DIAGRAMMATIC METHODS IN REPRESENTATION THEORY

Gus Lehrer

I shall give two sets of 3 lectures, which give two contexts in which abstract algebras may be realised as diagram algebras, in which multiplication becomes concatenation of diagrams according to certain rules. A good model is the realisation of the fundamental group of a configuration space as a braid group, with the familiar pictures. In addition to being useful for computation and providing suggestions for theorems, these diagrammatic realisations can lead to new insights. The two topics are as follows.

1. A diagrammatic approach to the affine Hecke algebras of classical type

Topics to be discuussed include

• The affine Temperley-Lieb algebra of type Ã_{n−1}. I shall discuss its definition, its relation to the affine Hecke algebras ((un)extended), its representation theory from the cellular viewpoint, its cell modules and homomorphisms between them, and their composition factors.

• Composition factors for the 2-step nilpotent cell modules of the affine Hecke algebra of type Ã_{n−1}. I shall show how the previous workk can be used to determine these (this is recent work of John Graham and myself). We then have the task of identifying cell modules with the standard modules of Bernstein-Zelevinsky, Rogawski and Kazhdan-Lusztig. This involves moving with facility between the Bernstein and Coxeter presentations of the Hecke algebras, which is achieved via diagrams.

• Braid group realisations for fundamental groups of configuration spaces of type B_{n} and Ã_{n−1}. The relationship between these is the key to this approach. It is the most important part of this section.

• Lifting representations from the Temperley-Lieb algebra of type B_{n} (sometimes called the “blob algebra”). The key to the comparison of standard modules with cell modules is that the affine Hecke algebra and the Temperley-Lieb algebra of type B_{n} are both quotients of a large diagram algebra. We’ll use this to complete the proof of the identification of cell modules with standard modules (which will depend on some work of Ariki).

Typeset by AMS-TEX

2. Diagrammatic endomorphism alebras of classical and quantum types and Schur-Weyl-Brauer duality.

Topics to be discussed include

• The classical case: the Brauer algebra. I shall show how decomposition of the tensor powers of the natural module for a classical (orthogonal or symplectic) group leads naturally to the definition of Brauer’s now celebrated algebra of diagrams, and how this may be interpreted in terms of invariant theory.

• The quantum case: the Birman-Wenzl algebra. I shall define the B-W algebra, show how it is related to both the Hecke and Brauer algebras, and show how to interpret it as an algebra of endomorphisms of an action of the quantum group sl_{q} on tensor space.

• Endomorphisms of tensor powers of certain highest weight modules. I shall show how the endomorphism algebras of such spaces may be studied by means of a new algebra, which comes both in classical and quantised form. This theorem, which is joint work with Ruibin Zhang, includes both the above examples. A new example is a diagrammatic approach to the tensors on the 7-dimensional representation of the Lie algebra of type G_{2}. This representation has been studied in the context of geometric invariant theory by G. Schwartz.