Notes written by Arun Ram, 2005

I am hoping that 2005 will be a productive year. The following are incomplete notes that I have worked on this year. I thank the Centre Bernoulli at the EPFL Lausanne and the Max-Planck-Institut fur Mathematik in Bonn for support and hospitality during this work. This research is also partially supported by the National Science Foundation and a Vilas Associates award at the University of Wisconsin--Madison.

Some of this material is based upon work supported by the National Science Foundation under Grant No. 0353038. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

The following is the start of what will hopefully someday be a book on Combinatorial Representation Theory

Crystals

• The fundamental data and Weyl groups Weylgps5.16.05.pdf, 3 pages.
• Schur functions schurfcns5.16.05.pdf, 5 pages.
• Classical Lie algebras clLiealgs7.22.05.pdf, 11 pages.
• Crystals pathmodel5.16.05.pdf, 8 pages.
• Column strict tableaux cstrtab7.18.05.pdf, 5 pages.
• Abstract crystals abscryst7.13.05.pdf, 15 pages.
• The affine Weyl group affWeyl12.14.05.pdf, 6 pages.
• The affine Hecke algebra affhecke12.14.05.pdf, 7 pages.
• The graded Hecke algebra
• The nil affine Hecke algebra
• The spherical Hecke algebra
• The center of the affine Hecke algebra cntraffhke7.18.05.pdf, 1 page.
• Demazure operators
• Root operators
• Spherical functions
• The rings Z[P] and Z[P]^W zpnzpw7.18.05.pdf, 3 pages.
• Fock space
• Verma crystals
• Quasitriangular and ribbon structure on crystals (R-matrix, T_{w_0}, and projection onto the invariants)
• Multisegments multisegs7.01.05.pdf, 13 pages.

Quantum groups

• Hopf algebras Hpfalgs4.26.05.pdf, 11 pages.
• Quasitriangular Hopf algebras
• Drinfeld-Jimbo quantum groups
• Knizhnik-Zamolodchikov KZ7.25.05.pdf, 5 pages.
• Yangians

Translation

• The categories O and O_int
• Verma modules and Ext
• Contravariant forms and determinants
• Vanishing filtrations
• The Steinberg tensor product theorem

Quantum group representations

• R-matrices and the Casimir
• The integral quantum group
• Canonical and PBW bases
• Shapovalov bases
• finite dimensional semisimple Lie algebras
• affine Lie algebras
• The Virasoro algebra virasoro7.11.05.pdf, 8 pages.

Almost semisimple algebras

• Matrix algebras
• generalized matrix algebras genmatalgs4.23.05.pdf, 3 pages.
• structure of generalized matrix algebras gmastructure4.23.05.pdf, 8 pages.
• convolution algebras almssalgs7.13.05.pdf, 24 pages.
• Quasihereditary algebras
• standardly stratified algebras
• cellular algebras

Reflection groups

• A start refgpdef7.12.05.pdf, 12 pages.
• Examples refgpexs7.13.05.pdf, 12 pages.
• Polynomial invariants
• Affine and graded Hecke algebras
• Tantalizers

Hecke algebras

• The Hecke algebra at q=0 0Hecke7.04.05.pdf, 9 pages.
• Kazhdan-Lusztig bases KLpolys7.13.05.pdf, 7 pages.
• Dihedral Hecke algebras

Diagram algebras

• Affine braid groups of classical type affbdgpGLn7.22.05.pdf, 12 pages.
• The symmetric group and the Brauer algebra Skbrauer7.22.05.pdf, 6 pages.
• The groups G_{H,1,n} and G_{H,H/K,n} GH1k7.22.05.pdf, 11 pages.
• G_{r,p,n} tanallizers tanabealgs7.22.05.pdf, 14 pages.
• Affine Hecke and BMW algebras
• Temperley-Lieb algebras TLmurphys7.22.05.pdf, 7 pages.
• Towers and Casimirs
• Markov traces
• Schur functors

(2) The reflection group G_4, refgps1.21.05.pdf

(3) Dihedral groups G_{r,r,2}, refgps1.21.05.pdf

(4) The path model, pathmodel1.15.05.pdf

(5) Yangians, ygnsBonn2.08.05.pdf

(6) Convolution algebras, convalgs1.24.05.pdf

(7) Basic constructions, basicconst2.16.05.pdf

(8) Temperley-Lieb algebras, TLalg2.13.05.pdf

(9) Homological algebra, homalg2.16.05.pdf .

(10) Kazhdan-Lusztig polynomials KLpolys2.28.05.pdf .