Alexander Kleshchev

Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) cyclotomic Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki’s categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial ℤ-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic.

Linear and projective representations of symmetric groups

Preface Part I. Linear Representations: 1. Notion and generalities 2. Symmetric groups I 3. Degenerate affine Hecke algebra 4. First results on Hn modules 5. Crystal operators 6. Character calculations 7. Integral representations and cyclotomic Hecke algebras 8. Functors e and f 9. Construction of Uz and irreducible modules 10. Identification of the crystal 11. Symmetric groups II Part II. Projective Representations: 12. Generalities on superalgebra 13. Sergeev superalgebras 14. Affine Sergeev superalgebras 15. Integral representations and cyclotomic Sergeev algebras 16. First results on Xn modules 17. Crystal operators fro Xn 18. Character calculations for Xn 19. Operators e and f 20. Construction of Uz and irreducible modules 21. Identification of the crystal 22. Double covers References Index.

Graded decomposition numbers for cyclotomic Hecke algebras

Abstract In recent joint work with Wang, we have constructed graded Specht modules for cyclotomic Hecke algebras. In this article, we prove a graded version of the Lascoux–Leclerc–Thibon conjecture, describing the decomposition numbers of graded Specht modules over a field of characteristic zero.

A proof of the Mullineux conjecture

A partition λ of a positive integer n is a sequence λ1 λ2 λm 0 of integers such that ∑λi n. For a positive integer p, a partition λ λ1 λ2 λm (or its Young diagram) is called p-regular if it does not have p or more equal parts, i.e. if there does not exist t m p 1 with λt λt 1 λt p 1. Let F be a field of characteristic p 0. It is well known that irreducible representations of the symmetric group Sn over F are naturally parametrized by pregular partitions of n (cf. for example [9, 12]). If λ is such a partition we denote the corresponding irreducible module by Dλ. Let sgnn be the one-dimensional sign representation of Sn over F; i.e., sgnn F as a vector space and g f sgn g f for any g Sn f F . Here sgn g is just the sign of the permutation g. It is clear that for any irreducible Dλ, the tensor product Dλ sgnn is also irreducible. The problem, usually called the problem of Mullineux, is to find the p-regular partition μ such that Dλ sgnn Dμ. Put Dλ sgnn Dn λ In this way a bijection bn on the set Pn of p-regular partitions of n is defined for each positive integer n, and the problem is:

Graded Specht modules

Abstract Recently, the first two authors have defined a ℤ-grading on group algebras of symmetric groups and more generally on the cyclotomic Hecke algebras of type G(l, 1, d). In this paper we explain how to grade Specht modules over these algebras.

Representations of shifted Yangians and finite -algebras

Introduction Shifted Yangians Finite W-algebras Dual canonical bases Highest weight theory Verma modules Standard modules Character formulae Notation Bibliography.

Highest Weight Theory for Finite W-Algebras

We define analogues of Verma modules for finite W-algebras. By the usual ideas of highest weight theory, this is a first step toward the classification of finite-dimensional irreducible modules. We also introduce an analogue of the BGG category . Motivated by known results in type A, we then formulate some precise conjectures in the case of nilpotent orbits of standard Levi type.

Modular representations of the supergroup Q(n), I☆

The representation theory of the algebraic supergroup Q(n) has been studied quite intensively over the complex numbers in recent years, especially by Penkov and Serganova [18, 19, 20] culminating in their solution [21, 22] of the problem of computing the characters of all irreducible finite dimensional representations ofQ(n). The characters of one important family of irreducible representations, the so-called polynomial representations, had been determined earlier by Sergeev [24], exploiting an analogue of Schur-Weyl duality connecting polynomial representations of Q(n) to the representation theory of the double covers Ŝn of the symmetric groups. In [2], we used Sergeev’s ideas to classify for the first time the irreducible representations of Ŝn over fields of positive characteristic p > 2. In the present article and its sequel, we begin a systematic study of the representation theory of Q(n) in positive characteristic, motivated by its close relationship to Ŝn. Let us briefly summarize the main facts proved in this article by purely algebraic techniques. Let G = Q(n) defined over an algebraically closed field k of characteristic p 6= 2, see §§2-3 for the precise definition. In §4 we construct the superalgebra Dist(G) of distributions on G by reduction modulo p from a Kostant Z-form for the enveloping superalgebra of the Lie superalgebra q(n,C). This provides one of the main tools in the remainder of the paper: there is an explicit equivalence between the category of representations of G and the category of “integrable” Dist(G)supermodules (see Corollary 5.7). In §6, we classify the irreducible representations of G by highest weight theory. They turn out to be parametrized by the set X p (n) = {(λ1, . . . , λn) ∈ Z | λ1 ≥ · · · ≥ λn with λi = λi+1 only if p|λi}. For λ ∈ X+ p (n), the corresponding irreducible representation is denoted L(λ), and is constructed naturally as the simple socle of an induced representation H(λ) := indBu(λ) where B is a Borel subgroup of G and u(λ) is a certain irreducible representation of B of dimension a power of 2. The main difficulty here is to show that H0(λ) 6= 0 for λ ∈ X+ p (n), which we prove by exploiting the main result of [2] classifying the irreducible polynomial representations of G: so ultimately the proof that H0(λ) 6= 0 depends on a counting argument involving p-regular conjugacy classes in Ŝn.

HOMOGENEOUS REPRESENTATIONS OF KHOVANOV-LAUDA ALGEBRAS

We construct irreducible graded representations of simply laced Khovanov�Lauda algebras which are concentrated in one degree. The underlying combinatorics of skew shapes and standard tableaux corresponding to arbitrary simply laced types has been developed previously by Peterson, Proctor and Stembridge. In particular, the Peterson�Proctor hook formula gives the dimensions of the homogeneous irreducible modules corresponding to straight shapes.

Parabolic Presentations of the Yangian Y (gl n )

We introduce some new presentations for the Yangian associated to the Lie algebra These presentations are parametrized by tuples of positive integers summing to n. At one extreme, for the tuple (n), the presentation is the usual RTT presentation of Yn. At the other extreme, for the tuple (1n), the presentation is closely related to Drinfeld’s presentation. In general, the presentations are useful for understanding the structure of the standard parabolic subalgebras of Yn.