
Research Interests
The representation theory of the symmetric groups, and the closely related cyclotomic Hecke algebras of type A, was transformed when Brundan and Kleshchev's discovered of a grading on these algebras following work of KhovanovLauda and Rouquier. The graded theory is harder than the "classical" approach to this subject but it reveals deeper underlying features of the representation theory which we could not see before. The underlying problems are still the same  we want to compute the (graded) dimensions and the (graded) decomposition numbers of these algebras  but there is now more structure to work with. There are indications that this new perspective may furnish us with the tools to finally answer these questions. This theory is intimately connected with the representation theory of affine Hecke algebras and quantum groups; there are also ramifications for the representation theory of the symmetric groups and finite reductive groups.
Other active interests include:
 Cyclotomic Hecke algebras, complex reflection groups and their braid groups.
 The qSchur algebras and cyclotomic qSchur algebras.
 Algebraic combinatorics, especially that associated with tableaux, Fock spaces and canonical bases.
 The theory of (graded) cellular algebras.
 Affine Hecke algebras.
 Quantum groups, canonical bases, and crystal graphs.
 KazhdanLusztig polynomials and cell representations.
 Coxeter groups and groups of Lie type, and their representation theory.
I am a member of the Algebra research group.
Publications
Alex and Erika (joint with M.)
Preprints  or see the arXiv and MathSciNet

Cyclotomic quiver Hecke algebras of type A
Abstract
This is a DRAFT chapter based on a series of lectures that I gave at the National University of Singapore in April 2013. They survey the representation theory of the cyclotomic Hecke algebras of type A with an emphasis on understanding the KLR grading and the connections between the "classical" ungraded representation theory and the rapidly emerging graded theory. They are fairly selfcontained and they try to give a leisurely introduction to these algebras, with many examples and calculations don't appear elsewhere. We make extensive use of the interactions between the ungraded and graded representation thory and try to explain what the grading gives us which we didn't have before. Combinatorics and cellular algebra techniques are used extensively, with a few results from geometry and 2representation theory being quoted. Highlights include a complete description of the semisimple KLR algebras of type A using just the KLR relations, extensive discussion about graded Specht modules, a proof of the ArikiBrundanKleshchev graded categorification theorem using the graded branchung rules, a cellular algebra approach to adjustment matrices and a (possibly optimistic) conjecture for the graded dimensions of the simple modules. . 
Seminormal forms and cyclotomic quiver Hecke algebras of type A.
Abstract
This paper shows that the cyclotomic quiver Hecke algebras of type~A, and the gradings on these algebras, are intimately related to the classical seminormal forms. We start by classifying all seminormal bases and then give an explicit ``integral'' closed formula for the Gram determinants of the Specht modules in terms of the combinatorics which utilizes the KLR gradings. We then use seminormal forms to give a deformation of the KLR algebras of type~A. This makes it possible to study the cyclotomic quiver Hecke algebras in terms of the semisimple representation theory and seminormal forms. As an application we construct a new distinguished graded cellular basis of the cyclotomic KLR algebras of type A. With Jun Hu. 
Cyclotomic quiver Schur algebras I: linear quivers.
Abstract
We define a graded quasihereditary covering for the cyclotomic quiver Hecke algebras R of type A when e=0 (the linear quiver) or e≥ n. We show that these algebras are quasihereditary graded cellular algebras by giving explicit homogeneous bases for them. When e=0 we show that the KLR grading on the quiver Hecke algebras is compatible with the gradings on parabolic category O previously introduced in the works of Beilinson, Ginzburg and Soergel and Backelin. As a consequence, we show that when e=0 our graded Schur algebras are Koszul over field of characteristic zero. Finally, we give an LLTlike algorithm for computing the graded decomposition numbers of the quiver Schur algebras in characteristic zero when e=0. With Jun Hu.
