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Algebra-Geometry-Combinatorics Seminar

Event Type
Seminar/Symposium
Sponsor
Algebra-Geometry-Combinatorics Seminar
Location
Altgeld Hall 143
Date
Oct 31, 2024   3:00 - 3:50 pm  
Views
25
Originating Calendar
Algebra, Geometry, and Combinatorics Seminar

Speaker: Abigail Price (UIUC)

Title: A brief introduction to the representation theory of $SL_n$, or why you should care about doughnuts

Abstract: $\mathbb{R}^n$ 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.

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