Geordie Williamson
Papers and preprints:
  1. Koszul duality for Kac-Moody groups and characters of tilting modules
    Joint with Pramod Achar, Shotaro Makisumi and Simon Riche.
    We establish modular Koszul duality relating parity and tilting sheaves on the flag variety, and deduce a character formula for indecomposable tilting sheaves. This gives a proof of the combinatorial part of the conjecture with Riche in "Tilting modules and the p-canonical basis" below. Another remarkable consequence is that the Hecke category of a (Kac-Moody) flag variety "knows" the Hecke category of the Langlands dual flag variety.

  2. Billards and tilting characters for SL3
    Joint with George Lusztig.
    Tilting modules for algebraic groups are fascinating and mysterious objects. Understanding their characters for GLn is equivalent to understanding decomposition numbers for symmetric groups. In this paper we advocate a "generational philosophy" for attacking this problem, and formulate a precise conjecture for SL3. Remarkably, these characters appear to be governed by a discrete dynamical system, that looks like billiards bouncing in equilateral triangles!

  3. Free-monodromic mixed tilting sheaves on flag varieties
    Joint with Pramod Achar, Shotaro Makisumi and Simon Riche.
    This is part of a big project to understand Koszul duality in the modular context. A first goal is a proof of a character formula for tilting modules in terms of the p-canonical basis, which was conjectured with Riche (see our tilting manifesto below). Here we define a category which is an algebraic version of "free monodromic tilting sheaves" considered by Bezrukavnikov-Yun. We work very hard to show that this category has a monoidal structure. In a sequel (to appear soon) we use these results to establish modular Koszul duality, and prove our character conjecture.

  4. The anti-spherical category
    Joint with Nicolas Libedinsky.
    Clasically, one considers the anti-spherical (or "polynomial") module for the affine Hecke algebra. We explain how to categorify its natural analogue for any Coxeter group using diagrammatics and Soergel calculus. In doing so we prove that (signed) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients. Much of our motivation for studying this category in detail comes from my joint work with S. Riche on tilting modules (see below).

  5. Algebraic representations and constructible sheaves
    Notes from my Takagi lecture in Tokyo, November 2016.
    I discuss what is known and not known about characters of simple and tilting modules for algebraic groups. The emphasis is on Lusztig's conjecture and ideas coming from constructible and perverse sheaves. I also briefly discuss the main ideas behind a conjecture with S. Riche (see "Tilting modules and the p-canonical basis" below).

  6. Relative hard Lefschetz for Soergel bimodules
    Preliminary version, joint with Ben Elias.
    We show the analogue of the relative hard Lefschetz theorem for Soergel bimodules. This implies the unimodality of the structure constants of the multiplication of the Kazhdan-Lusztig basis. It also has some interesting consequences for the structure of the tensor categories associated by Lusztig to any two sided cell in a Coxeter system.

  7. The Hodge theory of the Hecke category
    ECM prize lecture. To appear in the ECM proceedings.
    We survey how Soergel bimodules give rise to "Hodge theory" in three distinct ways (global, relative and local). We also briefly discuss other instances of "combinatorial Hodge theory".

  8. The Hodge theory of the Decomposition Theorem (after de Cataldo and Migliorini)
    Seminaire Bourbaki, No 1115.
    An attempt at a motivated introduction to de Cataldo and Migliorini's beautiful Hodge theoretic proof of the Decomposition Theorem.

  9. Tilting modules and the p-canonical basis
    Preliminary version, joint with Simon Riche.
    Our manifesto on tilting modules: we conjecture that the (diagrammatic) Hecke category acts on the principal block via wall-crossing functors. We prove our conjecture for the general linear group using the theory of 2-Kac-Moody actions. Remarkably (for me) the conjecture allows one to completely describe the principal block in terms of the Hecke category. As a corollary we derive character formulas for simple and tilting modules in terms of the p-canonical basis. This paper also ties up a few loose ends, including proving that the diagrammatic Hecke category has an alternative geometric realisation via parity sheaves.

