Thomas Lam's Publications and Preprints

  1. Tiling with commutative rings
    An expository article explaining an approach to tiling problems using commutative algebra, written for the Harvard undergraduate math journal.
    Harvard College Mathematics Review, 2 (2008), 60-55.
    [ps] [pdf]

  2. Combinatorial Hopf algebras and Towers of Algebras (with Nantel Bergeron and Huilan Li)
    We prove that a tower of algebras A = {A_n} whose Grothendieck groups give rise to graded dual Hopf algebras via induction and restriction must have dimension dim(A_n) = r^n n!.
    Proc. FPSAC, 2008, to appear.
    arxiv: 0710.3744

  3. Total positivity for cominuscule Grassmannians (with Lauren Williams)
    We study the totally non-negative cells of cominuscule Grassmannians. In particular, we define and study Le-diagrams.
    New York J. of Math., to appear.
    arxiv: 0710.2932

  4. Schubert polynomials for the affine Grassmannian of the symplectic group (with Anne Schilling and Mark Shimozono)
    Using Schur P and Q-functions, we define and study Schubert polynomials for the affine Grassmannian of the symplectic group.
    arxiv: 0710.2720

  5. Combinatorial Hopf algebras and K-homology of Grassmannians (with Pavlo Pylyavskyy)
    Motivated by work of Buch on set-valued tableaux, we define and study six "K-theoretic" combinatorial Hopf algebras.
    International Mathematics Research Notices, 2007 (2007), rnm 125, 48 pages.
    arxiv: 0705.2189

  6. Quantum cohomology of G/P and homology of affine Grassmannian (with Mark Shimozono)
    We prove an unpublished result of Dale Peterson showing that quantum and affine homology Schubert calculi are equivalent.
    arxiv: 0705.1386

  7. Dual graded graphs for Kac-Moody algebras (with Mark Shimozono)
    We define and study a family of dual graded graphs with vertex set equal to the Weyl group of a Kac-Moody algebra.
    Algebra and Number Theory, 1 (2007), 451-488.
    arxiv: math.CO/0702090

  8. Temperley-Lieb Pfaffinants and Schur Q-positivity conjectures (with Pavlo Pylyavskyy)
    We define and study a pfaffian analogue of immanants, focusing on Temperley-Lieb pfaffinants.
    arxiv: math.CO/0612842

  9. Signed differential posets and sign-imbalance
    We study a signed analogue of differential posets, and relate them to sign-imbalance.
    J. Comb. Theory Series A, to appear.
    arxiv: math.CO/0611296

  10. On domino insertion and Kazhdan-Lusztig cells in type B_n (with Cedric Bonnafe, Meinolf Geck, and Lacri Iancu)
    We give a conjectural characterization of all Kazhdan-Lusztig cells for B_n with unequal parameters, via domino insertion.
    Progress in Math (Lusztig Birthday Volume), Birkhauser, to appear.
    arxiv: math.RT/0609279

  11. P-partition products and fundamental quasi-symmetric function positivity (with Pavlo Pylyavskyy)
    We show that certain differences of products of P-partition generating functions are positive combinations of fundamental quasi-symmetric functions.
    Adv. Appl. Math., to appear.
    arxiv: math.CO/0609249

  12. Affine insertion and Pieri rules for the affine Grassmannian (with Luc Lapointe, Jennifer Morse, and Mark Shimozono)
    We prove an affine insertion algorithm, obtaining Pieri rules for the (co)homology of the affine Grassmannian as a consequence.
    Memoirs of the AMS, to appear.
    arxiv: math.CO/0609110

  13. On Sjostrand's skew sign-imbalance identity
    We give a quick proof of a sign-imbalance identity due to Sjostrand.
    [ps] [pdf] arxiv: math.CO/0607516

  14. Schubert polynomials for the affine Grassmannian
    We identify the (co)homology Schubert basis of the affine Grassmannian as the (dual)k-Schur functions. The files here are an extended abstract which appeared in Proc. FPSAC San Diego, 2006; the arXiv version is the real one.
    J. Amer. Math. Soc., 21 (2008), 259-281.
    [ps] [pdf] arxiv: math.CO/0603125

  15. A Little bijection for affine Stanley symmetric functions (with Mark Shimozono)
    David Little developed a combinatorial algorithm to study the Schur-positivity of Stanley symmetric functions and the Lascoux-Sch\"{u}tzenberger tree. We generalize this algorithm to affine Stanley symmetric functions.
    Seminaire Lotharingien de Combinatoire, 54A (2006), B54Ai.
    arxiv: math.CO/0601483

