Class 18.735: Topics in Algebra
Fall 2000

Lectures: MWF 11-12 in 12-122
Instructor: Anne Schilling, 2-279, x3-3214,
Office hours: After class or by appointment.
Text: We will use a variety of texts and original papers. Copies of the most important papers will be provided in class. Some books (such as I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press 1995, 2nd edition; W. Fulton, Young tableaux: with applications to representation theory and geometry, Cambridge University Press 1997) are on reserve in the library.
Presentation: During the course of the semester each student should present one topic in form of a 20-30 minute talk and a handout for the other students. Topics will be suggested in class, but if you already have a topic in mind which fits into the framework of the class, that is fine too.

Content of the lectures:

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

  • Overview of topics to be discussed
  • Ring of symmetric functions
  • Partitions

  • Lecture 2: September 8
  • Monomial symmetric functions
  • Elementrary symmetric functions
  • Complete symmetric functions
  • Powers sums
  • Schur functions

  • Lecture 3: September 11
  • (Skew) semi-standard Young tableaux
  • (Skew) Schur functions - combinatorial definition
  • Kostka numbers
  • Scalar product on the ring of symmetric functions (to be continued)

  • Lecture 4: September 13
  • Scalar product (continued)
  • Properties of Schur functions
  • Classical definition of Schur functions as quotients of determinants

  • Lecture 5: September 15
    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"

    Lecture 6: September 18
  • Proof of the Cauchy and dual Cauchy identity
  • Definition of Macdonald symmetric functions

  • 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.

    Lecture 7: September 20
  • Existence of Macdonald symmetric functions

  • Lecture 8: September 22
    Presentation on the Littlewood-Richardson rule by Tony and Peter
    Reference: W. Fulton, Young tableaux

    Lecture 9: September 25
  • Existence of Macdonald symmetric functions (continued)

  • Lecture 10: September 27
  • Duality of the Macdonald symmetric functions
  • Macdonald symmetric functions at special values for q and t

  • Lecture 11: September 29
  • Skew Macdonald symmetric functions
  • Specialization

  • Lecture 12: October 2
  • Macdonald-Kostka polynomials
  • Kostka polynomials

  • Lecture 13: October 4
    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

    Lecture 14: October 11
  • Cyclage
  • Cocharge, charge

  • Lecture 15: October 13
  • Embeddings of cyclage graphs

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

    Lecture 16: October 16
  • Sketch of the proof of the Theorem by Lascoux and Schuetzenberger that Kostka polynomials are given by the generating function of semi-standard tableaux with charge statistics
  • Morris recurrence relations

  • 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.

    Lecture 17: October 18
  • Axiomatic definition of crystal graphs
  • Examples
  • Quantized universal enveloping algebra

  • Lecture 18: October 20
    Hitchhiker's guide to Lie algebras

    Lecture 19: October 23
  • Finite-dimensional representations of U_q(sl_2)
  • Integrable modules
  • Local bases
  • Crystal bases

  • Lecture 20: October 25
  • From crystal bases to crystal graphs
  • Existence and uniqueness of crystals (statement only)
  • Tensor products

  • Lecture 21: October 27
    Ezra Miller will talk about the n! conjecture/theorem

    Lecture 22: October 30
    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.

    Lecture 23: November 1
  • Littlewood-Richardson rule in terms of crystals for type A and C

  • 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.

    Lecture 24: November 3
  • Unrestricted paths
  • Classically restricted paths
  • Littlewood-Richardson (LR) tableaux

  • Lecture 25: November 6
  • Cyclage for LR tableaux

  • References:
    M. Shimozono, preprint math.QA/9804037
    A. Schilling and S.O. Warnaar, Commun. Math. Phys. 202 (1999) 359-401.

    Lecture 26: November 8
    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

    Lecture 26: November 13
  • Isomorphisms of crystals
  • Explicit expression for cocharge of LR tableaux/energy function on crystals for type A

  • References:
    M. Shimozono, preprint math.QA/9804039
    A. Schilling and S.O. Warnaar, Commun. Math. Phys. 202 (1999) 359-401.

    Lecture 27: November 15
  • Outline of the proof coenergy on paths = cocharge on LR tableaux

  • Lecture 28: November 17
  • Affine crystals for type A

  • Lecture 29: November 20
    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

    Lecture 30: November 22
  • Energy function in terms of e_0
  • More on affine crystals of type A

  • November 24: no class; Thanksgiving Holiday

    Lecture 31: November 27
  • embedding of C_n to A_{2n-1} crystals for general \Lambda
  • affine C_n^{(1)} crystals B(\Lambda_r) (to be continued)

  • Lecture 32: November 29
  • affine C_n^{(1)} crystals (continued)

  • Lecture 33: December 1
  • rigged configurations
  • fermionic formula for Kostka polynomials (to be continued)

  • Lecture 34: December 4
  • bijection between semistandard tableaux and rigged configurations
  • fermionic formula for Kostka polynomials

  • Handout

    Lecture 35: December 6
    Presentation on the path model for representations by Kevin
    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.

    Lecture 36: December 8
  • fermionic formulas associated with simple Lie algebras

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

    Lecture 37: December 11
    Presentation on ribbon tableaux by Frederic
    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.

    Lecture 38: December 13
  • explicit formula for generating functions of unrestricted paths in type A and single columns
  • discussion of open problems