  10. On torsion in the intersection cohomology of Schubert varieties
    Journal of Algebra 475 (2017), 207-228, (Sandy Green memorial issue).
    I give a geometric proof that the torsion in the local integral intersection cohomology of Schubert varieties for the general linear group grows exponentially in the rank. The point of this paper is that the geometry of the situation is very simple. This is a geometric version of "Torsion Explosion" (see below).

  11. The p-Canonical Basis for Hecke Algebras
    Joint with Thorge Jensen, in "Categorification in Geometry, Topology and Physics", Contemp. Math., 583 (2017) 333-361.
    We give an introduction to the p-canonical basis of Hecke algebras. We establish its basic properties, describe algorithms for its calculation, give examples in low rank and explain some connections to the rational rerpresentation theory of algebraic groups.

  12. On the character of certain tilting modules
    Joint with George Lusztig, submitted.
    We give a closed formula for the characters of certain indecomposable tilting modules which can be written as iterated tensor products of Frobenius twists of "fundamental" tilting modules. The formula relies on a conjectural stabilization of certain tilting characters for large p, which might already be known and follows from a conjecture with Riche (see above).

  13. Soergel calculus and Schubert calculus
    Joint with Xuhua He. To appear in Lusztig's 70th birthday volume.
    We use Kostant-Kumar's nil Hecke ring to give simple closed formulas for certain numbers occuring in calculations in Soergel bimodules. These numbers determine how Soergel bimodules decompose modulo p, and finding efficient ways of calculating them has important applications in modular representation theory. In particular, one can use this formula to rederive certain examples that involve complicated calculations otherwise. This formula also finds applications in "Schubert calculus and torsion" below.

  14. Local Hodge theory of Soergel bimodules
    To appear in Acta Mathematica.
    We prove the local hard Lefschetz theorem and Hodge-Riemann bilinear relations for Soergel bimodules. This is the extra ingredient needed for Soergel bimodules to deduce an algebraic proof of the Jantzen conjectures (on Jantzen filtrations on Verma modules for complex semi-simple Lie algebras). Most of the ideas are adapted from the global case (see "Hodge theory of Soergel bimodules" below), in particular ideas of de Cataldo and Migliorini provide the scaffolding of the argument. However certain aspects are considerably more technical.

  15. Diagrammatics for Coxeter groups and their braid groups
    Joint with Ben Elias, will appear in Quantum topology.
    We give a generators and relations description of the 2-groups associated to Coxeter groups and their braid groups. This gives nice criteria for a braid group to act on a category in terms of generalized Zamolodchikov relations, which can (and should?) be thought of as higher braid relations.

  16. A reducible characteristic variety in type A
    In Representations of reductive groups: in honor of the 60th birthday of David A. Vogan Jnr.
    We give an example of a characteristic cycle of an intersection cohomology D-module of a Schubert variety in SL(12)/B which has two components in the same two-sided cell. This implies an example as in the title of the paper, answering a question which has been around since the early 1980s. We came across this example trying to answer a related question, namely whether the image of the p-canonical basis in an irreducible representation of the Hecke algebra of the symmetric group always coincides with the Kazhdan-Lusztig basis. Alas it doesn't!

  17. Parity sheaves and tilting modules
    This is joint with Daniel Juteau and Carl Mautner, Ann. Sci. ENS.
    We show that the geometric Satake equivalence relates parity sheaves and tilting modules under explicit and mild restrictions on the characteristic of the field of coefficients. This allows geometric proofs of the stability of tilting modules under tensor product and restriction to a Levi subgroup (under the same bounds). Recently, Achar and Rider have used this result to give a proof of the Mirkovic-Vilonen conjecture in almost all cases.

