Yegor Zenkevich (Edinburgh)

Quantum toroidal algebras and physics

Quantum toroidal algebras are quantum deformations of double loop algebras. They have some remarkable properties and rich representation theory. I will briefly introduce quantum toroidal algebras and explain their relations to integrable systems, gauge theories and string theory.

Theresa Ortscheidt (Glasgow)

Hecke Algebras at Roots of Unity

The Iwahori-Hecke algebra of type A can be seen as a q-deformation of the symmetric group algebra. At generic values of q its representation theory is similar to that of the symmetric group and can be understood combinatorially in terms of Young diagrams and Littlewood-Richardson coefficients. However, when q is a root of unity, the Hecke algebra is no longer semi-simple. Goodman and Wenzl introduced a ring in terms of a particular class of Hecke representations at roots of unity, which can be realised as a quotient of the ring of symmetric polynomials.  I will discuss briefly some of the open problems associated with finding a new combinatorial description of the structure coefficients of this ring.