Factorization of classical characters twisted by roots of unity

TL;DR

This paper provides a unified method to factorize irreducible characters of classical groups evaluated at roots of unity, characterizing when these values are nonzero and expressing them as products of smaller group characters.

Contribution

It introduces a uniform approach for all classical groups to analyze character evaluations at roots of unity, including new characterizations and product formulas for $z$-asymmetric partitions.

Findings

Character values are nonzero for $z$-asymmetric partitions.

Nonzero character values factor into smaller classical group characters.

Infinitely many $z$-asymmetric $t$-cores exist for $t \\geq z+2$.

Abstract

For a fixed integer $t \geq 2$, we consider the irreducible characters of representations of the classical groups of types A, B, C and D, namely $\text{GL}_{tn}, \text{SO}_{2tn+1}, \text{Sp}_{2tn}$ and $\text{O}_{2tn}$, evaluated at elements $\omega^k x_i$ for $0 \leq k \leq t-1$ and $1 \leq i \leq n$, where $\omega$ is a primitive $t$'th root of unity. The case of $\text{GL}_{tn}$ was considered by D. J. Littlewood (AMS press, 1950) and independently by D. Prasad (Israel J. Math., 2016). In this article, we give a uniform approach for all cases. In this article, we give a uniform approach for all cases. We also look at $\text{GL}_{tn+1}$ where we specialize the elements as before and set the last variable to $1$. In each case, we characterize partitions for which the character value is nonzero in terms of what we call $z$-asymmetric partitions, where $z$ is an integer which depends on the group. Moreover, if the character value is nonzero, we prove that it factorizes into characters of smaller classical groups. The proof uses Cauchy-type determinant formulas for these characters and involves a careful study of the beta sets of partitions. We also give product formulas for general $z$-asymmetric partitions and $z$-asymmetric $t$-cores. Lastly, we show that there are infinitely many $z$-asymmetric $t$-cores for $t \geq z+2$.