  18. Kazhdan-Lusztig conjectures and shadows of Hodge theory
    This is joint with Ben Elias and will appear in the proceedings for the Arbeitstagung in memory of Hirzebruch.
    We give a gentle and motivated introduction to Soergel modules and their "Hodge theory". It is an expanded version of a talk I gave at the Arbeitstagung in memory of Hirzebruch. It can be seen as providing geometric background for our paper The Hodge theory of Soergel bimodules below.

  19. Appendix to Modular perverse sheaves on flag varieties I: tilting and parity sheaves by Pramod Achar and Simon Riche.
    This is joint with Pramod Achar and Simon Riche, Ann. Sci. ENS.
    This paper and it sequel provides a (beautiful!) Koszul duality between tilting modules and parity sheaves (building on work of Bezrukavnikov and Yun in the characteristic zero setting). In the appendix we establish some basic properties of modular tilting sheaves on the flag variety.

  20. Schubert calculus and torsion explosion
    (with an appendix by Kontorovich, McNamara and Williamson), to appear in Journal of the AMS.
    We observe that certain numbers occurring in Schubert calculus for SLn also occur as entries in intersection forms controlling decompositions of Soergel bimodules and parity sheaves in higher rank. These numbers grow exponentially in the rank. This observation gives many counter-examples to Lusztig's conjecture on the characters of simple rational modules for SLn over a field of positive characteristic. We also explain how to use our results to get counter-examples to the James conjecture.

  21. Soergel calculus
    Joint with Ben Elias, Representation Theory 20 (2016), 295-374.
    The category of Soergel bimodules provides the most concrete incarnation of the Hecke category, the basic object of Kazhdan-Lusztig theory. We present the monoidal category of Soergel bimodules by generators and relations. We give a diagrammatic treatment of Libedinsky's "light leaves" morphisms, and show that they give a basis for morphisms. This allows us to give a new proof of Soergel's classification of the indecomposable Soergel bimodules.

  22. On cubes of Frobenius extensions
    Joint work with Ben Elias and Noah Snyder, in Representation theory - current trends and perspectives, EMS Ser. Congr. Rep., Eur. Math. Soc., Zurich, 2016.
    We prove some relations between induction and restriction functors for hypercubes of Frobenius extensions. We discovered these relations whilst trying to understand singular Soergel bimodules.

  23. The Hodge theory of Soergel bimodules
    (with Ben Elias), Annals of Mathematics, (2) 180 (2014), no. 3, 1089-1136..
    In geometric situations Soergel bimodules can be realised as the equivariant intersection cohomology of Schubert varieties, and hence have interesting real Hodge theory (hard Lefschetz, Hodge-Riemann bilinear relations etc). Inspired by work of de Cataldo and Migliorini giving Hodge theoretic proofs of the decomposition theorem we prove that these Hodge theoretic properties always hold for Soergel bimodules, whether they come from Schubert varieties or not! This gives structures strong enough to deduce Soergel's conjecture, and hence the Kazhdan-Lusztig positivity conjecture. The result can also be used to give the first algebraic proof of the Kazhdan-Lusztig conjectures on characters of simple highest weight modules over complex semi-simple Lie algebras.

  24. On an analogue of the James conjecture.
    Representation Theory 18, 15-27 (2014).
    We give a counterexample to the most optimistic analogue of the James conjecture for simply laced Khovanov-Lauda-Rouquier algebras. The basic idea is to recycle counterexamples known for Schubert varieties (due mostly to Braden and Polo). The bridge is provided by recent results of Maksimau. There are interesting connections to the reducibility of the characteristic variety, using work of Kashiwara and Saito and a result with Vilonen below.

  25. Modular Koszul duality
    (with Simon Riche and Wolfgang Soergel), Compositio Math. 150, No. 2, 273-332 (2014).
    Classical Koszul duality (due to Beilinson, Ginzburg and Soergel) relates category O and the derived category of Bruhat constructible sheaves of complex vector spaces on the flag variety. Modular Koszul duality relates "modular category O" (a subquotient of rational representations of a reductive group) and the derived category of constructible sheaves on the flag variety, this time with coefficients of positive characteristic. The key difficulty (which turns a simple idea into a sixty page paper) is establishing the formality of the dg-algebra of extensions of parity sheaves on the flag variety.

