Dimensions of compositions modulo a prime

TL;DR

This paper investigates the dimensions of projective indecomposable modules of the 0-Hecke algebra modulo a prime p, linking combinatorial ribbon numbers to congruence classes and extending results to other Coxeter groups.

Contribution

It introduces a method to count ribbon numbers modulo p and extends the analysis of module dimensions to broader Coxeter group contexts.

Findings

Counted ribbon numbers in each congruence class modulo p.

Extended the congruence analysis to other finite Coxeter groups.

Connected combinatorial ribbon numbers to algebraic module dimensions.

Abstract

The (ordinary) representation theory of the symmetric group is fascinating and has rich connections to combinatorics, including the Frobenius correspondence to the self-dual graded Hopf algebra of symmetric functions. The $0$-Hecke algebra (of type $A$) is a deformation of the group algebra of the symmetric group, and its representation theory has an analogous correspondence to the dual graded Hopf algebras of quasisymmetric functions and noncommutative symmetric functions. Macdonald used the hook length formula for the number of standard Young tableaux of a fixed shape to determine how many irreducible representations of the symmetric group have dimensions indivisible by a prime $p$. In this paper, we study the dimensions of the projective indecomposable modules of the $0$-Hecke algebra modulo $p$; such a module is indexed by a composition and its dimension is given by a ribbon number, i.e., the cardinality of a descent class. Applying a result of Dickson on the congruence of multinomial coefficients, we count how many ribbon numbers belong to each congruence class modulo $p$. We also extend the result to other finite Coxeter groups.