A supercharacter table decomposition via power-sum symmetric functions

TL;DR

This paper introduces a novel factorization of the supercharacter table for unipotent upper triangular matrices over finite fields, using a $q$-deformed power-sum basis of symmetric functions in noncommutative variables.

Contribution

It presents a new $q$-deformation of a power-sum basis and a corresponding $AB$-factorization of the supercharacter table, linking symmetric functions and supercharacter theory.

Findings

Derived an $AB$-factorization of the supercharacter table.

Introduced a $q$-deformed power-sum basis for noncommutative symmetric functions.

Computed the determinant of the supercharacter table.

Abstract

We give an $AB$-factorization of the supercharacter table of the group of $n\times n$ unipotent upper triangular matrices over $\FF_q$, where $A$ is a lower-triangular matrix with entries in $\ZZ[q]$ and $B$ is a unipotent upper-triangular matrix with entries in $\ZZ[q^{-1}]$. To this end we introduce a $q$ deformation of a new power-sum basis of the Hopf algebra of symmetric functions in noncommutative variables. The factorization is obtain from the transition matrices between the supercharacter basis, the $q$-power-sum basis and the superclass basis. This is similar to the decomposition of the character table of the symmetric group $S_n$ given by the transition matrices between Schur functions, monomials and power-sums. We deduce some combinatorial results associated to this decomposition. In particular we compute the determinant of the supercharacter table.