On its original publication, this book provided the first elementary treatment of representation theory of finite groups of Lie type in book form. This second edition features new material to reflect the continuous evolution of the subject, including entirely new chapters on Hecke algebras, Green functions and Lusztig families. The authors cover the basic theory of representations of finite groups of Lie type, such as linear, unitary, orthogonal and symplectic groups. They emphasise the Curtis–Alvis duality map and Mackey's theorem and the results that can be deduced from it, before moving on to a discussion of Deligne–Lusztig induction and Lusztig's Jordan decomposition theorem for characters. The book contains the background information needed to make it a useful resource for beginning graduate students in algebra as well as seasoned researchers. It includes exercises and explicit examples.

Representations of finite groups of Lie type

Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, Dugger-Shipley, ..., Toen and Toen-Vaquie.

On differential graded categories

We present a fast algorithm for computing the global crystal basis of the basic
 
$$U_q (\widehat{\mathfrak{s}\mathfrak{l}}_n )$$

 -module. This algorithm is based on combinatorial techniques which have been developed for dealing with modular representations of symmetric groups, and more generally with representations of Hecke algebras of typeA at roots of unity. We conjecture that, upon specializationq→1, our algorithm computes the decomposition matrices of all Hecke algebras at an
th root of 1.

/pdf/hecke-algebras-at-roots-of-unity-and-crystal-bases-of-4ghb3cy6mo.pdf

Hecke algebras at roots of unity and crystal bases of quantum affine algebras

We suggest a purely algebraic construction of the spin representation of an infinite-dimensional orthogonal Lie. al- gebra (sections 1 and 2) and a corresponding group (section 4). From this we deduce a construction of all level-one highest-weight representations of orthogonal affine Lie algebras in terms of cre- ation and annihilation operators on an infinite-dimensional Grass- mann algebra (section 3). We also give a similar construction of the level-one representations of the general linear affine. Lie al- gebra in an infinite-dimensional "wedge space." Along these lines we construct the corresponding representations of the universal central extension of. the group SL,(k(t,t-1)) in spaces of sections of line bundles over infinite-dimensional homogeneous spaces (sec- tion 5).

Spin and wedge representations of infinite-dimensional Lie algebras and groups (Clifford algebra/spin representation/affine Kac-Moody Lie algebra/highest weight representation/line bundle)

Quantum deformations of GLn

Quantum Weyl reciprocity relates the representation theory of Hecke algebras of type A with that of q-Schur algebras. This paper establishes that Weyl reciprocity holds integrally (i. e., over the ring ℤ[q, q− 1] of Laurent polynomials) and that it behaves well under base-change. A key ingredient in our approach involves the theory of tilting modules for q-Schur algebras. New results obtained in that direction include an explicit determination of the Ringel dual algebra of a q-Schur algebra in all cases. In particular, in the most interesting situation, the Ringel dual identifies with a natural quotient algebra of the Hecke algebra.

Quantum Weyl Reciprocity and Tilting Modules

Quasi-hereditary algebras can be viewed as a Lie theory approach to the theory of finite dimensional algebras. Motivated by the existence of certain nice bases for representations of semisimple Lie algebras and algebraic groups, we will construct in this paper nice bases for (split) quasi-hereditary algebras and characterize them using these bases. We first introduce the notion of a standardly based algebra, which is a generalized version of a cellular algebra introduced by Graham and Lehrer, and discuss their representation theory. The main result is that an algebra over a commutative local noetherian ring with finite rank is split quasi-hereditary if and only if it is standardly fullbased. As an application, we will give an elementary proof of the fact that split symmetric algebras are not quasi-hereditary unless they are semisimple. Finally, some relations between standardly based algebras and cellular algebras are also discussed.

/pdf/based-algebras-and-standard-bases-for-quasi-hereditary-1l7aapwfgw.pdf

Based algebras and standard bases for quasi-hereditary algebras

Two-parameter quantum linear groups and the hyperbolic invariance of q-Schur algebras

Canonical bases for irreducible representations of quantum GLn

We use the monomial basis theory developed by Deng and Du to present an elementary al- gebraic construction of the canonical bases for both the Ringel-Hall algebra of a cyclic quiver and the positive part U + of the quantum affine sln. This construction relies on analysis of quiver representa- tions and the introduction of a new integral PBW-like basis for the Lusztig Z(v, v 1 )-form of U + .

https://web.maths.unsw.edu.au/~jied/papers/ddx9.pdf

Jie Du

Papers

Quantum Weyl Reciprocity and Tilting Modules

Based algebras and standard bases for quasi-hereditary algebras

Two-parameter quantum linear groups and the hyperbolic invariance of q-Schur algebras

Canonical bases for irreducible representations of quantum GLn

Generic Extensions and Canonical Bases for Cyclic Quivers