Andrew Mathas

The University of Sydney · School of Mathematics and Statistics

Weighted KLRW algebras are diagram algebras generalizing KLR algebras. This paper undertakes a systematic study of these algebras culminating in the construction of homogeneous affine cellular bases in affine types A and C , which immediately gives cellular bases for the cyclotomic quotients of these algebras. In addition, we construct subdivision...

This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer’s cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a “cellular algebra Clifford theory” for the skew cellular algebras that arise as fixed point subalgebras of cellular...

This paper initiates a systematic study of the cyclotomic KLR algebras of affine types $A$ and $C$. We start by introducing a graded deformation of these algebras and the constructing all of the irreducible representations of the deformed cyclotomic KLR algebras using content systems and a generalisation of the Young's seminormal forms for the symm...

This paper proves a “positive” Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type A and uses it to obtain new bounds on decomposition numbers. Quite remarkably, our results are proved entirely inside the cyclotomic Hecke algebras. Our positive sum formula shows that, in the Grothendieck group, the Jantzen sum formul...

This paper constructs homogeneous affine sandwich cellular bases of weighted KLRW algebras in types $B$, $A^{(2)}$, $D^{(2)}$ and finite subquivers. Our construction immediately gives homogeneous sandwich cellular bases for the finite dimensional quotients of these algebras. Since weighted KLRW algebras generalize KLR algebras, we also obtain the b...

Weighted KLRW algebras are diagram algebras generalizing KLR algebras. This paper undertakes a systematic study of these algebras culminating in the construction of homogeneous affine cellular bases in finite or affine types A and C, which immediately gives cellular bases for the cyclotomic quotients of these algebras. In addition, we construct sub...

We prove a ``positive'' Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type~$A$. That is, in the Grothendieck group, we show that the sum of the pieces of the Jantzen filtration is equal to an explicit non-negative linear combination of modules $E^\nu_{f,e}$, which are modular reductions of simple modules for closely...

This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer's cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a ``cellular algebra Clifford theory'' for the skew cellular algebras that arise as fixed point subalgebras of cellul...

This paper computes the irreducible characters of the alternating Hecke algebras, which are deformations of the group algebras of the alternating groups. More precisely, we compute the values of the irreducible characters of the semisimple alternating Hecke algebras on a set of elements indexed by minimal length conjugacy class representatives and...

The main result of this paper shows that, over large enough fields of characteristic different from $2$, the alternating Hecke algebras are $\mathbb{Z}$-graded algebras that are isomorphic to fixed-point subalgebras of the quiver Hecke algebra of the symmetric group $\mathfrak{S}_n$. As a special case, this shows that the group algebra of the alter...

This paper proves that the restriction of a Specht module for a (degenerate or non-degenerate) cyclotomic Hecke algebra, or KLR algebra, of type A has a Specht filtration.

Gordon James proved that the socle of a Weyl module of a classical Schur algebra is a sum of simple modules labelled by $p$-restricted partitions. We prove an analogue of this result in the very general setting of "Schur pairs". As an application we show that the socle of a Weyl module of a cyclotomic $q$-Schur algebra is a sum of simple modules la...

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.

This is a DRAFT chapter based on a series of lectures that I gave at the National University of Singapore in April 2013. The notes 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 em...

This paper classifies the blocks of the truncated $q$-Schur algebras of type $A$ which have as weight poset an arbitrary cosaturated set of partitions.

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 uti...

We define a graded quasi-hereditary covering for the cyclotomic quiver Hecke algebras $\R$ of type $A$ when $e=0$ (the linear quiver) or $e\ge n$. 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...

The graded Specht module $S^\lambda$ for a cyclotomic Hecke algebra comes with a distinguished generating vector $z^\lambda\in S^\lambda$, which can be thought of as a "highest weight vector of weight $\lambda$". This paper describes the {\em defining relations} for the Specht module $S^\lambda$ as a graded module generated by $z^\lambda$. The firs...

This paper constructs an explicit homogeneous cellular basis for the cyclotomic Khovanov–Lauda–Rouquier algebras of type A.

Brundan, Kleshchev, and Wang have introduced a Z-grading on the Specht modules of the degenerate and nondegenerate cyclotomic Hecke algebras of type G(l,1,n). In this paper, we show that the induced Specht modules have an explicit filtration by shifts of graded Specht modules. This proves a conjecture of Brundan, Kleshchev, and Wang. To prove these...

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 pr...

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. Comment: 30 pages

Let Hn 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 Hn. This paper shows that the induced Specht module S(μ)Hn⊗Hn+1 has an explicit Specht filtration.

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 sho...

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...

The paper studies the modular representation theory of the cyclotomic Hecke algebras of type $G(r,p,n)$ with $(\varepsilon,q)$-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 $1\le s\le r$ and $1\le m...

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...

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 b...

Nazarov [Naz96] 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 ra...

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 Poincaré polynomial, confirming a conjecture of Uno's (J. Algebra 149 (1992) 287).

We prove a q-analogue of the row and column removal theorems for homomorphisms between Specht modules proved by M. Fayers and the first author [J. Pure Appl. Algebra 185, No. 1-3, 147-164 (2003; Zbl 1061.20011)]. These results can be considered as complements to James and Donkin’s row and column removal theorems for decomposition numbers of the sym...

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.

Recently, there has been progress in determining the representation type of the Hecke algebras of finite Weyl groups. We report on these results.

This paper is an attempt to compute the decomposition numbers of the blocks of the symmetric group which have "small defect"; that is, blocks of weight smaller than the characteristic. We present various methods for computing such decomposition numbers and use these as support for a conjecture which relates decomposition numbers in different charac...

this paper shows that G() is diagonalizable if and only if G( ) is diagonalizable, where is the partition conjugate to . Moreover, if G() is divisibly diagonalizable (that is, G() is equivalent to a diagonal matrix diag(d 1 ; : : : ; dm ) such that d i divides d i+1 , for 1 i < m), then so is G( ). In this case we can speak of elementary divisors a...

