Andrew Mathas

I am a professor of pure mathematics and an ARC Australian professorial fellow in the School of Mathematics and Statistics at the University of Sydney.

Andrew Mathas School of Mathematics and Statistics F07 University of Sydney, NSW 2006. Australia Office    Carslaw 718 +61 2 9351 6058 (W) +61 2 9351 4534 (Fax)

Research Interests

My main area of interest and expertise is the representation theory of Coxeter groups, Iwahori-Hecke algebras and Schur algebras. Recently my work has focused on the modular representation theory of the Ariki-Koike algebras and the associated cyclotomic q-Schur algebras. 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 q-Schur algebras and cyclotomic q-Schur algebras.
• Combinatorics of symmetric groups and Hecke algebras.
• The theory of (graded) cellular algebras.
• Affine Hecke algebras.
• Quantum groups, canonical bases, and crystal graphs.
• Kazhdan-Lusztig 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 Schur algebras I: linear quivers. Abstract
  We define a graded quasi-hereditary 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 quasi-hereditary 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 LLT-like algorithm for computing the graded decomposition numbers of the quiver Schur algebras in characteristic zero when e=0. With Jun Hu.
• 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., to appear. 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 l-splittable 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., to appear. With Jun Hu.

Book

• Iwahori-Hecke algebras and Schur algebras of the symmetric group, Abstract
  This book gives a fully self-contained introduction to the modular representation theory of the Iwahori-Hecke algebras of the symmetric groups and of the associated q-Schur 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