Book

IwahoriHecke algebras and Schur algebras of the symmetric group,
Abstract
This book gives a fully selfcontained introduction to the modular representation theory of the IwahoriHecke algebras of the symmetric groups and of the associated qSchur algebras. The main landmarks that we reach are the classification of the simple modules and the blocks of these algebras. Along the way the theory of cellular algebras is developed and an analogue of Jantzen's sum formula is proved. Combinatorial motifs pervade the text, with standard and semistandard tableaux being used to index explicit (cellular) bases. These bases are particularly well adapted to the representation theory. This results in clean and elegant proofs of most of the basic results about these algebras. In the final chapter we give a survey of some recent and exciting developements in the field and discuss open problems.
The book should be accessible to advanced graduate students and also useful to researchers in the field. University lecture series, 15, Amer. Math. Soc., 1999. Errata
Papers
The versions on the arXiv may differ slightly from the published articles. Cyclotomic CarterPayne homomorphisms,
Abstract
We construct a new family of homomorphisms between (graded) Specht modules of the quiver Hecke algebras of type~$A$. These maps have many similarities with the homomorphisms constructed by Carter and Payne in the special case of the symmetric groups, although the maps that we obtain are both more and less general than these. Representation Theory, to appear. With Sinéad Lyle. 
Blocks of the truncated qSchur algebras of type A,
Abstract
This paper classifies the blocks of the truncated qSchur algebras of type A which have as weight poset an arbitrary cosaturated set of partitions. "Algebraic and Combinatorial Approaches to Representation Theory", Cont. Math., 602 (2013), 123141, Amer. Math. Soc. With Marcos Soriano.  Universal graded Specht modules for cyclotomic Hecke algebras,
Abstract
The graded Specht module S^{λ} for a cyclotomic Hecke algebra comes with a distinguished generating vector z^{λ} in S^{λ}, which can be thought of as a ``highest weight vector of weight λ''. This paper describes the defining relations for the Specht module S^{λ} as a graded module generated by z^{λ}. The first three relations say precisely what it means for z^{λ} to be a highest weight vector of weight λ. The remaining relations are homogeneous analogues of the classical Garnir relations The homogeneous Garnir relations, which are simpler than the classical ones, are associated with a remarkable family of homogeneous operators on the Specht module which satisfy the braid relations. Proc. Lond. Math. Soc., 105 (2012), 12451289. With Alexander Kleshchev and Arun Ram. 
Morita equivalences of cyclotomic Hecke algebras of type G(r,p,n) II:
the (ε,q)separated case,
Abstract
The paper studies the modular representation theory of the cyclotomic Hecke algebras of type G(r,p,n) with (ε,q)separated parameters. We show that the decomposition numbers of these algebras are completely determined by the decomposition matrices of related cyclotomic Hecke algebras of type G(s,1,m), where 1≤ s≤ r and 1≤ m≤ n. Furthermore, the proof gives an explicit algorithm for computing these decomposition numbers meaning that the decomposition matrices of these algebras are now known in principle.
In proving these results, we develop a Specht module theory for these algebras, explicitly construct their simple modules and introduce and study analogues of the cyclotomic Schur algebras of type G(r,p,n) when the parameters are (ε,q)separated.
The main results of the paper rest upon two Morita equivalences: the first reduces the calculation of all decomposition numbers to the case of the lsplittable decomposition numbers} and the second Morita equivalence allows us to compute these decomposition numbers using an analogue of the cyclotomic Schur algebras for the Hecke algebras of type G(r,p,n). Proc. Lond. Math. Soc., 104 (2012), 865926. With Jun Hu.  Graded induction for Specht modules,
Abstract
Recently Brundan, Kleshchev and Wang introduced a Zgrading on the Specht modules of the degenerate and nondegenerate cyclotomic Hecke algebras of type G(l,1,n). In this paper we show that induced Specht modules have an explicit filtration by shifts of graded Specht modules. This proves a conjecture of Brundan, Kleshchev and Wang. International Mathematics Research Notices, 2012 (2012), 12301263. With Jun Hu. 
CarterPayne homomorphisms and Jantzen filtrations,
Abstract
We prove a qanalogue of the CarterPayne theorem in the case where the differences between the parts of the partitions are sufficiently large. We identify a layer of the Jantzen filtration which contains the image of these CarterPayne homomorphisms and we show how these homomorphisms compose. J. Alg. Comb., 32 (2010), 417457. With Sinéad Lyle. 
