Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 17 September 2013

Contents

Introduction
Part I
1.	What is Combinatorial Representation Theory? What is representation theory? Main questions in representation theory Answers should be of the form $\dots$
2.	Answers for $S_n,$ the symmetric group
3.	Answers for $GL(n,ℂ),$ the general linear group
4.	Answers for finite dimensional complex semisimple Lie algebras $𝔤$
Part II
5.	Generalizing the $S_n$ results Definitions Notes and references for answers to the main questions
6.	Generalizations of $GL(n,ℂ)$ results Partial results for further generalizations
Appendix A
A1.	Basic Representation Theory
A2.	Partitions and tableaux
A3.	The flag variety, unipotent varieties, and Springer theory for $GL(n,ℂ)$
A4.	Polynomial and rational representations of $GL(n,ℂ)$
A5.	Schur-Weyl duality and Young symmetrizers
A6.	The Borel-Weil-Bott construction
A7.	Complex semisimple Lie algebras
A8.	Roots, weights and paths
Appendix B
B1.	Coxeter groups, groups generated by reflections, and Weyl groups
B2.	Complex reflection groups
Partial results for $G(r,1,n)$
B3.	Hecke algebras and “Hecke algebras” of Coxeter groups
B4.	“Hecke algebras” of the groups $G(r,p,n)$
B5.	The Iwahori-Hecke algebras $H_k\left(q\right)$ of type $A$
Partial results for $H_k\left(q\right)$
B6.	The Brauer algebras $B_k\left(x\right)$
Partial results for $B_k\left(x\right)$
B7.	The Birman-Murakami-Wenzl algebras $BM{W}_k(r,q)$
Partial results for $BM{W}_k(r,q)$
B8.	The Temperley-Lieb algebras $T{L}_k\left(x\right)$
Partial results for $T{L}_k\left(x\right)$
B9.	Complex semisimple Lie groups

Notes and references

This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.

Key words and phrases. Algebraic combinatorics, representations.

Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..

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