• Graded induction for Specht modules, Abstract
  Recently Brundan, Kleshchev and Wang introduced a Z-grading on the Specht modules of the degenerate and non-degenerate 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), 1230-1263. With Jun Hu.
• Carter-Payne homomorphisms and Jantzen filtrations, Abstract
  We prove a q-analogue of the Carter-Payne 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 Carter-Payne homomorphisms and we show how these homomorphisms compose. J. Alg. Comb., 32 (2010), 417-457. With Sinéad Lyle.
• Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A, Abstract
  This paper constructs an explicit homogeneous cellular basis for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A over a field.Adv. Math., 225 (2010), 598-642. With Jun Hu.
• A Specht filtration of an induced Specht module, Abstract
  Let H_n be a (degenerate or non-degenerate) 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)⊗_HnH_n+1 has an explicit Specht filtration. J. Algebra, 322 (2009), 893-902.
• 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), 169-195. 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), 450-487. 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 R-algebra A which is equipped with a family of JM-elements we give a general technique for constructing orthogonal bases for A, and for all of its irreducible representations, when the JM-elements 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 A-module is equal to a product of certain structure constants coming from the seminormal basis of A. In the non-separated 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 Cayley-Hamilton theorem. J. Reine Angew. Math., 619 (2008), 141-173. 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), 854-878. With Sinéad Lyle.
• Cyclotomic Nazarov-Wenzl 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 rⁿ(2n-1)!! (when Ω is admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov-Wenzl 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), 47-134. 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 q-Schur 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 `non-abelian defect group' case. Finally, we apply these results to show that certain Specht modules are irreducible. Math. Z., 252 (2006), 511-531. With Gordon James and Sinéad Lyle.
• Row and column removal theorems for homomorphisms of Specht modules and Weyl modules, Abstract
  We prove a q-analogue 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 q-Schur algebra. J. Alg. Comb., 22 (2005), 151-179. With Sinéad Lyle.
• Elementary divisors of Specht modules, Abstract
  This paper shows that the Gram matrix for the Specht module S(μ) for the Iwahori-Hecke 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 q-hooks lengths of μ. Finally, we determine the elementary divisors of the Gram matrices of the hook partitions. European J. Combinatorics, 26 (2005), 943-964. With Matthias Künzer. (Special issue showcasing algebraic combinatorics.)
• Matrix units and generic degrees for the Ariki-Koike algebras, Abstract
  This paper uses the Murphy basis of the Ariki-Koike 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 Ariki-Koike algebras. Finally, we use these idempotents to compute the generic degrees of the Ariki-Koike algebras. J. Algebra, 281 (2004), 695-730.
• 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), 566-612. 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), 17-25. With Susumu Ariki.
• The representation theory of the Ariki-Koike and cyclotomic q-Schur 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), 261-320.
• The representation type of Hecke algebras of type B, Abstract
  This paper determines the representation type of the Iwahori-Hecke algebras of type B when q≠±1. In particular, we show that a single parameter non-semisimple Iwahori-Hecke 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), 134-159. With Susumu Ariki.
• Tilting modules for cyclotomic Schur algebras, Abstract
  This paper investigates the tilting modules of the cyclotomic q-Schur algebras, the Young modules of the Ariki-Koike 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 Ariki-Koike algebras; as far as we know this is new even for Coxeter groups of type B. J. Reine Angew. Math., 562 (2003), 137-169.
• Equating decomposition numbers for different primes, Abstract
  This paper shows that certain decomposition numbers for the Iwahori-Hecke algebras of the symmetric groups and the q-Schur 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), 599-614. With Gordon James.
• Morita equivalences of Ariki-Koike algebras, Abstract
  We prove that every Ariki--Koike algebra is Morita equivalent to a direct sum of tensor products of smaller Ariki--Koike algebras which have q-connected parameter sets. A similar result is proved for the cyclotomic q-Schur algebras. Combining our results with work of Ariki and Uglov, the decomposition numbers for the Ariki--Koike algebras defined over fields of characteristic zero are now known in principle. Math. Z., 240 (2002), 579-610. With Richard Dipper,
• The Jantzen sum formula for cyclotomic q-Schur algebras, Abstract
  The cyclotomic q-Schur algebra was introduced by Dipper, James and Mathas, in order to provide a new tool for studying the Ariki-Koike algebra. We prove an analogue of Jantzen's sum formula for the cyclotomic q-Schur algebra. Among the applications is a criterion for certain Specht modules of the Ariki-Koike algebras to be irreducible. Trans. Amer. Math. Soc., 352 (2000), 5381-5404. 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), 601-623. 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 2-restricted and not 2-regular. Bull. Lond. Math. Soc., 31 (1999), 457-462. With Gordon James.
• The Murphy operators and the centre of the Iwahori-Hecke 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 self--orthogonal then the centre of the Iwahori--Hecke algebra of the symmetric group is precisely the set of symmetric polynomials in the Murphy operators. J. Alg. Comb., 9 (1999), 295-313.
• Cyclotomic q-Schur algebras, Abstract
  This paper introduces the Cyclotomic q-Schur algebra, which is a quasi-hereditary cover of the Hecke algebra of the complex reflection group of type G(r,1,n). The cyclotomic q-Schur algebras are a natural generalization of the q-Schur algebras. We construct a cellular basis for these algebras, a complete set of simple modules and show that they are quasi-hereditary algebras. Math. Zeitschrift, 229 (1998), 385-416. 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 Ariki-Koike bases of these algebras are also quasi-symmetric. J. Algebra, 205 (1998), 275-293. 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 q-Schur algebra. We construct Weyl modules for S and obtain, as factor modules, a family of irreducible S-modules over any field. Proc. Lond. Math. Soc., 77 (1998), 327-361. With Richard Dipper and Gordon James.
• Simple modules of Ariki-Koike algebras, Abstract
  In this note we classify the simple modules of the Ariki--Koike 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), 383-396.
• A q-analogue of the Jantzen-Schaper theorem, Abstract
  In this paper we prove an analogue of Jantzen's sum formula for the q-Weyl modules of the q-Schur algebra and, as a consequence, derive the analogue of Schaper's theorem for the q-Specht modules of the Hecke algebras of type A. We apply these results to classify the irreducible q-Weyl modules and the irreducible (e-regular) q-Specht 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), 241-274. 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 2-regular 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 2-regular partitions with `enormous 2-cores' are Littlewoord-Richardson coefficients. J. Algebra, 184 (1996), 102-158. With Gordon James.
• On the left cell representations of Iwahori--Hecke algebras of finite Coxeter groups, Abstract
  On the left cell representations of Iwahori-Hecke algebras of finite Coxeter groups In this paper we investigate the left cell representations of the Iwahori-Hecke 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 Kazhdan-Lusztig 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), 475-488.
• Some generic representations, W-graphs, and duality, Abstract
  This paper begins by generalising the notion of `W--graph' to show that the W-graph data determines not one but four closely related representations of the generic Hecke algebra of an arbitrary Coxeter group. Canonical `Kazhdan-Lusztig 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), 322-353.
• A q-analogue of the Coxeter complex, Abstract
  In this paper a q-analogue 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), 831-848.

Gap packages

• Specht A Gap package for calculating decomposition matrices of Hecke algebras of type A.
• Murphy These programs implement the Murphy basis of the Iwahori-Hecke algebra of the symmetric group using Chevie, version 3.4.

Some mathematics links

• Abstract algebra on line
• Some related mathematicians
• Women Mathematicians
• Young Mathematicians Network
• Group Pub Forum
• FDList
• Mathematics servers
• TeX

Mathematics preprint archives

• Groups, Representations and Cohomology
• Recent preprints in representation theory
• Mathematics ArXiv (Adelaide)
  (see also the UC Davis site)
• eprint archive

If I were a Springer-Verlag Graduate Text in Mathematics, I would be J.-P. Serre's Linear Representations of Finite Groups. My creator is a Professor at the College de France. He has previously published a number of books, including Groupes Algebriques et Corps de Classes, Corps Locaux, and Cours d'Arithmetique (A Course in Arithmetic, published by Springer-Verlag as Vol. 7 in the Graduate Texts in Mathematics).

Which Springer GTM would you be? The Springer GTM Test