Thomas Lam's Publications and Preprints

  1. Total positivity, Schubert positivity, and Geometric Satake (with Konstanze Rietsch)
    We compare three notions (total, Schubert, Mirkovic-Vilonen) of positivity on the centralizer subgroup of a principal nilpotent element, in a complex simple algebraic group. We explicitly parametrize this positive part, establishing a conjecture of Rietsch.
    arxiv: 1203.1682

  2. Alcoved Polytopes II (with Alex Postnikov)
    We continue our study of polytopes arising from affine Coxeter arrangements. In particular, we give compatible definitions of descent numbers, major indices, and a q-analogue of Weyl's formula for the order of a Weyl group.
    arxiv: 1202.4015

  3. Positroid Varieties: Juggling and Geometry (with Allen Knutson and David Speyer)
    We study positroid varieties, the intersections of cyclically rotated Schubert varieties in the Grassmannian. This paper supercedes "Positroid Varieties I: Juggling and Geometry" which has been broken up into this paper, and "Projections of Richardson Varieties".
    arxiv: 1111.3660

  4. From double quantum Schubert polynomials to k-double Schur functions via the Toda lattice (with Mark Shimozono)
    We show that k-double Schur functions and quantum double Schubert polynomials can be obtained from each other by an explicit rational substitution. The main new ingredient is an explicit computation of Kostant's solution to the Toda lattice in terms of equivariant Schubert classes.
    arxiv: 1109.2193

  5. Quantum double Schubert polynomials represent Schubert classes (with Mark Shimozono)
    We show that quantum double Schubert polynomials represent equivariant quantum Schubert classes, and define parabolic analogues.
    Proc. AMS, to appear.
    arxiv: 1108.4958

  6. Projected Richardson varieties and affine Schubert varieties (with Xuhua He)
    We compare projections of Richardson varieties from G/B to G/P with Schubert varieties in the affine flag variety. Two comparisons are made: a combinatorial one on the level of the closure order poset, and a cohomological comparison of the corresponding classes.
    arxiv: 1106.2586

  7. Equivariant Pieri Rule for the homology of the affine Grassmannian (with Mark Shimozono)
    We give a Chevalley formula for equivariant homology in the affine Grassmannian of an arbitrary simple algebraic group. We give an explicit Pieri formula for the affine Grassmannian of type A.
    Journal of Algebraic Combinatorics , to appear.
    arxiv: 1105.5154

  8. k-Double Schur functions and equivariant (co)homology of the affine Grassmannian (with Mark Shimozono)
    The Schubert bases of the torus-equivariant homology and cohomology rings of the affine Grassmannian of the special linear group are realized by new families of symmetric functions called k-double Schur functions and affine double Schur functions.
    arxiv: 1105.2170

  9. Inverse problem in electrical cylindrical networks (with Pavlo Pylyavskyy)
    We study the inverse Dirichlet-to-Neumann problem for cylindrical electrical networks, and define an "electrical-R-matrix" which conjecturally solves the inverse problem.
    SIAM J. of Applied Math., 72 (2012), 767-788.
    arxiv: 1104.4998

  10. Electrical networks and Lie theory (with Pavlo Pylyavskyy)
    We introduce a new class of "electrical" Lie algebras and Lie groups which act on electrical networks.
    arxiv: 1103.3475

  11. A Markov chain on the symmetric group which is Schubert positive? (with Lauren Williams)
    We study a multivariate Markov chain on the symmetric group and conjecture that the invariant distribution is Schubert positive.
    Experimental Mathematics, to appear.
    arxiv: 1102.4406

  12. The shape of a random affine Weyl group element, and random core partitions
    We show that large random affine Weyl group elements almost surely have one of finitely many "shapes". In the type A case, one obtains that the shape of a random n-core is a piecewise-linear graph, in the sense of Vershik and Kerov.
    arxiv: 1102.4405

  13. Loop symmetric functions and factorizing matrix polynomials
    A survey of results obtained with Pavlo Pylyavskyy, concerning the ring of loop symmetric functions, and relations to networks on surfaces, total positivity, crystal graphs, and discrete integrable systems.
    Conference proceedings of ICCM 2010, Beijing, to appear.
    arxiv: 1012.1262

  14. Box-basket-ball systems (with Pavlo Pylyavskyy and Reiho Sakamoto)
    We define a new discrete solitonic system, which consists of boxes, baskets, and balls. We classify the solitons and study their scattering.
    arxiv: 1011.5930

  15. From quantum Schubert polynomials to k-Schur functions via the Toda lattice (with Mark Shimozono)
    We show that quantum Schubert polynomials and k-Schur functions can be obtained from each other via a rational substitution.
    Math Research Letters, to appear.
    arxiv: 1010.4047

  16. Projections of Richardson Varieties (with Allen Knutson and David Speyer)
    We prove that projected Richardson varieties are normal, Cohen-Macaulay, have rational singularities, and are exactly the compatibly Frobenius-split subvarieties of G/P with respect to the standard splitting. We also show that parabolic projections of the order complex of a Bruhat interval is shellable.
    Crelle's journal, to appear.
    arxiv: 1008.3939

  17. Crystals and total positivity on orientable surfaces (with Pavlo Pylyavskyy)
    We study a model of combinatorial networks on orientable surfaces, and apply this to the theory of geometric crystals, and to total positivity.
    Selecta Math., to appear.
    arxiv: 1008.1949

