A Morris recursion rule for values of the spin characters of wreath products

Abstract

The classical Murnaghan-Nakayama rule is a recursive formula for computing the values of complex irreducible characters of the symmetric group $S_n$. Alun Morris proved a recursive formula for evaluating the values of irreducible spin characters of ${\tilde{S}}_n$, where ${\tilde{S}}_n$ is one of the Schur covers of $S_n$ defined by ${\tilde{S}}_n:=⟨t_1,t_2,\cdots ,t_{n-1},z|z^2=1,t_i^2=z,(t_i{t}_{i+1}{)}^3=z,t_i{t}_j=z{t}_j{t}_i\text{if}|i-j|>1⟩$. A recursive formula for evaluating the values of complex irreducible characters of the wreath product $G\wr S_n$, where $G$ is a finite group, was proved by J. Stembridge. In this article, we state and prove a recursive formula to compute the values of the irreducible spin characters of the wreath product $G\wr {\tilde{S}}_n$. For the convenience of implementing these recursive formulas, we extend the notion of 0-1 boundary sequence of a Young diagram to shifted Young diagrams.