Graded cellular bases for the cyclotomic KhovanovLaudaRouquier algebras of type A,
Abstract
This paper constructs an explicit homogeneous cellular basis for the cyclotomic KhovanovLaudaRouquier algebras of type A over a field.Adv. Math., 225 (2010), 598642. With Jun Hu. 
A Specht filtration of an induced Specht module,
Abstract
Let H_{n} be a (degenerate or nondegenerate) Hecke algebra of type G(l,1,n), defined over a commutative ring R with one, and let S(μ) be a Specht module for H_{n}. This paper shows that the induced Specht module S(μ)⊗_{Hn}H_{n+1} has an explicit Specht filtration. J. Algebra, 322 (2009), 893902. 
Morita equivalences of the cyclotomic Hecke algebras of type G(r,p,n),
Abstract
We prove a Morita reduction theorem for the cyclotomic Hecke algebras H_{r,p,n}(q,Q) of type G(r,p,n) with p>1 and n≥ 3. As a consequence, we show that computing the decomposition numbers of H_{r,p,n}(q,Q) reduces to computing the p'splittable decomposition numbers of the cyclotomic Hecke algebras H_{r',p',n'}(q,Q'), where 1≤ r'≤ r, 1≤ n'≤ n, p' divides p and where the parameters Q' are contained in a single (ε',q)orbit and ε' is a primitive p'th root of unity. J. Reine Angew. Math., 628 (2009), 169195. With Jun Hu. 
Cyclotomic Solomon Algebras,
Abstract
This paper introduces an analogue of the Solomon descent algebra for the complex reflection groups of type G(r,1,n). As with the Solomon descent algebra, our algebra has a basis given by sums of `distinguished' coset representatives for certain `reflection subgroups'. We explicitly describe the structure constants with respect to this basis and show that they are polynomials in r This allows us to define a deformation, or qanalogue, of these algebras which depends on a parameter q We determine the irreducible representations of all of these algebras and give a basis for their radicals. Finally, we show that the direct sum of cyclotomic Solomon algebras is canonically isomorphic to a concatenation Hopf algebra. Adv. Math., 219 (2008), 450487. With Rosa Orellana. 
Seminormal forms and Gram determinants for cellular algebras,
Abstract
This paper develops an abstract framework for constructing ``seminormal forms'' for cellular algebras. That is, given a cellular Ralgebra A which is equipped with a family of JMelements we give a general technique for constructing orthogonal bases for A, and for all of its irreducible representations, when the JMelements separate A. The seminormal forms for A are defined over the field of fractions of R. Significantly, we show that the Gram determinant of each irreducible Amodule is equal to a product of certain structure constants coming from the seminormal basis of A. In the nonseparated case we use our seminormal forms to give an explicit basis for a block decomposition of A.
In the appendix Marcos shows, under much weaker assumptions, that the theory of seminormal forms rests, ultimately, on the CayleyHamilton theorem. J. Reine Angew. Math., 619 (2008), 141173. With an appendix by Marcos Soriano. 
Blocks of cyclotomic Hecke algebras,
Abstract
This paper classifies the blocks of the cyclotomic Hecke algebras of type G(r,1,n) over an arbitrary field. Rather than working with the Hecke algebras directly we work instead with the cyclotomic Schur algebras. The advantage of these algebras is that the cyclotomic Jantzen sum formula gives an easy combinatorial characterization of the blocks of the cyclotomic Schur algebras. We obtain an explicit description of the blocks by analyzing the combinatorics of `Jantzen equivalence'. Adv. Math., 216 (2007), 854878. With Sinéad Lyle. 
Cyclotomic NazarovWenzl algebras,
Abstract
Nazarov introduced an infinite dimensional algebra, which he called the affine Wenzl algebra, in his study of the Brauer algebras. In this paper we study certain `cyclotomic quotients' of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank r^{n}(2n1)!! (when Ω is admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic NazarovWenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Wenzl algebra (when Ω is admissible). Nagoya Math. J., 182 (2006), 47134. With Susumu Ariki and Hebing Rui. (Special issue in honour of George Lusztig.) 