  16. A combinatorial generalization of the Boson-Fermion correspondence
    We explain the ubiquity of tableaux and Pieri and Cauchy identities for many families of symmetric functions, using representations of Heisenberg algebras.
    Math. Res. Letters, 13 (2006), 377-392.
    [ps] [pdf] arxiv: math.CO/0507341

  17. Cell Transfer and Monomial Positivity (with Pavlo Pylyavaskyy)
    We show that certain differences of products of Schur functions are monomial positive and give a generalisation of this to arbitrary labelled posets.
    J. Alg. Combin., 26 (2007), 209-224.
    arxiv: math.CO/0505273

  18. Combinatorics of Ribbon Tableaux
    My Ph.D. Thesis written under the guidance of Richard Stanley. This contains my papers on ribbon Schur operators and ribbon tableaux and the Heisenberg algebra, together with a combinatorial generalisation of the Boson-Fermion correspondence.
    [ps] [pdf]

  19. Schur positivity and Schur log-concavity (with Alex Postnikov and Pavlo Pylyavskyy)
    We prove Schur positivity conjectures of Okounkov, of Lascoux, Leclerc and Thibon and of Fomin, Fulton, Li and Poon.
    Amer. J. Math., 129 (2007), 1611-1622.
    arxiv: math.CO/0502446

  20. Alcoved Polytopes I (with Alex Postnikov)
    We study certain polytopes arising from the affine Coxeter arrangement (in type A).
    Disc. and Comp. Geom., 38 (2007), 453-478 .
    arxiv: math.CO/0501246

  21. Affine Stanley Symmetric Functions
    We define and study a new family of symmetric functions which are affine analogues of Stanley symmetric functions.
    American J. of Math., 128 (2006), 1553-1586 .
    arxiv: math.CO/0501335

  22. Ribbon Schur Operators
    A new combinatorial approach to the ribbon tableaux generating functions and $q$-Littlewood Richardson coefficients of Lascoux, Leclerc and Thibon is suggested, following methods of Fomin and Greene.
    European J. of Combinatorics, 29 (2008), 343-359.
    arxiv: math.CO/0409463

  23. A note on graphs without short even cycles (with Jacques Verstraete)
    We show that any n-vertex graph without even cycles of length at most 2k has at most (1/2)n^{1 + 1/2} + O(n) edges.
    Electronic Journal of Combinatorics, 12/1 (2005), N5.
    arxiv: math.CO/0503623

  24. On symmetry and positivity for domino and ribbon tableaux
    We show the symmetry of LLT-ribbon functions, and describe the product of a Schur function and a domino function.
    Annals of Combinatorics, 9 (2005), 293-300.
    [ps] [pdf] arxiv: math.CO/0407184

  25. Affine Stanley Symmetric Functions (extended abstract)
    We define and study a new family of symmetric functions which are affine analogues of Stanley symmetric functions.
    Proceedings of FPSAC, 2005, Taormina.
    [ps] [pdf]

  26. Ribbon Tableaux and the Heisenberg Algebra
    We prove Pieri, Cauchy and Murnaghan-Nakayama formulae for the ribbon tableaux generating functions of Lascoux, Leclerc and Thibon. (The version on the arXiv is a longer, older version.)
    Mathematische Zeitschrift, 250 (2005), 685-710.
    [ps] [pdf] arxiv: math.QA/0310250

  27. Growth diagrams, domino insertion, and sign-imbalance
    We study some properties of domino insertion and settle Stanley's `2^{n/2}' conjecture on sign-imbalance.
    Journal of Combinatorial Theory Ser. A., 107 (2004), 87-115.
    [ps] [pdf] arxiv: math.CO/0308265

  28. Pieri and Cauchy formulae for Ribbon Tableaux
    This is an extended abstract of "Ribbon Tableaux and the Heisenberg Algebra", where I focus on more combinatorial aspects.
    Proceeedings of FPSAC, 2004, Vancouver.
    [ps] [pdf]

  29. A result on 2k-cycle free bipartite graphs
    A bipartite graph with parts of sizes N >= M and no cycles of length 2l, for all l \in [2,2k], has number of edges less than M^{1/2}N^{(k+1)/2k} + O(N).
    Australasian Journal of Combinatorics, 32 (2005), 163.
    [ps] [pdf]

  30. Graphs without cycles of even length
    We prove that a bipartite graph with parts of sizes M and N, having no cycles of even length less than or equal to 2(2k+1), has at most (NM)^{ \frac{k+1}{2k+1}} + O(N+M) edges.
    Bulletin of the Australian Mathematical Society, 63, (2001) 435-440.

  31. Graphs without cycles of even length
    Honours thesis for my B.Sc. at University of New South Wales (supervised by Professor Terence Tao), 2001.

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