Schedule-at-a-Glance

39 PRIMA 2013 AbstractsAlexander MolevUniversity of Sydney, Australiaalexander.molev@sydney.edu.auFor each simple Lie algebra g the vacuum module overthe corresponding affine Kac-Moody algebra has a vertexalgebra structure. For each Lie algebra g of classicaltype, we use explicit generators of the center of this vertexalgebra to produce explicit constructions of maximalcommutative subalgebras of the universal enveloping algebrasU(g).Cluster algebras and singular supports of perversesheavesHiraku NakajimaKyoto University, Japannakajima@kurims.kyoto-u.ac.jpWe propose an approach to Geiss-Leclerc-Schroer’s conjectureon the cluster algebra structure on the coordinatering of a unipotent subgroup and the dual canonical base.It is based on singular supports of perverse sheaves onthe space of representations of a quiver, which give thecanonical base.On orbits in double flag varieties for symmetricpairsHiroyuki OchiaiKyushu University, Japanochiai@imi.kyushu-u.ac.jpLet G be a connected, simply connected semisimple algebraicgroup over the complex number field, and let K bethe fixed point subgroup of an involutive automorphism ofG so that (G, K) is a symmetric pair. We take parabolicsubgroups P of G and Q of K respectively and considerthe product of partial flag varieties G/P and K/Q withdiagonal K-action, which we call a double flag variety fora symmetric pair. It is said to be of finite type if thereare only finitely many K-orbits on it. In this paper, wegive a parameterization of K-orbits on G/P × K/Q interms of quotient spaces of unipotent groups without assumingthe finiteness of orbits. As a result, we get severaluseful criteria for the finiteness of orbits. If one ofP ⊂ G or Q ⊂ K is a Borel subgroup, the finiteness of orbitsis closely related to spherical actions. In such cases,the criteria enable us to obtain a complete classificationof double flag varieties of finite type. As a consequence,we obtain classifications of K-spherical flag varieties G/Pand G-spherical homogeneous spaces G/Q.Elementary subalgebras of modular Lie algebrasJulia PevtsovaUniversity of Washington, USAjulia@math.washington.eduLet g be a p-Lie algebra. We call a subalgebra E of gelementary of rank r if it is an abelian Lie algebra withtrivial p-restriction of dimension r. For a fixed r we considera projective variety E(r, g) that parameterizes allelementary subalgebras of g of rank r. This variety is anatural generalization of the rank variety introduced byCarlson for elementary abelian p-groups and the supportvariety for Lie algebras of Friedlander and Parshall.We’ll identify this projective variety in various classicalcases. We’ll also show how representations of g with specialproperties lead to constructions of families of vectorbundles on E(r, g), thereby extending the study of “modulesof constant Jordan type" and their geometric applicationsto this more general context.Support varieties for reductive groupsPaul SobajeUniversity of Melbourne, Australiapaul.sobaje@unimelb.edu.auLet G be a reductive algebraic group over an algebraicallyclosed field of characteristic p, and denote by G (r) itsr-th Frobenius kernel. Each G (r) -module M has associatedto it a cohomological support variety, which canbe shown in most cases to be homeomorphic to a closedsubset of the variety of commuting r-tuples of p-nilpotentelements in the Lie algebra of G. We will give some ofthe details about this homeomorphism, and also discussexplicit computations in the case that M is the restrictionof either a simple G-module or a Weyl module.Decomposition numbers for the symmetricgroups and Schur algebrasKai Meng TanNational University of Singapore, Singaporetankm@nus.edu.sgThe complete determination of the decomposition numbersfor the symmetric groups and Schur algebras in positivecharacteristic p is a famous open problem; a completesolution of which does not seem to be forthcomingin the near future. In this talk, we present our recentresults which provide closed formulas for the decompositionnumber d λµ when the partition λ is obtained fromµ by moving some nodes whose p-residues are pairwisenon-adjacent.Hodge theory and representation theoryKari VilonenNorthwestern University, USAvilonen@northwestern.eduI will explain how Hodge theory can be used in the representationtheory of real groups to attack the problem ofthe unitary dual.Globalizing crystal basis for quantum superalgebrasWeiqiang WangUniversity of Virginia, USAww9c@virginia.eduCanonical basis (or global crystal basis) for quantumgroups and their integrable modules was introduced anddeveloped by Lusztig and subsequently by Kashiwara.We will present a construction for the first time, motivatedby categorification, of canonical basis for a classof quantum superalgebras and their integrable modules.This is joint work with Sean Clark and David Hill.Special Session 20Singularities in Geometry and TopologyThe topology of real suspension singularitiesof type fḡ + z nHaydée Aguilar CabreraNational Autonomous University of Mexico, Mexicolangeh@gmail.comIn this talk we present some results on the topology ofthe family of real analytic germs F : (C 3 , 0) → (C, 0)with isolated critical point at 0, given by F (x, y, z) =f(x, y)g(x, y)+z r , where f and g are holomorphic,r ∈ Z+