  26. Characteristic cycles and decomposition numbers
    (with Kari Vilonen), Math. Res. Let. 20, No. 2, 359-366 (2013).
    There are a number of false conjectures around characteristic cycles and decomposition numbers: e.g. Kazhdan and Lusztig's conjecture that characteristic varieties for Schubert varieties are irreducible; Kleshchev-Ram's conjecture that decomposition numbers for for KLR algebras are trivial in finite type; various people's hopes that stalks and costalks of IC sheaves on flag varieties have torsion only in bad characteristic. In this article we prove that the topological side (a decomposition number is non-trivial) implies the analytic side (the characteristic variety is reducible). This is a consequence of the trivial observation that the characteristic cycle of a sheaf commutes with base change of coefficients.

  27. Standard objects in 2-braid groups
    (with Nicolas Libedinsky), Proceedings of the LMS, (3) 109 (2014), no. 5, 1264--1280.
    The 2-braid group is a categorification of the braid group. It has been around for a few decades in highest weight representation theory (long before it acquired a name). A few years ago Khovanov showed that it may be used to construct HOMFLYPT homology, and since then there has been growing interest in its type A incarnation. Rouquier has emphasised that a study of the morphisms in the 2-braid group should have many applications in representation theory (new proofs of Kazhdan-Lusztig type conjectures, understanding of t-structures in modular representation theory, construction of Spetses, etc.) In this paper we make a first step in the study of morphisms spaces in 2-braid groups, namely we consider "standard" and "costandard" objects and show that they satisfy a vanishing condition conjectured by Rouquier.

  28. Kumar's criterion modulo p
    (with Daniel Juteau), Duke Mathematical Journal, 163 (2014), no. 14, 2617--2638.
    Here are some expository notes and here are some pictures of singularities intended to illustrate what's going on!
    We show that the numerator in Kumar's criterion for rational smoothness of Schubert varieties has a natural interpretation in terms of p-smoothness. We conjecture in certain cases the numerator calculates the order of the torsion subgroup of the link. One consequence is that certain parts of the equivariant multiplicity are in fact topological invariants (and not just invariants of the singularity with T-action).

  29. Singular Soergel bimodules
    IMRN 2011, No. 20, 4555-4632 (2011).
    Singular Soergel bimodules is a interesting 2-category which acts (or should act) in many representation theoretic situations. It also has an elementary definition in terms of certain rings of invariants, for the action of a Coxeter system on a polynomial ring. In this paper we classify the indecomposable singular Soergel bimodules, and prove that it categorifies the Schur algebroid, a natural algebroid generalising the Hecke algebra of a Coxeter system.

  30. Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties
    (with Peter Fiebig), will appear in Annales de l'Institut Fourier.
    A moment graph is a labelled graph which encodes the ``one-skeleton'' of certain algebraic torus actions on varieties. It is an amazing fact (usually referred to as the localisation theorem) that one can recover a lot of cohomological information about a variety from its moment graph. In this paper we show that the moment graph can be used to calculate the stalks of ``parity sheaves'' (see below). We use this result, together with an algebraic result of Fiebig, to deduce the p-smooth locus of Schubert varieties. We also apply this to representation theory and show that moment graphs can be used to calculate the weight spaces of tilting modules.

  31. The geometry of Markov traces
    (with Ben Webster), Duke Mathematical Journal 160, No. 2, 401-419 (2011).
    We show that the Jones-Ocneanu trace on the Hecke algebra of type A evaluated on a Kazhdan-Lusztig basis element is a mixed Poincare polynomial of the B-conjugation equivariant cohomology of the corresponding intersection cohomology complex. This then gives a natural trace on Hecke algebras of finite type. We then show that this trace is equal to a trace defined by Gomi in 2006. This yields a simple geometric proof of Gomi's result, and provides a natural framework in which to interpret his definition. Another useful (and unexpected) biproduct of our investigations is a proof that the Hochschild homology of Soergel bimodules in finite type is free. (This had been observed previous by Rasmussen in Type A, but his proof doesn't generalise.)