Rouquier blocks,
Abstract
This paper investigates the Rouquier blocks of the Hecke algebras of the symmetric groups and the Rouquier blocks of the qSchur algebras. We first give an algorithm for computing the decomposition numbers of these blocks in the `abelian defect group case' and then use this algorithm to explicitly compute the decomposition numbers in a Rouquier block. For fields of characteristic zero, or when q=1 these results are known; significantly, our results also hold for fields of positive characteristic with q≠1. We also discuss the Rouquier blocks in the `nonabelian defect group' case. Finally, we apply these results to show that certain Specht modules are irreducible. Math. Z., 252 (2006), 511531. With Gordon James and Sinéad Lyle. 
Row and column removal theorems for homomorphisms of Specht
modules and Weyl modules,
Abstract
We prove a qanalogue of the row and column removal theorems for homomorphisms between Specht modules proved by Fayers and the first author. These results can be considered as complements to James and Donkin's row and column removal theorems for decomposition numbers of the symmetric and general linear groups. In this paper we consider homomorphisms between the Specht modules of the Hecke algebras of type A and between the Weyl modules of the qSchur algebra. J. Alg. Comb., 22 (2005), 151179. With Sinéad Lyle. 
Elementary divisors of Specht modules,
Abstract
This paper shows that the Gram matrix for the Specht module S(μ) for the IwahoriHecke algebra is diagonalizable if and only if the Gram matrix of S(μ') is diagonalizable and we show that the elementary divisors of these matrices differ by the product of the qhooks lengths of μ. Finally, we determine the elementary divisors of the Gram matrices of the hook partitions. European J. Combinatorics, 26 (2005), 943964. With Matthias Künzer. (Special issue showcasing algebraic combinatorics.) 
Matrix units and generic degrees for the ArikiKoike
algebras,
Abstract
This paper uses the Murphy basis of the ArikiKoike algebras to explicitly construct a complete set of primitive idempotents when these algebras are semisimple and q≠1. As a consequence, we obtain an explicit Wedderburn basis for the ArikiKoike algebras. Finally, we use these idempotents to compute the generic degrees of the ArikiKoike algebras. J. Algebra, 281 (2004), 695730. 
Symmetric group blocks of small defect,
Abstract
This paper attempts to compute the decomposition numbers of the Hecke algebras algebras of type A for blocks of weight 3. J. Algebra, 279 (2004), 566612. With Gordon James. 
Hecke algebras with a finite number of indecomposable modules,
Abstract
This paper is a survey of the recent progress in determining the representation type of the Hecke algebras of finite Weyl groups. Representation theory of algebraic groups and quantum groups , Adv. Studies Pure Math., 40 (2004), 1725. With Susumu Ariki. 
The representation theory of the ArikiKoike and
cyclotomic qSchur algebras,
Abstract
This paper is an extended review of the representation theory of Hecke algebras of type G(r,1,n) and the associated cyclotomic Schur algebras. It discusses developments in this area up to 2001. Representation theory of algebraic groups and quantum groups, Adv. Studies Pure Math., 40 (2004), 261320. 
The representation type of Hecke algebras of type B,
Abstract
This paper determines the representation type of the IwahoriHecke algebras of type B when q≠±1. In particular, we show that a single parameter nonsemisimple IwahoriHecke algebra of type B has finite representation type if and only if q is a simple root of the Poincarée polynomial, confirming a conjecture of Uno's. Adv. Math., 181 (2004), 134159. With Susumu Ariki. 
Tilting modules for cyclotomic Schur algebras,
Abstract
This paper investigates the tilting modules of the cyclotomic qSchur algebras, the Young modules of the ArikiKoike algebras, and the interconnections between them. The main tools used to understand the tilting modules are contragredient duality, and the Specht filtrations and dual Specht filtrations of certain permutation modules. Surprisingly, Weyl filtrations  which are in general more powerful than Specht filtrations  play only a secondary role. We also develop a theory of Young modules for the ArikiKoike algebras; as far as we know this is new even for Coxeter groups of type B. J. Reine Angew. Math., 562 (2003), 137169. 
