#### Room 217

#### A morning of combinatorics

#### Multiple Speakers

*The program will consist of talks by students working in algebraic combinatorics.*

9:30-10:15 - K. Seethalakshmi

Title: Sun Tzu’s Theorem for Partitions: Existence and Uniqueness

Abstract: Let s and t be co-prime numbers. The Sun Tzu map for s and t takes a given partition lambda to the pair (sigma, tau), where sigma is the s-core and tau is the t-core of lambda. We have shown this map is surjective and are currently enumerating the fibers of this map.

This is joint work with Steven Spallone.

10:15-11:00 - Neha Malik

Title: Frobenius-Schur theory for GL(2,q)

Abstract: One of the problems in representation theory is to find all the self-dual irreducible representations of a group G and then check whether they are orthogonal or symplectic. For the purpose of this presentation, we would briefly discuss about the Frobenius- Schur indicator of a character and how it plays a role in checking self-duality, orthogonality, etc. of an irreducible representation of a finite group. Our main focus will be to find the conditions for self-duality of some particular irreducible representations of GL(2,q) and do some calculations to show that such self-dual representations are all orthogonal.

11:30-12:15 - Jyotirmoy Ganguli

Title: Counting Spinorial Representations of Symmetric Groups

Abstract: Representations of symmetric groups $S_n$ can be considered as homomorphisms $\pi$ to $O(d, R)$, where $d$ is the degree of $\pi$. If the determinant of $\pi$ is trivial we

call it achiral. The group $O(V)$ has a non-trivial double cover $\mathrm{Pin}(V)$ with the covering map $\rho:\mathrm{Pin}(V)\to O(V)$. We say $\pi$ is spinorial if there exists a homomorphism $\widehat{\pi}:S_n\to \mathrm{Pin}(V)$ such that $\rho\circ\widehat(\pi)=\pi$. In an ongoing work with Dr. Steven Spallone, we found a criterion for whether $\pi$ is spinorial. One can use that to count the number of spinorial irreducible representations of $S_n$, which are parametrized by the partitions $\lambda$ of $n$. In this talk, I will present the counting for odd-dimensional, achiral, spinorial partitions of $S_n$.

12:15-13:00 - To be announced.

Done