  32. Parity sheaves
    (with Daniel Juteau and Carl Maunter), Journal of the AMS 27, No. 4, 1169-1212 (2014).
    You can see a video of Daniel talking about this work in Cambridge and here are some slides of a talk I gave in Durham.
    We introduce a new class of sheaves on certain varieties (the "parity sheaves") which we believe will be fundamental in attempts to use modular perverse sheaves in representation theory. We show that one may prove a decomposition theorem type result for certain maps (which we call "even"), and show the role played by certain intersection forms introduced in work of de Cataldo and Migliorini in determining the stalks of parity sheaves. We also give lots of examples. Probably the most important being that parity sheaves exist on the affine Grassmannian, and (under some moderate assumptions) correspond to tilting modules.

  33. A geometric construction of colored HOMFLYPT homology
    Joint with Ben Webster, to appear in Geometry and Topology.
    You can see a video of a talk I gave about this work in Cambridge.
    This paper continues Ben and my efforts to understand various link homology theories geometrically, in terms of constructible sheaves. We are primarily interested in Khovanov and Rozansky's triply graded HOMFLYPT homology, and a natural first question is what on earth do all the gradings mean?! The crucial point is that, on a non-proper algebraic variety one has a weight filtration before and after pushing to a point, which may be used to construct a triple grading. In this way we obtain a completely geometric construction of HOMFLYPT homology, as well as various "colored" generalisations.

  34. Perverse sheaves and modular representation theory
    This is joint with Daniel Juteau and Carl Maunter, Séminaires et Congrès 24-II (2012), 313-350.
    We give an overview of three applications of perverse sheaves in modular representation theory. The basic idea is to consider sheaves of k-vector spaces on complex algebraic varieties, where k is a field of positive characteristic. The corresponding categories of perverse sheaves behave like (and sometimes are actually equivalent to) categories arising in modular representation theory. Just as is the case for modular representations, these categories are difficult to understand. In order to try to convince the reader of this we give some calculations on nilpotent cones: things are already very interesting in sl_n for n = 2, 3 and 4!

  35. Modular intersection cohomology complexes on flag varieties
    (with an appendix by Tom Braden), Math. Zeitschrift 272, No. 3-4, 697-727 (2012).
    The software and W-graphs referred to in this paper are available here.
    For a fixed field k of positive characteristic almost nothing is known about intersection cohomology complexes on flag varieties with coefficients in k. In this article we present a combinatorial algorithm which, if successful, proves that they ''look the same'' as in characteristic 0. Our algorithm relies on the W-graph for which no general description is known. Thus we can only apply our techniques in small rank. Thanks to results of Soergel, we are able to conclude parts of the Lusztig conjecture on modular representations of reductive groups.
    In the appendix, Tom Braden gives some examples of torsion in the stalks or costalks of intersection cohomology complexes on Schubert varieties in type A7 and D4.

  36. A geometric model for Hochschild homology of Soergel bimodules
    (with Ben Webster), Geometry and Topology 12, No. 2, 1243-1263 (2008).
    Khovanov has constructed a knot invariant in the homotopy category of bigraded modules over a polynomial ring. This involves first constructing a complex of Soergel bimodules and then taking Hochschild homology. In this paper we show that all of this may be interpreted geometrically: each term in the complex may be viewed is the equivariant cohomology of a ''Bott-Samelson'' type space, and the maps in the complex are induced from maps between Bott-Samelson varieties. Using geometric techniques we are also able to give explicit descriptions of the Hochschild homology of certain ''smooth'' Soergel bimodules in type A.

PhD Thesis, Essays, Software etc: