Factorization of classical characters twisted by roots of unity: II

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.