Factorization of classical characters twisted by roots of unity: II

TL;DR

This paper extends previous work on the factorization of specialized irreducible characters of classical groups evaluated at roots of unity, providing new characterizations, factorizations, and a bijection related to $t$-core partitions.

Contribution

It generalizes earlier results to larger groups and new specializations, characterizes when characters vanish, and establishes factorizations into smaller group characters.

Findings

Characterization of partitions with nonzero characters

Factorization of characters into smaller group characters

Existence of infinitely many nonzero $t$-core partitions

Abstract

Fix natural numbers $n \geq 1$, $t \geq 2$ and a primitive $t^{\text{th}}$ root of unity $\omega$. In previous work with A. Ayyer (J. Alg., 2022), we studied the factorization of specialized irreducible characters of $\text{GL}_{tn}$, $\text{SO}_{2tn+1},$ $\text{Sp}_{2tn}$ and $\text{O}_{2tn}$ evaluated at elements to $\omega^j x_i$ for $0 \leq j \leq t-1$ and $1 \leq i \leq n$. In this work, we extend the results to the groups $\text{GL}_{tn+m}$ $(0 \leq m \leq t-1)$, $\text{SO}_{2tn+3}$, $\text{Sp}_{2tn+2}$ and $\text{O}_{2tn+2}$ evaluated at similar specializations: (1) for the $\text{GL}_{tn+m}(\mathbb{C})$ case, we set the first $tn$ elements to $\omega^j x_i$ for $0 \leq j \leq t-1$ and $1 \leq i \leq n$ and the remaining $m$ to $y, \omega y, \dots, \omega^{m-1} y$; (2) for the other three families, the same specializations but with $m=1$. The main results of this paper are a characterization of partitions for which these characters vanish and a factorization of nonzero characters into those of smaller classical groups. Our motivation is the conjectures of Wagh and Prasad (Manuscripta Math., 2020) relating the irreducible representations of $\text{Spin}_{2n+1}$ and $\text{SL}_{2n}$, $\text{SL}_{2n+1}$ and $\text{Sp}_{2n}$ as well as $\text{Spin}_{2n+2}$ and $\text{Sp}_{2n}$. Our proofs use the Weyl character formulas and the beta-sets of $t$-core partitions. Lastly, we give a bijection to prove that there are infinitely many $t$-core partitions for which these characters are nonzero.