Focused Research Group on "Affine Schubert Calculus: Combinatorial, geometric, physical, and computational aspects"

This project is funded by the National Science Foundation grants DMS-0652641, DMS-0652652, DMS-0652668, DMS-0652648 through the provisions of NSF 06-580, Focused Research Groups in the Mathematical Sciences (FRG), and is administered through the American Institute of Mathematics, Drexel University, UC Davis, and Virginia Tech.

This project concerns the development of a vast extension of Schubert calculus to affine Grassmannians and affine flag varieties, called ``affine Schubert calculus”. Classical Schubert calculus, a branch of enumerative algebraic geometry concerned with counting subspaces satisfying certain intersection conditions, is the outcome of the solution to Hilbert’s Fifteenth problem. In the modern formulation, Schubert calculus is usually interpreted in cohomology theories of homogeneous spaces, most notably flag varieties. The full development of affine Schubert calculus will solve long-standing open problems in Macdonald theory and have an impact on physical questions, such as generalizations of Wess-Zumino-Witten conformal field theory models and extensions of Calogero-Sutherland quantum mechanical models whose eigenfunctions are k-Schur functions. The new approach to affine Schubert calculus is made possible by the recent discovery of certain explicitly defined symmetric functions called k-Schur functions. The k-Schur functions, which arose in the study of the seemingly unrelated Macdonald theory, were recently shown to be connected to the geometry and topology of the affine Grassmanian. The novel combinatorics of k-Schur functions will be exploited to deduce formulae for various multiplicities, including intersection multiplicities in the affine Grassmannian and the affine flag manifold. Some of these multiplicities are known to occur in Macdonald theory and as Verlinde fusion coefficients for the WZW model.

This many-faceted project involves and ties together various problems from combinatorics, geometry, representation theory, physics, and computation. The main questions that will be addressed can be viewed from several points of view: a geometric perspective (questions such as “how many lines are there satisfying a number of generic intersection conditions?”), a combinatorial perspective (”how many elements are in given sets and what properties do these sets have?”), a physics perspective (”how do fields correlate?”), and computational aspects (”are there efficient algorithms for calculating these numbers or objects?”). The project is an international cooperative research venture, with core group members located in Canada, the United States, Chile, and France, and interdisciplinary, involving mathematicians, physicists, and computer scientists. Graduate students will receive professional training by direct involvement in the research and will benefit from interaction with the research team. A summer school at the Fields institute for graduate students is also planned at the conclusion of the project. The investigation is largely fueled by extensive computational experimentation. The robust implementation of algorithms derived from the project, will lead to the development of new packages for computer algebra systems. The dissemination of this new software through an open-source computational package, will not only advance the proposed research program but will also have an outreach impact on the mathematics, physics, and computer science communities.


Contributing Scientists


  • Anouk Bergeron-Brlek, University of Toronto
  • Nicolas Borie, Universite Paris-Sud (also visiting UC Davis)
  • Daniel Bravo, Universdad Talca
  • Cory Brunson, Virginia Tech
  • Steve Pon, UC Davis
  • Tom Denton, UC Davis
  • Qiang Wang, UC Davis


  • MuPAD meeting in Paris, June 20-22, 2007 (organized by Nicolas Thiery)
  • Meeting at Banff, September 14-16, 2007 (organized by Jennifer Morse, Anne Schilling, and Nicolas Thiery)
  • MSRI program on Combinatorial Representation Theory, January 15, 2008 to May 23, 2008 (organized by P. Diaconis, A. Kleshchev, B. Leclerc, P. Littelmann, A. Ram, A. Schilling, R. Stanley)
  • AMS meeting in Claremont McKenna, May 3-4, 2008 (organized by Anne Schilling and Mike Zabrocki)
  • FPSAC'2008 in Valpariso Chile, June 23-27, 2008 (organized by Luc Lapointe)
  • Workshop: Combinatorics and Physics of k-Schur functions at Drexel University, March 25-29, 2009 (organized by Jennifer Morse and Mike Zabrocki)
  • Workshop at AIM, May 25-29, 2009 (organized by Anne Schilling and Mark Shimozono)
  • *-Combinat 2009, International Sage Workshop on Free and Practical Software for Algebraic Combinatorics, July 25-29, 2009 (organized by Nicolas Thiery)
  • Sage Days 20, CIRM, France, February 22-26, 2010 (organized by Florent Hivert, Franco Saliola, Nicolas Thiery)
  • Sage Days 20.5, Fields Institute, Toronto, May 3-7, 2010 (organized by Nantel Bergeron, Franco Saliola, Mike Zabrocki)
  • Schubert Calculus Summer School and Workshop, Fields Institute, Toronto, July 7-10, 2010 (organized by Anne Schilling and Mike Zabrocki)



Geometric aspects

  • M. Kashiwara and M. Shimozono, Equivariant K-theory of affine flag manifolds and affine Grothendieck polynomials, Duke Math. J. 148 (2009), no. 3, 501-538 math.AG/0601563
  • T. Lam, M. Shimozono, Quantum cohomology of G/P and homology of affine Grassmannian, Acta Math., to appear arXiv:0705.1386
  • T. Lam, A. Schilling, M. Shimozono, Schubert Polynomials for the affine Grassmannian of the symplectic group, Mathematische Zeitschrift 264(4) (2010) 765-811 arXiv:0710.2720
  • T. Lam, P. Pylyavskyy, Total positivity for loop groups I: whirls and curls, preprint arXiv:0812.0840
  • T. Lam, P. Pylyavskyy, Total positivity in loop groups II: Chevalley generators preprint arXiv:0906.0610
  • T. Lam, P. Pylyavskyy, Intrinsic energy is a loop Schur function preprint arXiv:1003.3948
  • T. Lam, A. Schilling, M. Shimozono, K-theory Schubert calculus of the affine Grassmannian, Compositio Math., to appear arXiv:0901.1506
  • A. Knutson, T. Lam, D. E. Speyer, Positroid varieties I: juggling and geometry, preprint arXiv:0903.3694
  • T. Lam, Affine Schubert classes, Schur positivity, and combinatorial Hopf algebras, preprint arXiv:0906.0385

Combinatorial aspects

  • T. Lam, L. Lapointe, J. Morse, and M. Shimozono, Affine insertion and Pieri rules for the affine Grassmannian, Memoirs of the AMS, to appear arXiv:0609.5110
  • T. Lam, M. Shimozono, Dual graded graphs for Kac-Moody algebras, Algebra and Number Theory, 1 (2007), 451-488 arXiv:0702.5090
  • J.-C. Novelli, A. Schilling, The forgotten monoid, RIMS Kokyuroku Bessatsu B8 (2008) 71-83, arXiv:0706.2996
  • D. Bravo and L. Lapointe, A recursion formula for k-Schur functions, J. Combin. Theory Ser. A 116 (2009), no. 4, 918-935 arXiv:0709.4509
  • N. Bergeron, F. Descouens, M. Zabrocki, A non-commutative generalization of k-Schur functions, Discrete Math. 309 (2009), no. 16, 5092-5105 arXiv:0804.0944
  • M. Mishna, M. Zabrocki, Analytic aspects of the shuffle product, Dans Proceedings of the 25th Annual Symposium on the Theoretical Aspects of Computer Science, STACS 2008, Bordeaux, France (2008)arXiv:0802.2844
  • F. Descouens, H. Morita, Y. Numata, A combinatorial proof of factorization formulas for Macdonald polynomials preprint arXiv:0803.2311
  • N. Bergeron, F. Descouens, M. Zabrocki, A Filtration of (q,t)-Catalan numbers, Adv. in Appl. Math. 44 (2010), no. 1, 16-36 preprint arXiv:0806.3046
  • T. Lam, Quantized dual graded graphs, preprint arXiv:0808.0345
  • A. Bergeron-Brlek, C. Hohlweg, M. Zabrocki, Words and polynomial invariants of finite groups in non-commutative variables, preprint arXiv:0907.0814
  • J. Morse, Combinatorics of the K-theory of the affine Grassmannian, preprint arXiv:0907.0044
  • J.-C. Aval, N. Bergeron, H. Li, On Noncommutative Combinatorial Inverse System, preprint arXiv:0909.1112v1
  • J. Bandlow, A. Schilling, M. Zabrocki, The Murnaghan–Nakayama rule for k-Schur functions, Journal of Combinatorial Theory, Series A, 118(5) (2011) 1588-1607 preprint arXiv:1004.4886
  • S. Pon, Q. Wang, Promotion and evacuation on standard Young tableaux of rectangle and staircase shape, preprint arXiv:1003.2728
  • A. Garsia, N. Wallach, G. Xin, M. Zabrocki, Kronecker Coefficients via Symmetric Functions and Constant Term identities preprint
  • J. Bandlow, J. Morse: The expansion of Hall-Littlewood functions in the dual Grothendieck polynomial basis, DMTCS, to appear
  • A. Garsia, N. Wallach, G. Xin, M. Zabrocki, Hilbert Series of Invariants, Constant terms and Kostka-Foulkes Polynomials, Discrete Mathematics, Volume 309, Issue 16, 28 August 2009, Pages 5206-5230 preprint

