Further results for classical and universal characters twisted by roots of unity

TL;DR

This paper explores factorizations of classical and universal characters twisted by roots of unity, revealing new factorization properties and value restrictions at roots of unity.

Contribution

It demonstrates that characters of classical groups twisted by roots of unity factor into smaller group characters and establishes value bounds for universal characters at roots of unity.

Findings

Characters twisted by roots of unity factor into smaller group characters.

Certain Schur polynomials factor into linear pieces.

Universal characters take values only in {0, ±1, ±2} at roots of unity.

Abstract

We revisit factorizations of classical characters under various specializations, some old and some new. We first show that all characters of classical families of groups twisted by odd powers of an even primitive root of unity factorize into products of characters of smaller groups. Motivated by conjectures of Wagh and Prasad (Manuscr. Math. 2020), we then observe that certain specializations of Schur polynomials factor into products of two characters of other groups. We next show, via a detour through hook Schur polynomials, that certain Schur polynomials indexed by staircase shapes factorize into linear pieces. Lastly, we consider classical and universal characters specialized at roots of unity. One of our results, in parallel with Schur polynomials, is that universal characters take values only in $\{0, \pm 1, \pm 2\}$ at roots of unity.