### School of Mathematics and Statistics

### Gus Lehrer

#### The representation theory of affine

Temperley-Lieb algebras I & II

*Friday 29th May and Friday 5th June*

12-1pm, Carslaw 273

We define a sequence **TL**^{a}(n),
*n=0,1,2,3,...*, of infinite dimensional
algebras as the sets of endomorphisms of the objects in a certain category of
diagrams. These algebras are extended versions of
the Temperley-Lieb quotients of the affine Hecke algebras
of type *Ã*_{n}.
They have bases consisting of diagrams drawn without intersections on the
surface of a cylinder.

Using the methods
of cellular algebras, we construct certain finite dimensional representations of
these algebras, which we call "cell" or "Weyl" modules; these
come from "functors on the category of diagrams" and are therefore
constructed simultaneously for all **TL**^{a}(n).
There are canonical invariant bilinear forms which put pairs of the cell modules
in duality with each other
and all the irreducible **TL**^{a}(n)-modules are obtained as quotients of
the cell modules by the radicals of the forms.

By determining all
homomorphisms between the cell modules, we are able to determine
their decomposition matrices and from these to
deduce the dimensions of all the irreducibles.
We also give explicit formulae for the discriminants of the forms. The
representations
we construct may be interpreted as representations of the affine Hecke algebra
of type *A*; they therefore give explicit results about some of the
representations
of the affine Hecke algebra at roots of unity.

Our results may also be applied to
study related finite dimensional algebras such as the usual Temperley-Lieb
algebra or Jones' annular algebra. For these, our results concerning
discriminants give
precise criteria for semisimplicity as well as a complete discussion of their
modular representation theory, including the determination of the composition
factors,
with their multiplicities, of the cell modules. As a by-product of our explicit
determination
of the homomorphisms between the cell modules, we also obtain a closed formula
for the Jones (or augmentation) idempotent of the Temperley-Lieb algebra
which yields a presentation of
Jones' projection algebra when the Jones trace on the Temperley-Lieb algebra is
degenerate.

This is joint work with J. Graham.