Organizer: Mark Haiman
Office: 771 Evans
E-mail:
Phone: 642-4318
Office Hours: Wednesday 11:00-12:30

Time and place: Wednesdays 4:00-6:00, 939 Evans, jointly with Reshetikhin's "Lie algebras, combinatorics and geometry" seminar

Course Control Number: 55122

Purpose: This is a graduate research seminar in which students will read and present papers from the literature. The theme is q-analogs in the theory of symmetric functions and the associated combinatorics, representation theory and geometry.
Sample papers suitable for presentation in the seminar are listed in the bibliography below.

Some suggested topics for this semester: (1) Littelmann's path model and his recent work with Gaussent relating it to MV-cycles (2) Kashiwara: "Realization of crystals," and "Level zero fundamental reps...", (3) The k-Schur functions and their q-analogs of Lascoux, Lapointe and Morse, (4) Combinatorics of diagonal harmonics - see my paper with Haglund, Loehr, Remmel and Ulyanov.

Talks scheduled for this semester:

• Jan. 28 - Organizational meeting
• Feb. 4 - (1 and 2) Viswanath Sankaran: Littelmann paths and tensor product stabilization.
• Feb. 11 - (1 and 2) Tony Chiang: Littelmann path crystals.
• Feb. 18 - (1) Joel Kamnitzer: Quantum groups and finite fields. (2) Nick Proudfoot: Quiver varieties.
• Feb. 25 - (1) Sami Assaf and (2) Li-Chung Chen: k-Atoms, k-Schur functions and a refined Macdonald positivity conjecture.
• Mar. 3 - (1 and 2) Ben Webster: Quiver varieties and representation theory.
• Mar. 10 - (1 and 2) Mark Haiman: Diagonal coinvariants and the (n+1)^n-1 theorem.
• Mar. 17 - (1 and 2) Noam Shomron (Lyon/UCB): Geometric patterns in quantum and super representation theory.
• Mar. 31 - (1 and 2) Mikhail Khovanov (UC Davis): Matrix factorizations and link homology.
• Apr. 7 - (1 and 2) Andrei Okounkov (Princeton): Crystals and curves in 3-folds.
• Apr. 14 - (1 and 2) Joel Kamnitzer: Crystals, bubbled categories, and the moduli space of points on RP¹.
• Apr. 21 - (1 and 2) Iain Gordon (Glasgow): The Hilbert scheme of points on the plane and Cherednik algebras.
• Apr. 28 - (1 and 2) Joel Kamnitzer: Cohomology of Hilbert schemes and super Fock space.
• May 5 - (1 and 2) Vera Serganova: On BGG resolution.

Summary of talks from previous semesters:

Fall 2002

• Sami Assaf: Garsia-Procesi, II-III (continued from previous summer).
• Tony Chiang: Zabrocki's survey paper, II-III (continued from summer).
• Brian Rothbach: Shimozono-Weyman, I-IV
• Michelle Bylund: Lascoux-Leclerc-Thibon I-II
• Mark Haiman: Overview of the Springer correspondence for GL_n and its connection with Hall-Littlewood polynomials
• Li-Chung Chen: Kerov-Kirillov-Reshetikhin, I-II

• Alex Woo: (q,t)-Catalan numbers and rc-graphs
• Li-Chung Chen: Kirillov-Reshetikhin (III) and Kirillov-Schilling-Shimozono
• Michelle Bylund: Lascoux-Leclerc-Thibon III-V
• Ivan Cherednik (guest speaker): Double Affine Hecke Algebras and skew diagrams
• Tony Chiang: Shimozono-Zabrocki I-II
• Joel Kamnitzer: Crystal graphs and crystal bases I-III
• Mark Haiman: L-L-T and the character of diagonal harmonics

• Sami Assaf: Symmetry of LLT polynomials
• Peter Tingley: Intro to representations of affine algebras
• Peter Tingley: Crystal bases for quantum affine algebras I-III
• Brian Rothbach: The generalized q-Kirillov-Reshetikhin conjecture of HKOTTY, I-III
• Tony Chiang: Rigged configurations and affine crystals, I - II
• Sami Assaf: Affine Type A Crystal Structure on Tensor Products of Rectangles, Demazure characters, and Nilpotent Varieties, I-II
• Tony Chiang: Rigged configurations and affine crystals, III-IV
• Brian Rothbach: Virtual crystal's and Kleber's algorithm

Bibliography: Some general references on symmetric function theory and Hall-Littlewood and Macdonald polynomials are the textbooks

• Macdonald, Symmetric Functions and Hall-Littlewood polynomials, 2nd edition
• Stanley, Enumerative Combinatorics, volume II

• Combinatorics, symmetric functions and Hilbert schemes (to appear in Current Developments in Mathematics 2002). Links to the file in Postscript, PDF.

