A super Frobenius formula for the characters of Iwahori–Hecke algebras
In this article, we establish a super Frobenius formula for the characters of Iwahori–Hecke algebras. We define Hall–Littlewood supersymmetric functions in a standard manner to make supersymmetric functions from symmetric functions, and give some properties of supersymmetric functions. Based on Schur–Weyl reciprocity between Iwahori–Hecke algebras and the general quantum super algebras, which was obtained in Mitsuhashi [H. Mitsuhashi, Schur–Weyl reciprocity between the quantum superalgebra and the Iwahori–Hecke algebra, Algeb. Represent. Theor. 9 (2006), pp. 309–322.], we derive that the Hall–Littlewood supersymmetric functions, up to constant, generates the values of the irreducible characters of Iwahori–Hecke algebras at the elements corresponding to cycle permutations. Our formula in this article includes both the ordinary quantum case that was obtained in Ram [A. Ram, A Frobenius formula for the characters of the Hecke algebra, Invent. Math. 106 (1991), pp. 461–488.] and the classical super case.