Last update: 17 September 2013
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 $GL(n,\u2102),$ the general linear group |
4. | Answers for finite dimensional complex semisimple Lie algebras $\U0001d524$ |
Part II | |
5. |
Generalizing the ${S}_{n}$ results
Definitions Notes and references for answers to the main questions |
6. |
Generalizations of $GL(n,\u2102)$ 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 $GL(n,\u2102)$ |
A4. | Polynomial and rational representations of $GL(n,\u2102)$ |
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 |
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..