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Events for October 2011 – February 2012

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Oct. 04, 2011 2pm (MATH 350)	Lie Theory Vivek Dhand (MSU) X A finite ranked poset is called a symmetric chain order if it has a decomposition into rank-symmetric saturated chains. The existence of a symmetric chain decomposition is a rather strong combinatorial property which is related to the representation theory of the quantum group U_q(SL_2). Moreover, several interesting posets are conjectured to have symmetric chain decompositions. We show that if P is a symmetric chain order, then the quotient of P^n by the natural cyclic group action is also a symmetric chain order. In particular, if P is a single chain with k vertices, then the poset of n-bead k-ary necklaces is a symmetric chain order. Symmetric chain decomposition of necklace posets
Oct. 11, 2011 2pm (MATH 350)	Lie Theory Rahbar Virk (CU) X A pervading tenet in mathematics is that to study an `object' one should study the `functions' on the object. I will explain the meaning of this with some very elementary examples (involving sets and functions on them). In particular, I will show that several (= at least 2) algebras of interest in combinatorics and representation theory arise via such set-theoretic models. I will then explain, using the particular example of the Hecke algebra, how to upgrade such a `function theory' to give a geometric/categorical realization of the algebra. The upshot: several a priori deep properties (including the definition) of the Hecke algebra and Kazhdan-Lusztig polynomials will be seen to be `obvious'. The Geometric Hecke algebra
Oct. 18, 2011 2pm (MATH 350)	Lie Theory James WIlson (CSU) X Deciding that two groups are isomorphic is a clear task: simply provide an invertible homomorphism between the groups. On the other-hand, understanding why two groups are non-isomorphic takes many different forms. There is a family of groups of size $N$ with nearly exponential numbers of isomorphism types. Yet, these groups are virtually indistinguishable: they have the same character tables, all centralizers of non-central elements are identical, their automorphism groups break into at most 2 flavors and are represented isomorphically on the abelianizations of the groups, and several other typically discerning isomorphism invariants fail to help understand these groups. Surprisingly, despite all these similarities, it is possible to test if a two of such groups are isomorphic using only a logarithmic number of steps (compared to $N$ ). The difference between isomorphism and non-isomorphism
Nov. 01, 2011 2pm (MATH 350)	Lie Theory Justin Keller (CU) X The Jordan decomposition of characters is a key component in Lusztig's classification of the representations of finite groups of Lie type that arise as the fixed points of a reductive group. We review this construction with emphasis on the case of GL(n,q), where the unipotent characters may be classified independently of the base field. Jordan Decomposition of Characters
Nov. 15, 2011 2pm (MATH 350)	Lie Theory Scott Andrews (CU) X A Gelfand--Graev representation of a finite reductive group G is obtained by inducing from a certain representation of a maximal unipotent subgroup of G. Gelfand--Graev representations can be used to classify certain irreducible representations of G. By considering other unipotent subgroups of G, we can recover more irreducible representations of G. I will construct a generalization of the Gelfand--Graev representation and look at some applications. Generalized Gelfand-Graev Representations
Feb. 07, 2012 2pm (MATH 350)	Lie Theory Richard Green (CU) X There are remarkable invariant cubic and quartic forms associated with the simple Lie algebras of types $E6$ and $E7$ , respectively. I will explain what this means, and how these invariant forms may be understood in terms of Coxeter groups and the combinatorics of minuscule representations. Lie algebras and invariant forms
Feb. 14, 2012 2pm (MATH 350)	Lie Theory Tyrrell McAllister (Wyoming) X A weight ring in type A is the coordinate ring of the GIT quotient of the variety of flags in $ℂ^{n}$ modulo a twisted action of the maximal torus in $SL(n,C)$ . An historically important example is the ring $R$ of invariants of $n$ -tuples of points in the projective plane, modulo automorphisms of the plane. We show that any weight ring in type A is generated by elements of degree strictly less than the Krull dimension, which is at worst $O(n2)$ . In particular, we get a linear upper bound of $2n ? 8$ for the degree in which $R$ is generated. We also discuss a certain toric degeneration $R'$ of $R$ that has been used to study the invariants of $n$ -tuples on the projective line. We show that this toric degeneration $R'$ ceases to be useful for higher-dimensional projective spaces. In contrast to the linear upper bound on the degrees needed to generate $R$ , $R'$ has essential generators whose degrees are exponential in $n$ . This is joint work with Benjamin J. Howard. Degree bounds for type-A weight rings and Gelfand--Tsetlin semigroups
Feb. 21, 2012 2pm (MATH 350)	Lie Theory Connor Sempek (CU) X Given a finite set X, a cyclic group G, and a polynomial f with positive integer coefficients, the data (X, G, f) is said to exhibit the cyclic sieving phenomenon if the values of f evaluated at certain roots of unity reveal combinatorial information about actions of G on X. In the talk, this definition will be made precise, examples will be given, and more telling equivalent definitions will be considered. The Cyclic Sieving Phenomenon
Feb. 28, 2012 2pm (MATH 350)	Lie Theory Rahbar Virk (CU) X I will report on my current (and ongoing) work on understanding Hecke correspondences on the flag variety. Hecke correspondences are generalizations of incidence correspondences on P^1 to general flag varieties. Since the pioneering work of Beilinson-Bernstein and Kashiwara-Brylinski (on D-modules), Goresky-Macpherson and Beilinson-Bernstein-Deligne's work (on intersection cohomology), and Lusztig-Kazhdan and Lusztig-Vogan's work (in representation theory), it has been understood that the representation theory of reductive groups is controlled by the geometry and combinatorics of these Hecke correspondences. I will present my own contributions to this study. Time permitting I will explain how this ties into efforts by Nat Thiem and myself to understand and generalize Lusztig's theory of character sheaves. On Hecke Correspondences

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