Vyjayanthi Chari (University of California, Riverside)

Department Colloquium

Speaker: Vyjayanthi Chari (University of California, Riverside)

Title: Snakes, Ladders, and Verma Modules: hidden connections in representation theory

Abstract: Representation theory studies abstract algebraic objects by letting them act on vector spaces. Starting from its origins in the work of Frobenius and Burnside, we will introduce some of its central themes: the classification of irreducible representations, characters, and what happens when representations fail to decompose into irreducible pieces.

We will then explore four settings which developed largely independently over the last fifty years: the category $O$ of the Lie algebra $\mathfrak{gl}_r$, affine Hecke algebras, the p-adic groups $\operatorname{GL}_{N}(F)$, and quantum affine algebras. Functors due to Arakawa–Suzuki, Heiermann and Chari–Pressley connect them, and the same phenomena appear in each under different names: the “snake modules” of quantum affine algebras are the “ladder representations” of p-adic groups, and both are governed by the Verma modules of category $O$.

Motivated by the theory of cluster algebras of Hernandez and Leclerc, we will discuss the problem of finding prime and real representations, and introduce a new family, the alternating snake modules. Their characters are given by an explicit determinant, which yields new information in all four settings. This is based on joint work with Matheus Brito.