# Algebra Seminar

### Seminars are held on Mondays at 17:00 in B3.02

#### Organisers: Adam Thomas and Gareth Tracey

(To see the abstract and title of a talk in the past, click on the speakers name to expand it)

Term 2:

**15th January:*** Jonathan Gruber (York)*

**22nd January: ***David Stewart (Manchester)*

**29th January:*** Hong Yi Huang (Bristol)*

**5th February: ***Beth Romano (Kings College London)*

**12th February: ***Rudradip Biswas (Warwick)*

**19th February: ***Josh Maglione (Galway)*

**26th February: **Martina Balagovic (Newcastle)

Title: Quantum symmetric pairs coideal subalgebras

Abstract: I will explain the motivation, history, and some recent progress in the theory of quantum symmetric pairs and their representations. Quantum symmetric pairs are certain defomations of universal enveloping algebras of Lie algebras, different to the standard Drinfeld Jimbo quantum groups but realised as their coideals. Just like the construction of the universal R-matrix for quantum groups produces solutions of the Yang-Baxter equation and gives an action of the braid group of type A on the category of their finite dimensional representations, quantum symmetric pairs allow an analogous construction of a universal K-matrix, producing solutions of the reflection equation and giving an action of the braid group of type B. I will also explain some recent constructions in their representation theory, which is work in progress. All this is joint work with Stefan Kolb.

**4th March: **Paul Levy (Lancaster)

Title: Special pieces in exceptional Lie algebras

Abstract: In connection with the Springer correspondence, Lusztig defined an important subset of the nilpotent orbits in a simple Lie algebra, called the special orbits. To each special orbit is associated an open subset of its closure, called a special piece; the special pieces partition the nilpotent cone. A long-standing conjecture of Lusztig, open in exceptional types, is that each special piece is the quotient of a smooth variety by a certain finite group H. In this talk I will outline a proof of the conjecture. The first step is the establishment of a "local version" of the conjecture, which holds in a suitable transverse slice. In each case, the transverse slice is isomorphic to the quotient of a vector space by H. The local version allows us to establish smoothness of a certain H-cover of the special piece, therefore establishing the conjecture. Along the way, we observe various interesting symplectic quotient singularities appearing as transverse slices between nilpotent orbits in exceptional Lie algebras. This is joint work with Fu, Juteau and Sommers.

**11th March: **Eileen Pan (Warwick) *In B3.01 today.

Title: Some coset actions in G_2(q)

Abstract: The coset actions of almost simple groups have sparked much interest and a lot has been done by various authors, but there remains much to explore. In this talk we consider \(G \cong G_2(q)\) for some prime power \(q\) and let \(H \le G\) be a maximal-rank maximal subgroup of \(G\). We describe the double cosets \(\{HgH: g \in G\}\) and the corresponding intersections \(H \cap H^g\). This in turn gives us the suborbit representatives and subdegrees of the coset action of \(G\) on \(H \backslash G \). Along the way, we give a brief overview of some known results in algebraic and finite groups of exceptional Lie type and explain how they are used in approaching this problem.

#### Term 1:

**9th October: ***Christian Ikenmeyer (Warwick)*

**16th October: ***Stacey Law (Birmingham)*

**23rd October: ***Veronica Kelsey (Manchester)*

**30th October: ***Lucia Morotti (York)*

**6th November: ***Martin van Beek (Manchester)*

**13th November: ***Peiran Wu (St Andrews)*

**20th November: ***Matthew Chaffe (Birmingham)*

**27th November: ***Iulian Simion (Babeș-Bolyai University)*

**4th December: ***Tim Burness (Bristol)*