains 25 sections, 13 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Acyclic complexes lifting ribbon identities
  3. Koszulness
  4. Skew projective modules and branching
  5. Organization
  6. Combinatorics of compositions and ribbons
  7. Compositions and weak compositions
  8. Ribbons, generalized ribbons, and descents
  9. Standard ribbon tableaux and reading words
  10. Operations on compositions
  11. Symmetric, noncommutative symmetric, and quasisymmetric functions
  12. Symmetric functions
  13. Noncommutative symmetric functions
  14. Quasisymmetric functions
  15. Pairing between NSym and QSym
  16. ...and 10 more sections

Key Result

Theorem 4.4

For $\alpha \vDash n$, the corresponding $\mathcal{H}_n(0)$-module $\mathbf{P}_\alpha$ with the action described above is isomorphic to the projective indecomposable representation of $\mathcal{H}_n(0)$ corresponding to $\alpha$ defined by Norton Norton-Hecke.

Theorems & Definitions (47)

  • Example 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4: Shuffle product
  • Example 2.5
  • Example 3.1
  • Definition 4.1
  • Definition 4.2: heckeTab
  • Example 4.3
  • Theorem 4.4: heckeTab
  • ...and 37 more