Fall 2000

This class will cover topics in algebraic combinatorics and representation theory such as symmetric functions, tableaux combinatorics, crystal base theory, Littelmann paths, q-analogues of tensor product multiplicities, Demazure characters and fermionic formulas.

You can find a short review of each lecture and relevant material below.

(This information will be updated in the course of the semester).

**Lecture 1:** September 6

Presentation on the Robinson-Schensted-Knuth (RSK) correspondence by Carly and Sergi

Reference: A. Lascoux, B. Leclerc, J.-Y. Thibon, "The plactic monoid" in "Combinatorics on words"

References:

I. G. Macdonald, A new class of symmetric functions, Séminaire Lotharingien de Combinatoire, B20a (1988), 41pp.

I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Science Publ., 2nd edition, 1995.

I. G. Macdonald, Symmetric functions and orthogonal polynomials, University Lecture Series 12, AMS, Providence, 1997.

Presentation on the Littlewood-Richardson rule by Tony and Peter

Reference: W. Fulton, Young tableaux

Presentation on coplactic operations by Thao

Reference: A. Lascoux, B. Leclerc, J.-Y. Thibon, "The plactic monoid" in "Combinatorics on words", ch. 6.5

October 6: no class (make-up was September 25)

October 9: no class; Columbus day

Presentation on special entries of the Macdonald-Kostka matrix by Edward

Handout

Reference: J. R. Stembridge, "Some particular entries of the two-parameter Kostka matrix", Proc. Amer. Math. Soc. 121 (1994) 367-373.

References:

1. A. Lascoux and M. P. Schuetzenberger, Sur une conjecture de H. O. Foulkes, C.R. Acad. Sci. Paris 286A (1978) 323-324.

2. L. M. Butler, Subgroup lattices and symmetric functions, Memoirs Amer. Math. Soc., vol. 112, number 539, 1994.

Hitchhiker's guide to Lie algebras

Ezra Miller will talk about the n! conjecture/theorem

Presentation on crystal graphs for U_q(A_n)-modules and U_q(C_n)-modules by Etienne and Tilman

Reference: M. Kashiwara and T. Nakashima, Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Alg. 165 (1994) 295-345.

Reference: T. Nakashima, "Crystal base and a generalization of the Littlewood-Richardson rule for the classical Lie algebras", Commun. Math. Phys. 154 (1993) 215-243.

References:

M. Shimozono, preprint math.QA/9804037

A. Schilling and S.O. Warnaar, Commun. Math. Phys. 202 (1999) 359-401.

Presentation on an expression of the Kostka polynomials in terms of crystals by Brett

Reference: A. Lascoux, B. Leclerc, J.-Y. Thibon, Crystal graphs and q-analogues of weight multiplicities for the root system A_n, Lett. Math. Phys. 35 (1995) 359-374.

November 10: no class; Veterans day

References:

M. Shimozono, preprint math.QA/9804039

A. Schilling and S.O. Warnaar, Commun. Math. Phys. 202 (1999) 359-401.

Presentation on an embedding of type C crystals into type A crystals by Hugh

Reference: T. H. Baker, Zero actions and energy functions for perfect crystals, preprint

November 24: no class; Thanksgiving Holiday

Handout

Presentation on the path model for representations by Kevin

References:

P. Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math. 116 (1994) 329-346.

P. Littelmann, Paths and root operators in representation theory, Ann. Math. 142 (1995) 499-525.

Reference: G. Hatayama, A. Kuniba, M. Okado, T. Takagi, Y. Yamada, Remarks on fermionic formula, math.QA/9812022

Presentation on ribbon tableaux by Frederic

Reference:

A. Lascoux, B. Leclerc, J.-Y. Thibon, Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties, J. Math. Phys. 38 (1997) 1041-1068.

anne@math.mit.edu