A brief introduction to the representation theory of $SL_n$, or why you should care about doughnuts, Abigail Price (UIUC) -- October 31st, 2024

$\mathbb{R}^n$ and $\mathbb{C}^n$ can be regarded as both algebraic and geometric entities; as vector spaces they carry natural addition and scalar multiplication structures, and as as geometric objects they carry natural notions of distance, angle, volume, and orientation. $GL_n(k)$ is by definition composed of maps that preserve the vector space structure of $k^n$; if we ask for maps that also preserve geometric structure, we obtain the classical Lie groups. In this talk, we will discuss representations, or structure-preserving maps, of the Lie group $SL_n$, both in themselves and as they appear in other areas of mathematics (in particular, as they appear in Fourier analysis). We will then move towards a combinatorial classification of the representations of $SL_n$, focusing on how these representations can be constructed by studying $SL_n$-invariant properties of configurations of vectors in $n$-space.

The seminar co-organizers are Ian Cavey, Andrew Hardt, David Keating, and Alexander Yong.