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

Ayyer, A and Kumari, N (2022) Factorization of Classical Characters Twisted by Roots of Unity: Extended Abstract. In: Seminaire Lotharingien de Combinatoire (86).

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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 O2tn, 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.

Item Type:	Journal Article
Publication:	Seminaire Lotharingien de Combinatoire
Publisher:	Universitat Wien, Fakultat fur Mathematik
Additional Information:	The copyright for this article belongs to the Universitat Wien, Fakultat fur Mathematik.
Keywords:	classical groups; factorizations; generating functions; twisted characters; Weyl character formula
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	11 Jul 2023 06:43
Last Modified:	11 Jul 2023 06:43
URI:	https://eprints.iisc.ac.in/id/eprint/82424

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