MATH 711: REPRESENTATION THEORY OF SYMMETRIC GROUPS LECTURES BY PROF. ANDREW SNOWDEN; NOTES BY ALEKSANDER HORAWA These are notes from Math 711: Representation Theory of Symmetric Groups taught by Profes- sor Andrew Snowden in Fall 2017, LATEX'ed by Aleksander Horawa (who is the only person responsible for any mistakes that may be found in them). This version is from December 18, 2017. Check for the latest version of these notes at http://www-personal.umich.edu/~ahorawa/index.html If you find any typos or mistakes, please let me know at
[email protected]. The first part of the course will be devoted to the representation theory of symmetric groups and the main reference for this part of the course is [Jam78]. Another standard reference is [FH91], although it focuses on the characteristic zero theory. There is not general reference for the later part of the course, but specific citations have been provided where possible. Contents 1. Preliminaries2 2. Representations of symmetric groups9 3. Young's Rule, Littlewood{Richardson Rule, Pieri Rule 28 4. Representations of GLn(C) 42 5. Representation theory in positive characteristic 54 6. Deligne interpolation categories and recent work 75 References 105 Introduction. Why study the representation theory of symmetric groups? (1) We have a natural isomorphism V ⊗ W ∼ W ⊗ V v ⊗ w w ⊗ v We then get a representation of S2 on V ⊗V for any vector space V given by τ(v⊗w) = ⊗n w ⊗ v. More generally, we get a representation of Sn on V = V ⊗ · · · ⊗ V . If V is | {z } n times Date: December 18, 2017.