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Events for August – December 2011

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Aug. 30, 2011 2pm (MATH 350)	Lie Theory Fred Goodman (Iowa) X The Birman--Murakami--Wenzl algebra (or BMW algebra for short) plays an important role in knot theoretic invariants and in centralizer algebras for quantum groups. It may be defined either by (rather complicated) generators and relations, or as an algebra of diagrams with a pictorially defined multiplication. Affine and cyclotomic Hecke algebras are related to the Hecke algebras of type A in the same way that the wreath products Z \wr S_n and (Z/rZ) \wr S_n are related to the symmetric group S_n. The affine and cyclotomic BMW algebras discussed in this talk are BMW analogues of affine and cyclotomic Hecke algebras. Affine and cyclotomic BMW algebras
Sep. 06, 2011 2pm (MATH 350)	Lie Theory Richard Green (CU) X A problem in algebraic geometry is to compute Schubert intersection numbers. These count the points of intersection of generically translated Schubert varieties. For minuscule flag varieties such as Grassmannians, these numbers can be computed by a combinatorial rule called "jeu de taquin", which in the Grassmannian case is called the Littlewood--Richardson rule. This talk will describe this combinatorial rule and some of its key properties. No knowledge of algebraic geometry will be assumed. Generalized Littlewood--Richardson rules
Sep. 13, 2011 2pm (MATH 350)	Lie Theory Tyson Gern (CU) X Kazhdan--Lusztig polynomials arise in the context of Hecke algebras associated to Coxeter groups. The computation of these polynomials is very difficult for examples of even moderate rank. It known that the leading coefficients, \mu(x,w) all 0 or 1 when x is fully commutative in type A, but in type D there are certain "bad" elements that make \mu value computation difficult. I will investigate a projection of the Kazhdan--Lusztig basis of the Hecke algebra with the goal of computing \mu values involving these "bad" elements. A Projection of the Kazhdan--Lusztig Basis in Type D I
Sep. 20, 2011 2pm (MATH 350)	Lie Theory Tyson Gern (CU) X Kazhdan--Lusztig polynomials arise in the context of Hecke algebras associated to Coxeter groups. The computation of these polynomials is very difficult for examples of even moderate rank. It known that the leading coefficients, \mu(x,w) all 0 or 1 when x is fully commutative in type A, but in type D there are certain "bad" elements that make \mu value computation difficult. I will investigate a projection of the Kazhdan--Lusztig basis of the Hecke algebra with the goal of computing \mu values involving these "bad" elements. A Projection of the Kazhdan--Lusztig Basis in Type D II
Sep. 27, 2011 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X The Hopf algebra of symmetric functions in noncommuting variables NCSym(X) has an representation theoretic realization as a space of supercharacters of the finite unipotent upper-triangular matrices. In such a correspondence, the representation theory gives several distinguished sets of bases for NCSym(X). However, it is also natural to ask whether we learn anything about supercharacters from such a correspondence. This talk gives one possible answer to such a question, demonstrating how an obvious basis of NCSym(X) has a surprising representation theoretic interpretation on supercharacters. This is joint work with N. Bergeron. Symmetric functions and the representation theory of unipotent upper-triangular matrices
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

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