Generating functions for real character degree sums of finite general linear and unitary groups
We compute generating functions for the sum of the real-valued character degrees of the finite general linear and unitary groups, through symmetric function computations. For the finite general linear group, we get a new combinatorial proof that every real-valued character has FrobeniusSchur indicator 1, and we obtain some q-series identities. For the finite unitary group, we expand the generating function in terms of values of HallLittlewood functions, and we obtain combinatorial expressions for the character degree sums of real-valued characters with FrobeniusSchur indicator 1 or ?1.
generating functions real-valued characters finite general linear group frobeniusschur indicator q-series finite unitary group involutions halllittlewood polynomial