• N. Bergeron and A. M. Garsia, On certain spaces of harmonic polynomials. In Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991), Contemp. Math., 138, pp. 51--86, Amer. Math. Soc., Providence, RI, 1992.
• Vyjayanthi Chari and Michael Kleber, Symmetric Functions and Representations of Quantum Affine Algebras (math.QA/0011161) In "Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory." Contemporary Mathematics 297, AMS, 2002.
• (+) A. M. Garsia and C. Procesi, On certain graded Sn-modules and the q-Kostka polynomials. Adv. Math. 94 (1992), no. 1, 82--138.
• A. M. Garsia and Mike Zabrocki, Polynomiality of the q,t-Kostka Revisited, math.QA/0008199.
• S. Gaussent, P. Littelmann, LS-galleries, the path model and MV-cycles (preprint, 2003). Available from Littlemann's home page.
• J. Haglund, M. Haiman, N. Loehr, J. B. Remmel, A. Ulyanov, A combinatorial formula for the character of the diagonal coinvariants, math.CO/0310424.
• (+) G. Hatayama, Y. Koga, A. Kuniba, M. Okado, T. Takagi, Finite crystals and paths Combinatorial methods in representation theory (Kyoto, 1998), 113--132, Adv. Stud. Pure Math., 28, Kinokuniya, Tokyo, 2000, math.QA/9901082.
• G. Hatayama, A. N. Kirillov, A. Kuniba, M. Okado, T. Takagi and Y. Yamada, Character formulae of \hat(sl)sub>n-modules and inhomogeneous paths Nuclear Phys. B 536 (1999), no. 3, 575--616, math.QA/9802085.
• (+) G. Hatayama, A. Kuniba, M. Okado, T. Takagi, Z. Tsuboi, Paths, crystals and fermionic formulae MathPhys odyssey, 2001, 205--272, Prog. Math. Phys., 23, Birkhäuser Boston, Boston, MA, 2002, math.QA/0102113
• M. Kashiwara, Realizations of crystals, math.QA0202268.
• (+) S.-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima, A. Nakayashiki, Affine crystals and vertex models, Infinite analysis, Part A, B (Kyoto, 1991). Adv. Ser. Math. Phys. 16 (1992) 449--484.
• (+) S.-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima, A. Nakayashiki, Perfect crystals of quantum affine Lie algebras, Duke Math. J. 68 (1992), no. 3, 499-607.
• Masaki Kashiwara, Realizations of Crystals, Combinatorial and geometric representation theory (Seoul, 2001), 133--139, Contemp. Math., 325, Amer. Math. Soc., Providence, RI, 2003. math.QA/0202268
• Masaki Kashiwara, Level zero fundamental representations over quantized affine algebras and Demazure modules, math.QA/0309142
• (+) S. V. Kerov, A. N. Kirillov, and N. Yu. Reshetikhin, Combinatorics, the Bethe ansatz and representations of the symmetric group. (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155 (1986), Differentsialnaya Geometriya, Gruppy Li i Mekh. VIII, 50--64, 193; translation in J. Soviet Math. 41 (1988), no. 2, 916--924
• (+) Anatol N. Kirillov, Anne Schilling, and Mark Shimozono, A bijection between Littlewood-Richardson tableaux and rigged configurations (math.CO/9901037) Selecta Mathematica (N.S.) 8 (2002) 67-135.
• (+) A. N. Kirillov and N. Yu. Reshetikhin, The Bethe ansatz and the combinatorics of Young tableaux. (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155 (1986), Differentsialnaya Geometriya, Gruppy Li i Mekh. VIII, 65--115, 194; translation in J. Soviet Math. 41 (1988), no. 2, 925--955
• (+) Michael Kleber, Finite Dimensional Representations of Quantum Affine Algebras. (math.QA/9809087) Ph.D. dissertation, U.C. Berkeley, 1998.
• L. Lapointe, A. Lascoux, J. Morse, Tableau atoms and a new Macdonald positivity conjecture. (math.QA/0008073)
• L. Lapointe, J. Morse. Schur function analogs for a filtration of the symmetric function space. (math.CO/0111192)
• L. Lapointe, J. Morse. Schur function identities, their t-analogs, and k-Schur irreducibility. (math.CO/0111193)
• (+) Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon, Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and Unipotent Varieties. (q-alg/9512031) Sém. Lothar. Combin. 34 (1995).
• Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras. (q-alg/9503001) Comm. Math. Phys. 181 (1996), no. 1, 205--263.
• Bernard Leclerc and Jean-Yves Thibon, Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials Combinatorial methods in representation theory (Kyoto, 1998) Adv. Stud. Pure Math. 28 (2000) 155--220.
• P. Littelmann, A Littlewood-Richardson type rule for symmetrizable Kac-Moody algebras, Invent. Math. 116 (1994) 329-346.
• Peter Littelmann, Paths and root operators in representation theory Ann. of Math. 142, no. 3 (1995) 499-525.
• P. Littelmann, Bases for representations, LS-paths and Verma flags (preprint, 2003). Available from Littlemann's home page.
• (+) Masato Okado, Anne Schilling, Mark Shimozono, Virtual crystals and Kleber's algorithm (math.QA/0209082)
• (+) Masato Okado, Anne Schilling, Mark Shimozono, A crystal to rigged configuration bijection for nonexceptional affine algebras (math.QA/0203163)
• Anne Schilling, Rigged configurations and the Bethe ansatz (math-ph/0210014)
• (+) Mark Shimozono, Affine Type A Crystal Structure on Tensor Products of Rectangles, Demazure characters, and Nilpotent Varieties (math.QA/9804039)
• (+) Mark Shimozono and Mike Zabrocki, Hall-Littlewood vertex operators and generalized Kostka polynomials (math.QA/0001168), Advances in Math. 158 (2001) 66-85.
• (+) Mark Shimozono and Jerzy Weyman, Graded characters of modules supported in the closure of a nilpotent conjugacy class. (math.QA/9804036), to appear in European J. Combin.
• Geanina Tudose and Mike Zabrocki A q-analog of Schur's Q-functions (math.CO/0203046)
• (+) Mike Zabrocki, q-Analogs of symmetric function operators (math.QA/0008188) Discrete Math., to appear.
• Mike Zabrocki, Ribbon operators and Hall-Littlewood symmetric functions Advances in Math. 156 (2000) 33-43.