Matrices Connected with Brauer’s Centralizer Algebras

ACKNOW LEDGEM ENTS .......................................................................................................... ii
LIST OF FIGURES ....................................................................................................................... iv
CHAPTER
1. Introd uction .......................................................................................................................... 1
1. A Theorem of Schur ..................................................................................................... 1
2. Centralizer algebras for 0 n and Sp 2 n ..................................................................... 3
3. A tower of ideals in A^ ........................................................................................... 6
4. Matrices whose nullspaces encode the semisimple structure of AjX^ ................. 7
5. A combinatorial definition for 16
6. Some results about the matrices Y a / a « ..................................................................... 19
7. The algebra 20
2. Determinants of M x^ and .......................................................................... 25
1. Column permutations of standard matchings ........................................................ 25
2. Product formulas for and 31
3. Eigenvalues of Tk{x) and Tk{yi,..., yn) ................................................................. 32
4. The column span of P .............................................................................................. 34
5. Computation of det M x^ and 41
3. Combinatorial algorithms and the Littlewood-Richardson r u le ..................... 48
1. Robinson-Schensted-Knuth row insertion .............................................................. 50
2. Dual Knuth relations and equivalence ..................................................................... 51
3. Jeu de Taquin for standard tableaux ........................................................................ 53
4. The Littlewood-Richardson r u le .............................................................................. 54
5. A theorem of Dennis White ........................................................................................ 55
4. Jeu de Taquin for standard m atchings ....................................................................... 56
1. Definition of the algorithm ........................................................................................ 56
2. Jeu de Taquin preserves standardness .................................................................... 59
3. Dual Knuth equivalence with Jeu de Taquin for tableaux ................................. 65
4. The normal shape obtained via Jeu de Taquin .................................................... 70
5. Alternate statements of Theorems 2.23 and 2 .2 5 ................................................. 74
5. Remaining P roblem s .......................................................................................................... 75
BIBLIOGRAPHY ............................................................................................................................. 77