Equating decomposition numbers for different primes,
Abstract
This paper shows that certain decomposition numbers for the IwahoriHecke algebras of the symmetric groups and the qSchur algebras at different roots of unity in characteristic zero are equal. To prove our results we first establish the corresponding theorem for the canonical basis of the level one Fock space and then apply deep results of Ariki and Varagnolo and Vasserot. J. Algebra, 258 (2002), 599614. With Gordon James. 
Morita equivalences of ArikiKoike algebras,
Abstract
We prove that every ArikiKoike algebra is Morita equivalent to a direct sum of tensor products of smaller ArikiKoike algebras which have qconnected parameter sets. A similar result is proved for the cyclotomic qSchur algebras. Combining our results with work of Ariki and Uglov, the decomposition numbers for the ArikiKoike algebras defined over fields of characteristic zero are now known in principle. Math. Z., 240 (2002), 579610. With Richard Dipper, 
The Jantzen sum formula for cyclotomic qSchur algebras,
Abstract
The cyclotomic qSchur algebra was introduced by Dipper, James and Mathas, in order to provide a new tool for studying the ArikiKoike algebra. We prove an analogue of Jantzen's sum formula for the cyclotomic qSchur algebra. Among the applications is a criterion for certain Specht modules of the ArikiKoike algebras to be irreducible. Trans. Amer. Math. Soc., 352 (2000), 53815404. With Gordon James. 
The number of simple modules of the Hecke algebras of type G(r,1,n),
Abstract
This paper classifies the simple modules of the cyclotomic Hecke algebras of type G(r,1,n) and the affine Hecke algebras of type A in arbitrary characteristic. We do this by first showing that the simple modules of the cyclotomic Hecke algebras are indexed by the set of `Kleshchev multipartitions'. Math. Zeitschrift, 233 (2000), 601623. With Susumu Ariki.  The irreducible Specht modules in characteristic 2,
Abstract
In this paper we classify the irreducible Specht modules over a field of caharacteristic 2. In particular, we prove a conjecture of the first author from 1978 which says that S(2,2) is the only irreducible Specht module which is indexed by a partition which is not 2restricted and not 2regular. Bull. Lond. Math. Soc., 31 (1999), 457462. With Gordon James. 
The Murphy operators and the centre of the IwahoriHecke algebras of type A,
Abstract
In this paper we introduce a family of polynomials indexed by pairs of partitions and show that if these polynomials are selforthogonal then the centre of the IwahoriHecke algebra of the symmetric group is precisely the set of symmetric polynomials in the Murphy operators. J. Alg. Comb., 9 (1999), 295313.  Cyclotomic qSchur algebras,
Abstract
This paper introduces the Cyclotomic qSchur algebra, which is a quasihereditary cover of the Hecke algebra of the complex reflection group of type G(r,1,n). The cyclotomic qSchur algebras are a natural generalization of the qSchur algebras. We construct a cellular basis for these algebras, a complete set of simple modules and show that they are quasihereditary algebras. Math. Zeitschrift, 229 (1998), 385416. With Richard Dipper and Gordon James. 
Symmetric cyclotomic Hecke algebras,
Abstract
In this paper we prove that the generic cyclotomic Hecke algebras for imprimitive complex reflection groups are symmetric over any ring containing inverses of the parameters. For this we show that the determinant of the Gram matrix of a certain canonical symmetrizing form introduced by Bremke and Malle is a unit in any such ring. On the way we show that the ArikiKoike bases of these algebras are also quasisymmetric. J. Algebra, 205 (1998), 275293. With Gunter Malle.  The (Q,q)Schur Algebra,
Abstract
In this paper we use the Hecke algebra of type B to define a new algebra S which is an analogue of the qSchur algebra. We construct Weyl modules for S and obtain, as factor modules, a family of irreducible Smodules over any field. Proc. Lond. Math. Soc., 77 (1998), 327361. With Richard Dipper and Gordon James. 
Simple modules of ArikiKoike algebras,
Abstract
In this note we classify the simple modules of the ArikiKoike algebras when q=1 and also describe the classification of the simple modules for those algebras, together with the underlying computation of the canonical bases of the affine quantum group U. Proc. Pure Symp. Math., 63 (1998), 383396. 
