Further results for classical and universal characters twisted by 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.

arXiv comment28 pages, dedicated to Vyjyanthi Chari on the occasion of her 65th birthday, improved exposition, final version

References in corpus (6)

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• Bijective proofs of skew Schur polynomial factorizations
• Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order
• Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions
• Factorization of classical characters twisted by roots of unity
• Universal characters twisted by roots of unity