Dimensions of Compositions Modulo a Prime

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 (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 $p$ and extend the result to other finite Coxeter groups.