Let H_q(S_n) be the Iwahori-Hecke algebra of the symmetric group. This algebra is semisimple over the rational function field Q(q), where q is an indeterminate, and its irreducible representations over this field are q-analogues S_q(lambda) of the Specht modules of the symmetric group. The q-Specht modules have an "integral form" which is defined o...

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 S. Arik...

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, W...

This article is a comprehensive review of the representation theory of the Ariki-Koike algebras and the cyclotomic Schur algebras.

We compute the generic degrees of the Ariki--Koike algebras by first constructing a basis of matrix units in the semisimple case. As a consequence, we also obtain an explicit isomorphism from any semisimple Ariki--Koike algebra to the group algebra of the corresponding complex reflection group. 1.

The cyclotomic q q -Schur algebra was introduced by Dipper, James and Mathas, in order to provide a new tool for studying the Ariki-Koike algebra. We here prove an analogue of Jantzen’s sum formula for the cyclotomic q q -Schur algebra. Among the applications is a criterion for certain Specht modules of the Ariki-Koike algebras to be irreducible.

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...

The cyclotomic q-Schur algebra was introduced by Dipper, James and Mathas, in order to provide a new tool for studying the Ariki-Koike alge- bra. We here prove an analogue of Jantzen's sum formula for the cyclotomic q-Schur algebra. Among the applications is a criterion for certain Specht mod- ules of the Ariki-Koike algebras to be irreducible.

this paper we study an analogue of the q--Schur algebra for an arbitrary Ariki--Koike algebra. The Ariki--Koike algebra H is a cyclotomic algebra of type G(r; 1; n) [2], and it becomes the Iwahori--Hecke algebra of type A or B when r = 1 or 2 respectively. By working over a ring R which contains a primitive rth root of unity, and by specializing th...

Introduction Let n and r be integers with n 0 and r 1. Let R be a commutative ring with 1 and let q, Q 1 ; : : : ; Q r be elements of R with q invertible. The cyclotomic Hecke algebra H R;n = H R;n (q; fQ 1 ; : : : ; Q r g) of type G(r; 1; n) is the unital associative R--algebra with generators T 0 ; T 1 ; : : : ; TnGamma1 and relations (T 0 Gamma...

These notes give a fully self--contained introduction to the (modular) representation theory of the Iwahori--Hecke algebras and the q--Schur algebras of the symmetric groups. The central aim of this work is to give a concise, but complete, and an elegant, yet quick, treatment of the classification of the simple modules and of the blocks of these tw...

is irreducible in characteristic 2 if and only if (i) i Gamma i+1 j Gamma1 mod 2 l( i+1 Gamma i+2 ) for all i 1; or, (ii) 0 i Gamma 0 i+1 j Gamma1 mod 2 l( 0 i+1 Gamma 0 i+2 ) for all i 1, where 0 = ( 0 1 ; 0 2 ; : : : ) is the partition conjugate to ; or, (iii) = (2; 2). In part (i) of the Theorem, is necessarily 2--regular and the irreduci

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 in [3] is a unit in any such ring. On the way we show that the Ariki-...

this paper we use the cyclotomic q--Schur algebras to prove an analogue of the Jantzen--Schaper theorem for the Ariki--Koike algebras. Most of the argument is devoted to first proving an analogue of Jantzen's sum formula for the Weyl modules of the cyclotomic q--Schur algebra. The result for the Ariki--Koike algebras is then deduced by a Schur func...

In this paper we use the Hecke algebra of type $B$ to define a new algebra $\Sch$ which is an analogue of the q-Schur algebra. We construct Weyl modules for $\Sch$ and obtain, as factor modules, a family of irreducible $\Sch$-modules over any field.

In this paper we use a deep result of Ariki's to give a combinatorial algorithm for computing the decomposition matrices of the Ariki--Koike algebras H over fields of characteristic zero. As a corollary we obtain a classification of the irreducible H-- modules over an arbitrary field (for certain choices of the defining parameters).

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, define...

this paper we reduce Conjecture 1.1 to a purely combinatorial problem of showing that certain polynomials are orthogonal. To the best of our knowledge these polynomials have not appeared in the literature previously; it seems likely that they will be of independent interest. One of the reasons why the conjecture for H is more difficult to prove tha...

In this note we classify the simple modules of the Ariki--Koike algebras when q = 1 and also describe the classification for those algebras considered in [3, 14], together with the underlying computation of the computing canonical bases of an affine quantum group. In particular, this gives a classification of the simple modules of the Iwahori--Heck...

In this paper a q--analogue of the Coxeter complex of a finite Coxeter group W is constructed for the (generic) Hecke algebra H 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 H afforded by t...

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 Tw0 , where w 0 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...

In this paper we study the decomposition matrices of the Hecke algebras of typeAwithq=−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...

This paper begins by generalising the notion of a "W-graph" to show that the W-graph data determine 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,...

In this paper a q-analogue of the Coxeter complex of a finite Coxeter group W is constructed for the (generic) Hecke algebra 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 afforded by the top...

This paper classifies the blocks of the affine Hecke algebras of type A and the blocks of the cyclotomic Hecke algebras of type G(r, 1, n) over an arbitrary algebraically closed field. Rather than work-ing with the Hecke algebras directly we work instead with the cyclotomic Schur algebras. The advantage of these algebras is that the cyclotomic Jant...

Thesis (Ph. D. in mathematics)--University of Illinois at Chicago, 1993. Vita. Includes bibliographical references (leaves 127-129). Typescript (photocopy).