## Combinatorial Representation Theory

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 $GL\left(n,ℂ\right),$ 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 $GL\left(n,ℂ\right)$ 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 $GL\left(n,ℂ\right)$ A4. Polynomial and rational representations of $GL\left(n,ℂ\right)$ 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 $G\left(r,1,n\right)$ B3. Hecke algebras and “Hecke algebras” of Coxeter groups B4. “Hecke algebras” of the groups $G\left(r,p,n\right)$ 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 $BM{W}_{k}\left(r,q\right)$ Partial results for $BM{W}_{k}\left(r,q\right)$ 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..