Research

My research interests range from algebraic combinatorics, representation theory to mathematical physics. I have been interested in combinatorial properties of quantum algebras. Quantum algebras have their roots in two dimensional solvable lattice models in statistical mechanics. Kashiwara showed that at zero temperature the quantum algebras exhibit beautiful combinatorial properties. Mathematically these are formulated in the crystal base theory. Particularly nice crystals are Kirillov-Reshetikhin crystals; they are affine finite-dimensional crystals that can be used in the Kyoto path model to construct highest weight affine crystals. They are also in close relation to Demazure crystals as shown in work with Fourier and Shimozono. Together with Okado and Fourier, I recently showed that all Kirillov-Reshetikhin crystals of nonexceptional types exist and constructed them combinatorially exploiting certain automorphisms of the affine Dynkin diagrams. With Kirillov and Shimozono I have established a relation between crystal base theory and other combinatorial objects, called rigged configurations, which also arise from physics. A generalization of this result provides new fermionic formulas of the physical partition sums which, as follows by work of the Stony Brook group, encapture the statistics of the particle in the physics model (actually these statistics are similar to those found in the fractional quantum Hall effect; see here). In terms of q-series these yield generalizations of the famous Rogers-Ramanujan identities.

I am also part of the Focused Research Group on Affine Schubert Calculus: Combinatorial, geometric, physical, and computational aspects funded by the NSF. More information can be found at the FRG website.

I am especially fond of understanding relations between different areas in mathematics and mathematical physics. Often this helps to unravel hard problems in one area by exploiting the techniques of another area. I have also contributed to the open-source computer package Sage through Sage-Combinat by implementing crystal bases and Kirillov-Reshetikhin crystals (I previously contributed to MuPAD-Combinat).

• K-theory Schubert calculus of the affine Grassmannian
  with Thomas Lam and Mark Shimozono
  preprint arXiv:0901.1506 [math.CO]


• Perfectness of Kirillov-Reshetikhin crystals for nonexceptional types
  with Ghislain Fourier and Masato Okado
  Contemp. Math., to appear ( arXiv:0811.1604 [math.RT] )


• Kirillov-Reshetikhin crystals for nonexceptional types
  with Ghislain Fourier and Masato Okado
  Advances in Mathematics, to appear ( arXiv:0810.5067 [math.RT] )


• On the uniqueness of promotion operators on tensor products of type A crystals
  with Jason Bandlow and Nicolas Thiery
  Algebraic Combinatorics, to appear ( arXiv:0806.3131 [math.CO] )


• 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 [math.RT] )
  Hecke group algebras as degenerate affine Hecke algebras
  DMTCS proc AJ (2008) 611-624
  with Florent Hivert and Nicolas Thiery


• Schubert Polynomials for the affine Grassmannian of the symplectic group
  with Thomas Lam and Mark Shimozono
  Mathematische Zeitschrift, to appear ( arXiv:0710.2720 [math.CO] )


• The forgotten monoid
  with Jean-Christophe Novelli
  RIMS Kokyuroku Bessatsu B8 (2008) 71-83 ( arXiv:0706.2996 [math.CO] )


• Existence of Kirillov-Reshetikhin crystals for nonexceptional types
  with Masato Okado
  Representation Theory 12 (2008) 186-207 ( arXiv:0706.2224 [math.QA] )


• 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 [math.QA] )


• Demazure structure inside Kirillov-Reshetikhin crystals
  with Ghislain Fourier and Mark Shimozono
  J. Algebra 309 (2007) 386-404 ( math.QA/0605451 )


• X=M Theorem: Fermionic formulas and rigged configurations under review
  MSJ Memoirs 17 (2007) 75-104 (published by the Mathematical Society of Japan) ( math.QA/0512161 )


• Virtual crystal structure on rigged configurations
  Formal Power Series and Algebraic Combinatorics, San Diego, California 2006 ( pdf file )


• New fermionic formula for unrestricted Kostka polynomials
  with Lipika Deka
  J. Combinatorial Theory, Series A 113 (2006) 1435-1461 ( math.CO/0509194 )


• Crystal structure on rigged configurations
  International Mathematics Research Notices, Volume 2006, Article ID 97376, Pages 1-27 ( math.QA/0508107 )


• Non-unitary minimal models, Bailey's lemma and N=1,2 superconformal algebras
  with Lipika Deka
  Commun. Math. Phys. 260 (2005) 711-725 ( math-ph/0412084 )


• X=M for symmetric powers
  with Mark Shimozono
  J. Algebra 295 (2006) 562-610 ( math.QA/0412376 )


• A simple introduction to crystals B^{2,s} for Kirillov-Reshetikhin modules of type D_n^{(1)}
  with Philip Sternberg
  in: A. Kuniba, M. Okado (Eds.), RIMS Kokyuroku 1429 "Combinatorial Aspect of Integrable Systems", to appear. ( postscript file )


• Finite-dimensional crystals B^{2,s} for quantum affine algebras of type D_n^{(1)}
  with Philip Sternberg
  J. Alg. Combin. 23 (2006) 317-354 ( math.QA/0408113 )


• A bijection between type D_n^{(1)} crystals and rigged configurations
  J. Algebra 285 (2005) 292-334 ( math.QA/0406248 )


• Rigged configurations and the Bethe Ansatz
  in: B. Lulek, T. Lulek and A. Wal (Eds), "Symmetry and Structural Properties of Condensed Matter", Vol. 7, World Scientific, Singapore 2003; pp. 201-224 ( math-ph/0210014 )


• Virtual crystals and Kleber's algorithm
  with Masato Okado and Mark Shimozono
  Commun. Math. Phys. 238 (2003) 187-209 ( math.QA/0209082 )


