Schedule, Notes/Handouts & Videos

After brief introduction to algebraic supergroups, we concentrate on the example of the general linear supergroup GL(m|n).

We discuss in detail blocks in the category of finite-dimensional representations of  GL(m|n) and the Kazhdan-Lusztig theory and calculate the multiplicities of standard modules in indecomposable projective modules using categorification approach due to Brundan and weight diagrams of Brundan and Stroppel.

Then we talk about flag supermanifolds, Borel-Weil-Bott theory and support variety. If time permits I explain how these geometric methods are used to prove the Kac-Wakimoto conjecture about superdimension of an irreducible representation of GL(m|n).

The central object in geometric representation theory is a representation constructed from a variety, generally through a construction (like cohomology) that turns geometry into a vector space.  In the first talk, we describe the seminal example of Springer representations, including their geometry and combinatorics.  In the second, we move to other examples.  Throughout both talks we highlight themes and tools that recur across different geometric representations, as well as open questions (big and small).

After brief introduction to algebraic supergroups, we concentrate on the example of the general linear supergroup GL(m|n).

We discuss in detail blocks in the category of finite-dimensional representations of  GL(m|n) and the Kazhdan-Lusztig theory and calculate the multiplicities of standard modules in indecomposable projective modules using categorification approach due to Brundan and weight diagrams of Brundan and Stroppel.

Then we talk about flag supermanifolds, Borel-Weil-Bott theory and support variety. If time permits I explain how these geometric methods are used to prove the Kac-Wakimoto conjecture about superdimension of an irreducible representation of GL(m|n).