Date: Monday, February 11, 2013
Location: 3088 East Hall (4:00 PM to 5:00 PM)
Title: Chevalley's Theorems: Representations of Lie Groups and Invariants of Finite Groups
Abstract: Our main goal will be to discuss two of Chevalley's major theorems, bringing together ideas from representation theory, commutative algebra, algebraic geometry, and combinatorics. The first theorem relates the ring of class functions of a Lie group to the ring of functions on a maximal torus invariant under the action of the Weyl group. As one way to try to understand the representations of a Lie group is to understand the space in which characters live, it is then natural to ask (by way of Chevalley's first theorem) what the structure of the Weyl-invariant functions on a maximal torus is. Chevalley's second theorem gives a beautiful answer to this result: it is a polynomial ring!
For commutative algebraists: Let R be the polynomial ring of n indeterminates with complex coefficients and let G be a finite subgroup of GLn(C). The ring of indeterminates R^G is a Cohen-Macaulay ring. But when is R^G a polynomial ring? Equivalently, for algebraic geometers: When is the quotient variety C^n/G = A^m? There is a clean classification to such groups. In this discussion, we will also see the appearance of the regular representation of G, tying our discussion back to representation theory.
Speaker: Charlotte Chan
Institution: UM
Event Organizer:
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