The representation type of Ariki-Koike algebras and cyclotomic q-Schur algebras

We give a necessary and sufficient condition on parameters for Ariki-Koike algebras (resp. cyclotomic q-Schur algebras) to be of finite representation type.     

Clifford theory on rational Cherednik algebras of imprimitive groups

Ram and Rammage have introduced an automorphism and Clifford theory on affine Hecke algebras. Here we will extend them to cyclotomic Hecke algebras and rational Cherednik algebras.     

Hochschild cohomology and graded Hecke algebras

We develop and collect techniques for determining Hochschild cohomology of skew group algebras and apply our results to graded Hecke algebras. We discuss the explicit computation of certain types of invariants under centralizer subgroups, focusing on the infinite family of complex reflection groups to illustrate our ideas. Resulting formulas for Hochschild two-cocycles give information about deformations of and, in particular, about graded Hecke algebras. We expand the definition of a graded Hecke algebra to allow a nonfaithful action of on , and we show that there exist nontrivial graded Hecke algebras for , in contrast to the case of the natural reflection representation. We prove that one of these graded Hecke algebras is equivalent to an algebra that has appeared before in a different form.

     

Semisimple infinitesimal q-Schur algebras

In this paper, we shall classify the semisimple infinitesimal q-Schur algebras. Received: 2 May 2007, Revised: 27 September 2007     

Classification of graded Hecke algebras for complex reflection groups

The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V). This paper classifies all the algebras obtained by applying Drinfeld's construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld's construction. The classification shows that there exist algebras obtained from Drinfeld's construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups. Received: July 25, 2001     

On structures of modular adjacency algebras of Johnson schemes

In this paper, we consider algebras over a field of characteristic p, which are generated by adjacency algebras of Johnson schemes. If the algebra is semisimple, the structure is the same as that of the well-known Bose-Mesner algebras. We determine the structure of the algebra when it is not semisimple.     

Radical of weakly ordered semigroup algebras

We define the notion of weakly ordered semigroups. For this class of semigroups, we compute the radical of the semigroup algebras. This generalizes some results on left regular bands and on 0- Hecke algebras.     

Fractalized cyclotomic polynomials

For each prime power , we realize the classical cyclotomic polynomial as one of a collection of different polynomials in . We show that the new polynomials are similar to in many ways, including that their discriminants all have the form . We show also that the new polynomials are more complicated than in other ways, including that their complex roots are generally fractal in appearance.

     

Representations of affine Hecke algebras and based rings of affine Weyl groups

In this paper we show that the Deligne-Langlands-Lusztig classification of simple representations of an affine Hecke algebra remains valid if the parameter is not a root of the corresponding Poincaré polynomial. This verifies a conjecture of Lusztig proposed in 1989.

     

The modular branching rule for affine Hecke algebras of type A

For the affine Hecke algebra of type A at roots of unity, we make explicit the correspondence between geometrically constructed simple modules and combinatorially constructed simple modules and prove the modular branching rule. The latter generalizes work by Vazirani (2002) [22].     

New bases of some Hecke algebras via Soergel bimodules

For extra-large Coxeter systems (m(s,r)>3), we construct a natural and explicit set of Soergel bimodules D={Dw}_w∈W such that each Dw contains as a direct summand (or is equal to) the indecomposable Soergel bimodule Bw. When decategorified, we prove that D gives rise to a set {dw}_w∈W that is actually a basis of the Hecke algebra. This basis is close to the Kazhdan–Lusztig basis and satisfies a positivity condition.     

A survey on coefficients of cyclotomic polynomials

Cyclotomic polynomials play an important role in several areas of mathematics and their study has a very long history, which goes back at least to Gauss (1801). In particular, the properties of their coefficients have been intensively studied by several authors, and in the last 10 years there has been a burst of activity in this field of research. This concise survey attempts to collect the main results regarding the coefficients of the cyclotomic polynomials and to provide all the relevant references to their proofs.     

Representation Dimension and Quasi-hereditary Algebras

We study Auslander's representation dimension of Artin algebras, which is by definition the minimal projective dimension of coherent functors on modules which are both generators and cogenerators. We show the following statements: (1) if an Artin algebra A is stably hereditary, then the representation dimension of A is at most 3. (2) If two Artin algebras are stably equivalent of Morita type, then they have the same representation dimension. Particularly, if two self-injective algebras are derived equivalent, then they have the same representation dimension. (3) Any incidence algebra of a finite partially ordered set over a field has finite representation dimension. Moreover, we use results on quasi-hereditary algebras to show that (4) the Auslander algebra of a Nakayama algebra has finite representation dimension.     

All double coverings of cyclotomic fields

A double covering of a Galois extension K/F in the sense of [3] is an extension /K of degree ≤2 such that /F is Galois. In this paper we determine explicitly all double coverings of any cyclotomic extension over the rational number field in the complex number field. We get the results mainly by Galois theory and by using and modifying the results and the methods in [2] and [3]. Project 10571097 supported by NSFC     

Hecke algebras,

The Donald-Flanigan conjecture asserts that the integral group ring of a finite group can be deformed to an algebra over the power series ring with underlying module such that if is any prime dividing then is a direct sum of total matric algebras whose blocks are in natural bijection with and of the same dimensions as those of We prove this for using the natural representation of its Hecke algebra by quantum Yang-Baxter matrices to show that over localized at the multiplicatively closed set generated by and all , the Hecke algebra becomes a direct sum of total matric algebras. The corresponding ``canonical" primitive idempotents are distinct from Wenzl's and in the classical case (), from those of Young.

     

On multiplication formulas of affine q-Schur algebras

We provide a simple and conceptual proof of Du-Fu's multiplication formula of affine q-Schur algebras via Lusztig's formula. We use the multiplication formulas to provide a proof of the existence of generic affine Schur algebras, in return, and a formula of the generators under comultiplication.