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This seminar covers a wide range of topics in Lie theory including (but not limited to) groups of Lie type, Lie algebras, Coxeter groups and their Hecke algebras, algebraic groups, and quantum groups, with emphasis on combinatorial and representation theoretic properties. Further information about upcoming seminars and people in the math department can be found at the CU Math Department Home Page.
Tue, Oct. 1 2:30pm (MATH 3…
Richard Green (CU)
X
A root system for a simply laced finite Weyl group is a discrete collection of vectors in Euclidean space on which the group acts transitively. In some cases, the set of maximal sets of orthogonal positive roots has the structure of a graded partially ordered set, known as a quasiparabolic set. The talk will describe the rich combinatorial structure of these partially ordered sets.
This is joint work with Tianyuan Xu (University of Richmond).
The "extended tensor product" is a relatively new operation in the representation theory of finite groups which generalizes both the familiar tensor product of representations over the ground field as well as the tensor product of bimodules. It was introduced in 2008 by Serge Bouc and studied extensively by Robert Boltje and Philipp Perepelitsky in the early part of this decade. I will attempt to motivate and explain the definition of the extended tensor product. If I am successful I will go on to discuss new results in the theory of modular representations of finite groups that are proved using the extended tensor product construction. The material will be presented in the most elementary ways possible: a potential audience member should only know what a finite group is and what an action of a finite group on a finite set looks like.
The Extended Tensor Product Sponsored by the Meyer Fund
Wed, Oct. 16 3:30pm (MATH 3…
Tomoyuki Arakawa (Research Institute for Mathematical Sciences, Kyoto University)
X
Symplectic singularities, introduced by Beauville, appear in various aspects of representation theory. Additionally, symplectic singularities arise in the context of quantum field theory, particularly in the Higgs and Coulomb branches of three-dimensional theories, as well as in the Higgs branches of four-dimensional theories. Furthermore, the 4D/2D duality proposed by Beem et al. identifies vertex algebras as invariants for superconformal four-dimensional theories. It has been suggested that the Higgs branch of four-dimensional theories can be reconstructed as the associated variety of vertex algebras. As a result, all vertex algebras arising from four-dimensional theories are believed to be chiral quantizations of symplectic singularities. In this lecture, we will discuss such vertex algebras and their representation theory.
Classical W-algebras, which are reductions of enveloping algebra of affine Lie algebras, have been studied in connection with Hamiltonian integrable equations. By quantization method of Hamiltonian reduction called BRST cohomology, one can get (quantum) W-algebras which are algebraic structures in Toda field theories. Moreover, conjecturally, W-algebras can be embedded in supersymmetric field theories by considering supersymmetric W-algebras obtained by supersymmetric analogue of BRST. In this talk, I will try to avoid technical details but explain basic ideas of the constructions of classical/quantum W-algebras and supersymmetric W-algebras. For last 10 mins, I will also introduce recent results related to supersymmetric W-algebras.
Supersymmetry in W-algebras theory Sponsored by the Meyer Fund
Tue, Oct. 29 2:30pm (MATH 3…
Jen Gensler (CU)
X
Supercharacter theory is a construction that condenses the representation theory of an algebraic structure, hopefully without losing too much information about its representation theory while gaining combinatorial data. Nonnesting supercharacter theory considers the normal supercharacter theory of normal subgroups of UT_n associated to nonnesting set partitions or, equivalently, Dyck paths. In this talk I will define supercharacter theory and work through a motivating example leading into a discussion of results on the nonnesting supercharacter theory of UT_n.
The q,t-Catalan numbers can be described elegantly in terms of pairs of statistics on Dyck paths: area and bounce, or area and dinv. Mahonian permutation statistics, such as coinversion, disorder, and the sorting index, are some of the most thoroughly studied statistics on permutations. Using bijective methods, we prove new expressions of the q,t-Catalan numbers over subsets of permutations in terms of at least one Mahonian statistic. In the process, we develop new Catalan subsets of permutations, new bijections from Dyck paths to these subsets, and new permutation statistics.
