Related papers: Derived equivalences for symmetric groups and sl\_…

We show that each block of an alternating group over an arbitrary complete discrete valuation ring is splendidly Rickard equivalent to its Brauer correspondent. This provides new evidence for a refined version of Brou\'{e}'s abelian defect…

We construct an equivalence of categories from a strong categorical sl(2) action, following the work of Chuang-Rouquier. As an application, we give an explicit, natural equivalence between the derived categories of coherent sheaves on…

We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3 x 3-lemma and the snake lemma. We briefly discuss exact functors,…

For any abelian category \calC satsifying (AB5) over a separated, quasi-compact scheme S, we construct a stack of 2-groups \GL(\calC) over the flat site of S. We will give a concrete description of \GL(\calC) when \calC is the category of…

Let $\mathcal U$ be the 2-category associated with $\mathfrak{sl}_2$. We prove that a complex of 1-morphisms of $\mathcal U$ is null-homotopic if and only if its image in every simple 2-representation is null-homotopic. Under mild…

This paper is a continuation and a completion of [BoRo1]. We extend the Jordan decomposition of blocks: we show that blocks of finite groups of Lie type in non-describing characteristic are Morita equivalent to blocks of subgroups…

We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra.…

We prove the symmetrising trace conjecture of Brou\'e, Malle and Michel for the generic Hecke algebra associated to the exceptional irreducible complex reflection group $G_{13}$. Our result completes the proof of the conjecture for the…

In the representation theory of finite groups, Brou\'e's abelian defect group conjecture says that for any prime p if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block B of the…

We investigate the extensions of the Hecke algebras of finite (complex) reflection groups by lattices of reflection subgroups that we introduced, for some of them, in our previous work on the Yokonuma-Hecke algebras and their connections…

We strengthen the results of Boltje and Yilmaz regarding the Galois descent of equivalences of blocks of $p$-nilpotent groups and a result of Kessar and Linckelmann regarding Galois descent of splendid Rickard equivalences for blocks with…

We explicitly show that symmetric Frobenius structures on a finite-dimensional, semi-simple algebra stand in bijection to homotopy fixed points of the trivial SO(2)-action on the bicategory of finite-dimensional, semi-simple algebras,…

We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki's…

For a possibly twisted loop group $LG$, and any character sheaf of its Iwahori subgroup, we identify the associated affine Hecke category with a combinatorial category of Soergel bimodules. In fact, we prove such results for affine Hecke…

Starting with a k-linear or DG category admitting a (homotopy) Serre functor, we construct a k-linear or DG 2-category categorifying the Heisenberg algebra of the numerical K-group of the original category. We also define a 2-categorical…

We classify all $2$-blocks with abelian defect groups of rank $4$ up to Morita equivalence. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. An application…

For an abelian category, a category equivalent to its derived category is constructed by means of specific projective (injective) multicomplexes, the so-called homological resolutions.

Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of \v{S}penko and Van den Bergh, we construct equivalences between the derived categories of coherent sheaves…

We show that if two rings have equivalent derived categories then they have the same algebraic K-theory. Similar results are given for G-theory, and for a large class of abelian categories.

In this article, we study the modular representations of the special linear group of degree two over a finite field in defining characteristic. In particular, we study the automorphisms of derived category of representations. We have been…