Winter STARS Workshop 2021
Abstracts
Inna Entova-Aizenbud (Ben Gurion University)
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McKay Trees
Abstract: I will report on a joint work with A. Aizenbud on the possible shapes of McKay graphs.
Given a finite group G and its representation V, the corresponding McKay graph is a (directed) graph whose vertices are the irreducible representations of G; the number of edges between two vertices W, U of our graph is the multiplicity of U in the tensor product W\otimes V. Such graphs can be seen as a combinatorial tool to encode (part) of the data of the character ring of G.
I will give some background on these graphs and some of their uses, and then present our recent results on classification of McKay graphs in the shape of (unoriented) trees.
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Maria Gorelik (Weizmann Institute of Science),
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Root groupoid and relations in Kac-Moody superalgebras
Abstract: This talk will be a continuation of the talk by Vladimir Hinich.
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Thorsten Heidersdorf (University of Bonn),
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Indecomposable summands in tensor products
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Abstract: I will give an explicit description of the indecomposable summands in a tensor power
V^{\otimes r} where V denotes the standard representation of the orthosymplectic supergroup OSp(m|2n).
At the end I will ask what we know in general about the structure of the indecomposable summands in a tensor product decomposition L(\lambda) \otimes L(\mu) for irreducible representations of a supergroup such as GL(m|n).
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Vladimir Hinich (University of Haifa),
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Root groupoid and related Lie superalgebras.
Abstract: This is a report on a joint work with M. Gorelik and V. Serganova.
We present a construction of a groupoid of root-type data whose interesting components lead to construction of Lie superalgebras. This is another variation of the construction of Kac-Moody Lie algebra, but better fit to describe non-equivalent Borels ("odd reflections") as well as better justify the relations defining these superalgebras.
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Crystal Hoyt (Bar Ilan University),
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Representations of the Cartan Type Lie superalgebra W(infty)
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Abstract: The Lie superalgebra W(infty) is the direct limit of finite-dimensional Cartan type Lie superalgebras W(n) as n goes to infinity. In this talk, we will discuss Z-graded modules over W(infty). We introduce a category T_W of W(infty)-modules which is closely related to the category T_gl of tensor sl(infty)-modules introduced and studied by Dan-Cohen, Serganova and Penkov. We show that each simple module in T_W is isomorphic to the unique simple quotient of a module induced from a simple module in T_gl, and vice versa. This is joint work with Lucas Calixto.
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Vadim Schechtman (Institut de Mathématiques de Toulouse),
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PROBs and sheaves
Abstract: I propose to explain how certain universal bialgebra allows us to give a linear algebra description of categories of perverse sheaves over all symmetric powers of the complex plane, smooth along the diagonal stratification. This bialgebra is closely related to"contingency tables" introduced by Karl Pearson more than one hundred years ago.
This is a joint work with Mikhail Kapranov.
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Vera Serganova (UC Berkeley),
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Green's correspondence for algebraic supergroups
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TBA
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Alex Sherman (Ben Gurion University),
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Localization theorem for the Duflo-Serganova functor
Abstract: Suppose u is an odd, square-semisimple vector field on a smooth affine variety X. There are many situations where we want to compute DS_uC[X], i.e. the result of the DS functor corresponding to u on the space of functions on X. I will discuss a localization theorem which computes this when u is nondegenerate, in a specific sense, along its zero locus. The result also applies to equivariant vector bundles on X. There applications to splitting subgroups, symmetries of the DS functor, and more. Part of a joint project with Vera Serganova.
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