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 GLtn, SO2tn+1, Sp2tn and OE2tn,
evaluated at elements ωkxi for 0≤k≤t-1 and 1≤i≤n, where ω is a primitive t'th root of unity.
The case of GLtn 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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