The next talk in the Special Functions and Number Theory Seminar is below. As usual, we have requested the speaker to give something suitable for students and non-experts, so please feel free to give this announcement to interested students and colleagues.
Talk announcement
Speaker: Amritanshu Prasad, IMSc., Chennai.
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 of 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 nth 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.
