Papers and preprints:
PhD Thesis, Essays, Software etc:
- Lectures on the geometry and
modular representation theory of algebraic groups
Joint with Joshua Ciappara.
These are based on notes (taken by Josh) of some lectures I gave
last August at the Simons Centre in Stony Brook. The subject is the modular representation
theory of algebraic groups. This subject can be difficult to
penetrate. The goal of these notes is to move quickly to the
frontiers of research, with the hope that the interested reader
can orient themselves at the front.
- Kazhdan-Lusztig polynomials and subexpressions
Joint with Nicolas
We refine an idea of Deodhar, whose goal is a counting formula for Kazhdan-Lusztig polynomials. This is a consequence of a simple observation that one can use the solution of Soergel's conjecture to make ambiguities involved in defining certain morphisms between Soergel bimodules in characteristic zero (double leaves) disappear.
- Smith-Treumann theory and the linkage principle
Joint with Simon Riche.
In this paper we apply Treumann's "Smith theory for sheaves" in the context of the Iwahori-Whittaker model of the Satake category. We deduce two results in the representation theory of reductive algebraic groups over fields of positive characteristic: (a) a geometric proof of the linkage principle; (b) a character formula for tilting modules in terms of the p-canonical basis, valid in all blocks and in all characteristics.
- Modular representations and
Report for Current Developments in Mathematics.
This is document has three goals: the first few pages try to convince
the reader that modular representations are fascinating. We then
try to explain the "philosophy of deformations", and connect it
with Lusztig's "philosophy of generations". Finally, we explain
the connection between recent work of Leslie-Lonergan on Treumann-Smith theory and Hazi, on certain
localizations of the Hecke category. Hiding behind all of this are
- A simple character formula
Joint with Simon Riche.
We prove a character formula for simple modules for reductive algebraic groups in characteristic p in terms of the p-canonical basis in the periodic module (which we also define). This provides an analogue of Lusztig's conjecture, which is valid under reasonable bounds on p.
- Parity sheaves and the Hecke category
Report for Rio ICM.
The Hecke category is emerging as a fundamental object in representation theory. We give a motivated introduction to this category in both its geometric
(via parity sheaves) and diagrammatic (generators and relations) incarnations.
We also discuss the p-canonical basis and Koszul duality for the Hecke
- A non-perverse Soergel bimodule in type A
Joint with Nicolas
C. R. Math. Acad. Sci. Paris
355 (2017), no. 8, 853-858.
It is an interesting question as to how non-perverse parity sheaves can be.
One might hope that there are none on type A flag varieties, but this is not the case. This paper gives an example for p = 2.
McNamara has recently found examples for all primes. I suspect that one also has some form of "torsion explosion" for non-perversity,
but can't prove this (despite some effort).
- Koszul duality for Kac-Moody groups and characters of tilting modules
Joint with Pramod
Achar, Shotaro Makisumi and Simon Riche, Journal of the AMS, 32 (2019), no. 1, 261-310.
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.
- Billards and tilting
characters for SL3
14 (2018), 015.
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
- 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.
- The anti-spherical category
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).
- Algebraic representations and constructible sheaves
Notes from my Takagi lecture in Tokyo,
November 2016. Japanese Journal of Math, 12 (2017).
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
Relative hard Lefschetz for Soergel bimodules
Joint with Ben Elias, submitted.
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
- The Hodge theory of the Hecke category
ECM prize lecture. Proceedings of the 7th European Congress of Mathematics.
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".
The Hodge theory of the Decomposition Theorem (after de Cataldo and Migliorini)
Seminaire Bourbaki, No 1115, Astérisque No. 390 (2017)
An attempt at a motivated introduction to de Cataldo and
Migliorini's beautiful Hodge theoretic proof of the Decomposition Theorem.
Tilting modules and the p-canonical basis
Joint with Simon Riche, Astérisque 2018, no. 397.
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.
- On torsion in the intersection cohomology of Schubert varieties
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).
- 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.
- On the character of certain tilting modules
Joint with George Lusztig,
Sci. China Math.
61 (2018), no. 2, 295-298.
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).
- Soergel calculus and Schubert calculus
Joint with Xuhua He.
Bull. Inst. Math. Acad. Sin. (N.S.) 13 (2018), no. 3, 317-350.
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.
Hodge theory of Soergel bimodules
Acta Mathematica, 217 (2016).
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.
- Diagrammatics for Coxeter
groups and their braid groups
Joint with Ben
Elias, Quantum topology, 8 (2017).
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.
- 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!
- Parity sheaves and tilting modules
This is joint with Daniel
Juteau and Carl
Mautner, Ann. Sci. ENS(4) 49 (2016), no. 2, 257-275.
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.
Kazhdan-Lusztig conjectures and shadows of Hodge theory
This is joint with Ben
Elias. It appeared 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.
- 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.
Schubert calculus and torsion explosion
(with an appendix by Kontorovich, McNamara and
Williamson). Journal of the AMS, 30 (2017).
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.
Joint with Ben
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.
- On cubes of
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.
The Hodge theory of Soergel bimodules
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
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.
On an analogue of the James conjecture.
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.
Modular Koszul duality
Simon Riche and
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
Characteristic cycles and decomposition numbers
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
- Standard objects in
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.
- Kumar's criterion modulo p
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).
- 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.
- Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties
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
- The geometry of Markov traces
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.)
- Parity sheaves
Juteau and Carl
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.
A geometric construction of colored HOMFLYPT homology
Ben Webster, to
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"
- Perverse sheaves and
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!
- Modular intersection
cohomology complexes on flag varieties
(with an appendix by
272, No. 3-4, 697-727 (2012).
The software and W-graphs referred to in this paper are available
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.
geometric model for Hochschild homology of Soergel bimodules
Geometry and Topology 12, No. 2, 1243-1263 (2008).
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.
- Singular Soergel bimodules
This is my PhD thesis. It defines singular Soergel bimodules in a
general framework and classifies the indecomposable bimodules (generalising
results of Soergel). Soergel bimodules and their singular variants have
many applications (including the study of category O, equivariant
perverse sheaves and knots) and the list will probably grow in the
future. One exciting possibility is a tensor category of bimodules
which is equivalent to the representation ring of an (adjoint)
the Kazhdan-Lusztig basis of the Hecke
Algebra is a Cellular Basis
My honours essay at the University of Sydney supervised by Gus Lehrer.
In this essay I prove that the standard Kazhdan-Lusztig basis of the
Hecke algebra of the symmetric group is a cellular basis in the sense
of Graham and Lehrer. This involves lots of combinatorics centred
around the Robinson-Schensted correspondence. One of the corollaries of
cellularity is the fact that the cell
representations are irreducible.
Fundamental Example of
Bernstein and Lunts
have tried to write a motivated introduction to the equivariant
derived category, as well as provide the details of a proof of the
so-called "fundamental example". This relates the equivariant
cohomology of a torus stable subvariety of affine space to the
intersection cohomology of a projective variety (a quotient) and the
stalk at 0. The proof of the fundamental example is complete, except
assumption of the hard Lefschetz theorem for intersection cohomology.
This is the product of a project with David Kohel
at the University of
We have written software for Magma which calculates the formal group of
the Jacobian of a genus 2 curve.
Introduction to the
The product of a vacation scholarship at the University of New South
Wales under the supervision of Jie Du.
introduce the BMW-algebra and calculate some canonical bases in