• A crystal to rigged configuration bijection for nonexceptional affine algebras
  with Masato Okado and Mark Shimozono
  "Algebraic Combinatorics and Quantum Groups", Edited by N. Jing, World Scientific (2003), 85-124 ( math.QA/0203163 )


• A tensor product theorem related to perfect crystals
  with Masato Okado and Mark Shimozono
  J. Algebra 267 (2003) 212-245 ( math.QA/0111288 )


• q-Supernomial coefficients: From riggings to ribbons
  in: MathPhys Odyssey 2001, M. Kashiwara and T. Miwa (eds.), Birkhaeuser Boston, Cambridge, MA, 2002, pp. 437-454 ( math.CO/0107214 )


• Virtual crystals and fermionic formulas of type $D_{n+1}^{(2)}$, $A_{2n}^{(2)}$, and $C_n^{(1)}$
  with Masato Okado and Mark Shimozono
  Representation Theory 7 (2003) 101-163 ( math.QA/0105017 )


• Crystal bases and q-identities
  with Masato Okado and Mark Shimozono
  Contemp. Math. 291 (2001) 29-53 ( math.QA/0104268 )


• Branching formula for q-Littlewood--Richardson coefficients
  with Mark Shimozono and Dennis E. White
  Advances in Applied Mathematics 30 (2003) 258-272 ( postscript file )


• Fermionic formulas for level-restricted generalized Kostka polynomials and coset branching functions
  with Mark Shimozono
  Commun. Math. Phys. 220 (2001) 105-164 ( math.QA/0001114 )


• New expressions for level-restricted Kostka polynomials
  with Mark Shimozono
  in: Formal Power Series and Algebraic Combinatorics, D. Krob, A.A. Mikhalev, A.V. Mikhalev (eds.), 12th International conference, FPSAC'00, Moscow, Russia, June 2000, Proceedings, Springer Verlag Berlin/Heidelberg 2000, pp. 367-378. ( postscript file )


• A generalization of the $q$-Saalschütz sum and the Burge transform
  with S. Ole Warnaar
  in: Physical Combinatorics, M. Kashiwara and T. Miwa (eds.), Birkhaeuser Boston, Cambridge, MA, 2000, pp. 163-183. ( math.QA/9909044 )


• Conjugate Bailey pairs. From configuration sums and fractional-level string functions to Bailey's lemma
  with S. Ole Warnaar
  Contemp. Math. 297 (2002) 227-255 ( math.QA/9906092 )


• A bijection between Littlewood-Richardson tableaux and rigged configurations
  with Anatol N. Kirillov and Mark Shimozono
  Selecta Mathematica (N.S.) 8 (2002) 67-135 ( math.CO/9901037 )


• Various representations of the generalized Kostka polynomials
  with Anatol N. Kirillov and Mark Shimozono
  Seminaire Lotharingien de Combinatoire 42 [B42j] (1999) ( postscript file )


• Bosonic formula for level-restricted paths
  with Mark Shimozono
  Advanced Studies in Pure Mathematics 28, Combinatorial Methods in Representation Theory (2000) 305-325 ( math.QA/9812106 )


• An A$_2$ Bailey lemma and Rogers-Ramanujan-type identities
  with George E. Andrews and S. Ole Warnaar
  J. Amer. Math. Soc. 12 (1999) 677-702 ( math.QA/9807125 )


• Inhomogeneous lattice paths, generalized Kostka polynomials and A$_{n-1}$ supernomials
  with S. Ole Warnaar
  Commun. Math. Phys. 202 (1999) 359-401 ( math.QA/9802111 )


• Bailey flows and Bose-Fermi identities for the conformal coset models $(A_1^{(1)})_N\times (A_1^{(1)})_{N'}/(A_1^{(1)})_{N+N'}$
  with Alexander Berkovich, Barry M. McCoy, and S. Ole Warnaar
  Nucl. Phys. B499 [PM] (1997) 621-649 ( hep-th/9702026 )


• Supernomial coefficients, polynomial identities and $q$-series
  with S. Ole Warnaar
  The Ramanujan Journal 2 (1998) 459-494 ( q-alg/9701007 )


• Rogers-Schur-Ramanujan Type Identities for the $M(p,p')$ minimal models of Conformal Field Theory
  with Alexander Berkovich and Barry M. McCoy
  Commun. Math. Phys. 191 (1998) 325-395 ( q-alg/9607020 )


• A Higher-Level Bailey Lemma: Proof and Application
  with S. Ole Warnaar
  The Ramanujan Journal 2 (1998) 327-349 ( q-alg/9607014 )


• A Higher-level Bailey Lemma
  with S. Ole Warnaar
  Int. J. of Mod. Phys. B11 (1997) 189-195 ( q-alg/9604015 )


• $N=2$ Supersymmetry and Bailey Pairs
  with Alexander Berkovich and Barry M. McCoy
  Physica A 228 (1996) 33-62 ( hep-th/9512182 )


• Multinomials and Polynomial Bosonic Forms for the Branching Functions of the $\widehat{su}(2)_M \times \widehat{su}(2)_N / \widehat{su}(2)_{M+N}$ Conformal Coset Models
  Nucl. Phys. B467 (1996) 247-271 ( hep-th/9510168 )


• Polynomial Fermionic Forms for the Branching Functions of the Rational Coset Conformal Field Theories $\widehat{su}(2)_M \times \widehat{su}(2)_N / \widehat{su}(2)_{M+N}$
  Nucl. Phys. B459 (1996) 393-436 ( hep-th/9508050 )


• Multiplicative BRST renormalization of the $SU(2)$ Higgs model
  with Peter van Nieuwenhuizen
  Phys. Rev. D50 (1994) 967-970 ( hep-th/9408125 )