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Séminaire Lotharingien de Combinatoire, 86B.38 (2022), 12 pp.

# Arvind Ayyer and Nishu Kumari

# Factorization of Classical Characters Twisted by Roots of Unity: Extended Abstract

**Abstract.**
For a fixed integer *t*≥2, we consider the irreducible characters of representations of the classical groups of types A, B, C and D, namely GL_{tn}, SO_{2tn+1}, Sp_{2tn} and OE_{2tn},
evaluated at elements ω^{k}*x*_{i} for 0≤*k*≤*t*-1 and 1≤*i*≤*n*, where ω is a primitive *t*'th root of unity.
The case of GL_{tn} was considered by D. Prasad (Israel J. Math., 2016).
In this article, we give a uniform approach for all cases.
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 that 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 |*z*|≤*t*-2.

Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.

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