A qanalogue of the JantzenSchaper theorem,
Abstract
In this paper we prove an analogue of Jantzen's sum formula for the qWeyl modules of the qSchur algebra and, as a consequence, derive the analogue of Schaper's theorem for the qSpecht modules of the Hecke algebras of type A. We apply these results to classify the irreducible qWeyl modules and the irreducible (eregular) qSpecht modules, defined over any field. In turn, this allows us to identify all of the ordinary irreducible representations of the finite general linear group GL_n(q) which remain irreducible modulo a prime p not dividing~q. Proc. Lond. Math. Soc., 74 (1997), 241274. With Gordon James.  Hecke algebras of type A at q=1,
Abstract
In this paper we study the decomposition matrices of the Hecke algebras of type~\bf A with q=1 over a field of characteristic 0. We give explicit formulae for the columns of the decomposition matrices indexed by all 2regular partitions with 1 or 2 parts and an algorithm for calculating the columns of the decomposition matrix indexed by partitions with 3 parts. Combining these results we find all of the rows of the decomposition matrices which are indexed by partitions with at most four parts. All this is accomplished by means of a more general theory which begins by showing that the decomposition numbers in the columns of the decomposition matrices indexed by 2regular partitions with `enormous 2cores' are LittlewoordRichardson coefficients. J. Algebra, 184 (1996), 102158. With Gordon James.  On the left cell representations of IwahoriHecke
algebras of finite Coxeter groups,
Abstract
On the left cell representations of IwahoriHecke algebras of finite Coxeter groups In this paper we investigate the left cell representations of the IwahoriHecke algebras associated to a finite Coxeter group W. Our main result shows that T_{ω}, where ω is the element of longest length in W, acts essentially as an involution upon the canonical bases of a cell representation. We describe some properties of this involution, use it to further describe the left cells, and finally show how to realize each cell representation as a submodule of H. Our results rely upon certain positivity properties of the structure constants of the KazhdanLusztig bases of the Hecke algebra and so have not yet been shown to apply to all finite Coxeter groups. J. London Math. Soc., 54 (1996), 475488. 
Some generic representations, Wgraphs, and duality,
Abstract
This paper begins by generalising the notion of `Wgraph' to show that the Wgraph data determines not one but four closely related representations of the generic Hecke algebra of an arbitrary Coxeter group. Canonical `KazhdanLusztig bases' are then constructed for several families of ideals inside the Hecke algebra of a finite Coxeter system (W,S). In particular for each J⊆ S we construct the left cell module corresponding to the `top' left cell C^{J} as a submodule of the Hecke algebra and give a precise description of its canonical basis. In the case of the symmetric group it is shown that every irreducible representation arises as a top cell representation. Finally analogues of the representations considered are discussed for the case of an infinite Coxeter group. J. Algebra, 170 (1994), 322353.  A qanalogue of the Coxeter complex,
Abstract
In this paper a qanalogue of the Coxeter complex of a finite Coxeter group W is constructed for the (generic) Hecke algebra Image associated to W. It is shown that the homology of this chain complex, together with that of its truncations, vanishes away from top dimension. The remainder of the paper investigates the representations of Image afforded by the top homology modules of these complexes. In particular necessary and sufficient conditions are given for a specialisation of the Hecke algebra to decompose into a direct sum of its "truncation" representations. J. Algebra, 164 (1994), 831848.
Code
I am one of the developers for the Sagecombinat group, an opensource platform for computer calculations in algebraic combinatorics, which is part of the Sage project.I have contributed to the Gap 3 project, including to chevie. (I have to confess that I never liked Gap 4 so I have never used it. As Gap4 is not backwardly compatible my code does not run on it.) The following programs are included in Gap 3.4.4, which can be downloaded from Jean Michel's Gap 3 distribution.
 Specht A Gap package for calculating decomposition matrices of Hecke algebras of type A.
 Murphy These programs implement the Murphy basis of the IwahoriHecke algebra of the symmetric group using Chevie, version 3.4.
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