Algebra Research Group

Welcome to the pages of the Algebra group in the Mathematical Institute at Oxford. Here you will find information on our members, the seminars and other events we organise, news about us and the research networks we participate in. There are also lists of lecture courses related to our interests.

The research interests of the group span group theory, representation theory and algebraic aspects of geometry, among many other topics. For more detailed information on the people in our group and their individual research interests, please see our list of members.

If you are interested in undertaking graduate studies with us, please see the department's information for prospective graduate students. Post-doctoral positions and funding opportunities and faculty positions are listed on the Institute's vacancies page.

Details of the next scheduled seminar in each of the series we organise are listed below. For future events, please follow the link to each seminar's listings.

Algebra seminar

6 February 2018

It is well-known that nilpotent orbits in $\mathfrak{sl}_n(\mathbb C)$ correspond bijectively with the set of partitions of $n$, such that the closure (partial) ordering on orbits is sent to the dominance order on partitions. Taking dual partitions simply turns this poset upside down, so in type $A$ there is an order-reversing involution on the poset of nilpotent orbits. More generally, if $\mathfrak g$ is any simple Lie algebra over $\mathbb C$ then Lusztig-Spaltenstein duality is an order-reversing bijection from the set of special nilpotent orbits in $\mathfrak g$ to the set of special nilpotent orbits in the Langlands dual Lie algebra $\mathfrak g^L$.
It was observed by Kraft and Procesi that the duality in type $A$ is manifested in the geometry of the nullcone. In particular, if two orbits $\mathcal O_1<\mathcal O_2$ are adjacent in the partial order then so are their duals $\mathcal O_1^t>\mathcal O_2^t$, and the isolated singularity attached to the pair $(\mathcal O_1,\mathcal O_2)$ is dual to the singularity attached to $(\mathcal O_2^t,\mathcal O_1^t)$: a Kleinian singularity of type $A_k$ is swapped with the minimal nilpotent orbit closure in $\mathfrak{sl}_{k+1}$ (and vice-versa). Subsequent work of Kraft-Procesi determined singularities associated to such pairs in the remaining classical Lie algebras, but did not specifically touch on duality for pairs of special orbits.
In this talk, I will explain some recent joint research with Fu, Juteau and Sommers on singularities associated to pairs $\mathcal O_1<\mathcal O_2$ of (special) orbits in exceptional Lie algebras. In particular, we (almost always) observe a generalized form of duality for such singularities in any simple Lie algebra.

Representation Theory seminar

There are no seminars currently scheduled for this series.

An archive of previous events is also available.