University of Sydney Algebra Seminar

Arun Ram (University of Melbourne)

Tuesday 14 June, 12-1pm, Place: Carslaw 375

Combinatorics of the Loop Grassmannian

I will explain what the loop Grassmannian and the affine flag variety are and how to label their points. This labelling is a refinement of the labelling of crystal bases by Littelmann paths. I’ll show the picture which summarises the connection to the affine Hecke algebra and the spherical affine Hecke algebra. I’ll give a summary of the relationship between Mirkovic-Vilonen cycles and the crystal bases and explain how this is reflected in the path model indexing.

Thursday 16 June, 12-1pm, Place: Carslaw 375

Combinatorics of representations of affine Lie algebras

This will be a survey of my current understanding of the combinatorial representation theory of affine Lie algebras. For category O at negative level, Verma modules have finite composition series with decomposition numbers determined by Kazhdan-Lusztig polynomials. The structure of affine Weyl group orbits controls the pretty patterns. For category O at positive level, Verma modules have infinite compositions with decomposition numbers given by inverse Kazhdan-Lusztig polynomials, and at critical level, the patterns correspond to the periodic Kazhdan-Lusztig polynomials. I’ll also discuss parabolic category O. Finite dimensional modules (which are level 0) are indexed by Drinfeld polynomials and then there are various collections of smooth representations where our combinatorial understanding has increased greatly in recent years.

Friday 17 June, 12-1pm, Place: Carslaw 375

Combinatorics of affine Springer fibres

This talk will be a survey of the relation between affine Springer fibres and representations of the double affine Hecke algebra. I will likely focus on a favourite example of the elliptic homogeneous case where I can draw a nice picture illustrating how the affine Springer fibre is decomposed into cells indexed by connected components of complement of a hyperplane arrangement called the Shi arrangement (the same one that appears in the K-theory version of the Chevalley-Shephard-Todd theorem for reflection groups). These regions then correspond to a Macdonald polynomial basis of the corresponding representation of the double affine Hecke algebra.