  18. k-shape poset and branching of k-Schur functions (with Luc Lapointe, Jennifer Morse, and Mark Shimozono)
    We introduce a new class of partitions, called k-shapes, and use them to study branching of k-Schur functions.
    Memoirs of the AMS, to appear.
    arxiv: 1007.5334

  19. Stanley symmetric functions and Peterson algebras
    Lecture notes on affine Stanley symmetric functions for the summer school on affine Schubert calculus at the Fields Institute 2010.
    arxiv: 1007.2871

  20. Affine geometric crystals in unipotent loop groups (with Pavlo Pylyavskyy)
    The quotient by the R-matrix of products of the affine geometric crystal of type A corresponding to symmetric powers is constructed in the unipotent loop group.
    Representation Theory, to appear.
    arxiv: 1004.2233

  21. Intrinsic energy is a loop Schur function (with Pavlo Pylyavskyy)
    We identify the intrinsic energy function of tensor products of Kirillov-Reshetikhin crystals for symmetric powers of type A with the tropicalization of loop Schur functions.
    arxiv: 1003.3948

  22. Skew Littlewood-Richardson rules for Hopf algebras (with Aaron Lauve and Frank Sottile)
    We use Hopf algebra theory to prove "skew" versions of Littlewood-Richardson rules, establishing a conjecture of Assaf and McNamara.
    IMRN, 2010.
    arxiv: 0908.3714

  23. (Appendix in) A Pieri rule for skew shapes
    An algebraic proof of Assaf and McNamara's (who are the main authors) skew Pieri rule.
    J. Combin. Theory Ser. A, 118 (2011), 277-290.
    arxiv: 0908.0345

  24. Total positivity for loop groups II: Chevalley generators (with Pavlo Pylyavskyy)
    This is the second in a series of papers developing a theory of total positivity for loop groups. In this paper we study infinite products of Chevalley generators.
    arxiv: 0906.0610

  25. Affine Schubert classes, Schur positivity, and combinatorial Hopf algebras
    We suggest that Schubert classes for the affine Grassmannian of a simple algebraic group can be viewed as Schur positive symmetric functions.
    Bulletin of the LMS, 2011; doi:10.1112/blms/bdq110.
    arxiv: 0906.0385

  26. Positroid varieties I: juggling and geometry (with Allen Knutson and David Speyer)
    We study positroid varieties, the intersections of cyclically rotated Schubert varieties in the Grassmannian.
    arxiv: 0903.3694

  27. Combinatorial Hopf algebras and Towers of Algebras: Dimension, Quantization, and Functorality (with Nantel Bergeron and Huilan Li)
    We study the relations between combinatorial Hopf algebras, towers of algebras, and dual graded graphs. We prove a dimension formula, describe a quantization and a categorial version of our constructions. This paper is a full version of the similarly titled summary below.
    Algebras and Representation Theory, to appear.
    arxiv: 0903.1381

  28. K-theory Schubert calculus of the affine Grassmannian (with Anne Schilling and Mark Shimozono)
    We study the K-theory Schubert calculus of the affine Grassmannian of a simple algebraic group G. For the case G = SL_n, we identify the Schubert classes in K-(co)homology as K-k-Schur functions and affine stable Grothendieck polynomials.
    Compositio Mathematica, 146 (2010), 811-852.
    arxiv: 0901.1506

  29. Total positivity for loop groups I: whirls and curls (with Pavlo Pylyavskyy)
    We initiate the study of a theory of total positivity for loop groups.
    Adv. in Math., to appear.
    arxiv: 0812.0840

  30. Quantized dual graded graphs
    We study quantizations of dual graded graphs, and construct examples of them using Fibonacci posets, permutations, standard Young tableaux, and plane binary trees.
    Elec. J. Combin., 17 (2010), no.1, Research Paper 88, 11pp..
    arxiv: 0808.0345

  31. 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), 55-60.
    [ps] [pdf]

  32. 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!.
    Discrete Math. Theor. Comput. Sci. Proc., AJ, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, France (Proc. FPSAC 2008), 2008, 52-60.
    arxiv: 0710.3744

  33. 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., 14 (2008), 53-99.
    arxiv: 0710.2932

  34. 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.
    Math. Z., to appear.
    arxiv: 0710.2720

  35. 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

  36. 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.
    Acta. Math., 204 (2010), 49-90.
    arxiv: 0705.1386

  37. 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

  38. 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.
    Adv. in Math., 218 (2008), 1654-1684.
    arxiv: math.CO/0612842

  39. 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, 115 (2008), 466-484.
    arxiv: math.CO/0611296

  40. 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.
    Representation theory of algebraic groups and quantum groups (Nagoya 2006; eds. A. Gyoja et al.) , 33-54, Progress in Math. 284, Birkhauser, 2010.
    arxiv: math.RT/0609279

  41. 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., 40 (2008), 271-294.
    arxiv: math.CO/0609249

  42. 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, 208 (2010), no. 977.
    arxiv: math.CO/0609110

  43. 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

  44. 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

  45. 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

  46. 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

  47. 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

  48. 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]

  49. 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

  50. 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

  51. 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

  52. 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

  53. 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

  54. 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

  55. 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]

  56. 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

  57. 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

  58. 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]

  59. 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]

  60. 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.

  61. 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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