## University of Sydney Algebra Seminar

# Representation Theory Day 2015

A one day workshop on August 21, held at the University of Sydney. We will have five lectures and an informal dinner in the evening. For more information contact the organisers Andrew Mathas (andrew.mathas@sydney.edu.au) and Oded Yacobi (oded.yacobi@sydney.edu.au).

## Confirmed speakers:

1) Shrawan Kumar (University of North Carolina)

2) Anthony Licata (Australian National University)

3) Peter McNamara (University of Queensland)

4) Elizabeth Milićević (Haverford College)

5) Catharina Stroppel (University of Bonn)

## Location:

Morning session - Carslaw 275

Afternoon session - Carslaw 175

## Schedule:

9:30-9:55 -- Coffee and tea in School Lounge (7th floor of Carslaw)

Morning session:

10:00-10:55 -- Kumar

11:00-11:55 -- Milićević

Afternoon session:

2:00-2:55 -- McNamara

3:00-3:55 -- Licata

4:00-4:55 -- Stroppel

Evening session:

5:00-?? -- Drinks and dinner (stay tuned for details)

## Abstracts:

#### Kumar: A study of saturated tensor cones for symmetrizable Kac-Moody algebras

This is joint work with M. Brown. Let \(\mathfrak{g}\) be a symmetrizable Kac-Moody Lie algebra with standard Cartan subalgebra \(\mathfrak{h}\) and the Weyl group \(W\). Let \(P_+\) be the set of dominant integral weights. For \(\lambda\in P_+\) let \(L(\lambda)\) be the integrable, highest weight (irreducible) representation of \(\mathfrak{g}\) with highest weight \(\lambda\). For a positive integer \(s\), define the saturated tensor semigroup as
\[
\Gamma_s:=\{(\lambda_1,...,\lambda_s,\mu)\in P_+^{s+1}:\exists N\geq 1 \text{with } L(N\mu)\subset L(N\lambda_1)\otimes\cdots\otimes L(N\lambda_s) \}.
\]
The aim of this work is to begin a systematic study of \(\Gamma_s\) in the infinite dimensional symmetrizable Kac-Moody case. In this work, we produce a set of necessary inequalities satisfied by \(\Gamma_s\). These inequalities are indexed by products in \(H^*(G^{min}/B;\mathbb{Z})\) for \(B\) the standard Borel subgroup, where \(G^{min}\) is the 'minimal' Kac-Moody group with Lie algebra \(\mathfrak{g}\). The proof relies on the Kac-Moody analogue of the Borel-Weil theorem and Geometric Invariant Theory (specifically the Hilbert-Mumford index). In the case that \(\mathfrak{g}\) is affine of rank \(2\), we show that these inequalities are necessary and sufficient. We further prove that any integer \(d>0\) is a saturation factor for \(A_1^{(1)}\) and \(4\) is a saturation factor for \(A_2^{(2)}\).

#### Licata: ADE braid groups and finite dimensional algebras

We explain how the representation theory of quivers gives rise to some interesting metrics on braid groups of type ADE. This is joint work with Hoel Queffelec.

#### McNamara: Bases of canonical type and cluster algebras

A categorification of a universal enveloping algebra or its
quantum analogue tends to produce a basis of "canonical type" which
certain characteristic properties. The dual of these bases are perfect
bases of (quantum) cluster algebras. It is expected that these bases
contain all cluster monomials. We will talk about the extent to which
these expectations have been proved.

#### Milićević: Orbits in affine flag varieties

Flag varieties are often studied by decomposing them into orbits of various special subgroups. This principle is also fruitful in the case of the affine flag variety, which is the quotient of a reductive algebraic group over a field of Laurent series. In this talk, we will discuss a combinatorial tool due to Parkinson, Ram, and Schwer for visualizing the unipotent orbits inside of the complete affine flag variety. This alcove walk model has applications to questions in algebraic geometry, analytic number theory, and representation theory.

#### Stroppel: VW-algebras to Brauer algebras and Khovanov algebras of type D

In this talk I will explain how one can deduce properties like gradings, Koszulity, and decomposition numbers of the Brauer algebras from categorification results. This is joint work with Michael Ehrig.