Character Polynomials and their Moments

Abstract: A polynomial in a sequence of variables can be regarded as a class function on every symmetric group when the $i$th variable is interpreted as the number of $i$-cycles. Many nice families of representations of symmetric groups have characters represented by such polynomials.

We introduce two families linear functionals of this space of polynomials -- moments and signed moments. For each $n$, the moment of a polynomial at $n$ gives the average value of the corresponding class function on the $n$th symmetric group, while the signed moment gives the average of its product by the sign character. These linear functionals are easy to compute in terms of binomial bases of the space of polynomials.

We use them to explore some questions in the representation theory of symmetric groups and general linear groups. These explorations lead to interesting expressions involving multipartite partition functions and some peculiar variants.

classical analysis and ODEscombinatoricsnumber theory

Audience: researchers in the topic

Special Functions and Number Theory seminar

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