Thursday, August 27, 2020

Amritanshu Prasad (IMSc, Chennai): August 27, 2020 at 3:55-5:00 PM (IST)

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.

The regular semester is beginning soon.  In case you have anything to present which you are currently excited about, please do volunteer for a talk in this seminar. Further, we would like to include an agenda for some expository talks, perhaps a series leading to a research problem. Any ideas in these directions are welcome!

Talk announcement

Speaker: Amritanshu Prasad, IMSc., Chennai.

Title: Character polynomials and their moments

When: Thursday, August 27, 3:55 PM - 5:00 PM IST (GMT + 5:30)

Where: Zoom: Please write to sfandnt@gmail.com to get the link to the zoom talk. It is best to write a day in advance.  (The link will be open at 3:30 PM for the organizers to test their systems)

Tea or Coffee: Please bring your own.

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 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.

Gaurav Bhatnagar (Ashoka University), Atul Dixit (IIT, Gandhinagar), Krishnan Rajkumar (JNU)