Algebraic and representation theoretic aspects

  • L. Lapointe, J. Morse, Quantum cohomology and the $k$-Schur basis, Trans. Amer. Math. Soc. 360 (2008) 2021-2040 math.CO/0501529
  • A. Schilling, Combinatorial structure of Kirillov-Reshetikhin crystals of type D_n(1), B_n(1), A_{2n-1}(2), J. Algebra 319 (2008) 2938-2962, arXiv:0704.2046
  • M. Okado, A. Schilling, Existence of Kirillov-Reshetikhin crystals for nonexceptional types, Representation Theory 12 (2008) 186-207 arXiv:0706.2224
  • G. Fourier, M. Okado, A. Schilling, Kirillov-Reshetikhin crystals for nonexceptional types, Advances in Mathematics 222 Issue 3 (2009) 1080-1116 arXiv:0810.5067
  • G. Fourier, M. Okado, A. Schilling, Perfectness of Kirillov-Reshetikhin crystals for nonexceptional types, Contemp. Math. 506 (2010) 127-143 arXiv:0811.1604
  • B. Jones, A. Schilling: Affine structures and a tableau model for E_6 crystals, preprint arXiv:0909.2442
  • N. Bergeron, T. Lam, H. Li, Combinatorial Hopf algebras and Towers of Algebras, Discrete Math. Theor. Comput. Sci. Proc (2008) 52–60 arXiv:0710.3744
  • N. Bergeron, T. Lam, H. Li, Combinatorial Hopf algebras and Towers of Algebras: Dimension, Quantization, and Functorality, preprint arXiv:0903.1381
  • F. Hivert, N. M. Thiéry, The Hecke group algebra of a Coxeter group and its representation theory, Journal of Algebra 321 (2009), 2230–2258 arXiv:0711.1561
  • F. Hivert, A. Schilling, N. M. Thiéry, Hecke group algebras as degenerate affine Hecke algebras, DMTCS proc AJ (2008) 611-624
  • F. Hivert, A. Schilling, N. Thiéry, Hecke group algebras as quotients of affine Hecke algebras at level 0, Journal of Combinatorial Theory, Series A 116 (2009) 844–863 arXiv:0804.3781
  • F. Hivert, A. Schilling, N. M. Thiery, The biHecke monoid of a finite Coxeter group, DMTCS, to appear arXiv:0912.2212
  • J. Bandlow, A. Schilling, N. Thiery, On the uniqueness of promotion operators on tensor products of type A crystals, J. Algebraic Combinatorics 31 (2010) 217-251 arXiv:0806.3131
  • Q. Wang, A. Schilling, Promotion operator on rigged configurations of type A, The Electronic Journal of Combinatorics 17(1) (2010) R24 arXiv:0908.2458[math.CO]
  • C. Lecouvey, M. Okado, M. Shimozono, Affine crystals, one-dimensional sums and parabolic Lusztig q-analogues, preprint arXiv:1002.3715
  • M.-C. David, N. M. Thiéry, Exploration of finite dimensional Kac algebras and lattices of irreducible intermediate subfactors, preprint arXiv:0812.3044[math.QA]
  • A. Brown, S. van Willigenburg, M. Zabrocki, Expressions for Catalan Kronecker Products, Pacific J. Math, Vol. 248 (2010), No. 1, 31–48 preprint arXiv:0809.3469
  • T. Lam, A. Lauve, F. Sottile, Skew Littlewood-Richardson rules from Hopf algebras, preprint arxiv:0908.3714
  • T. Denton, A Combinatorial Formula for Orthogonal Idempotents in the 0-Hecke Algebra of S_N, DMTCS, to appear (accepted for the proceedings of FPSAC 2010)


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