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Events for December 2008 – April 2009

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Dec. 01, 2008 2pm (MATH 350)	Lie Theory Dan Daly (DU) X This talk will be an introduction to permutation patterns. A permutation p in Sn is said to contain a pattern permutation q in Sm if there exists a subsequence of p that is order-isomorphic to q. If p does not contain q, then p avoids q. One of the central questions in the study of patterns is: given a pattern p in Sm, how many permutations in Sn avoid p? This question has a beautiful answer for the case m=3 and we will prove the result in this case. When m=4, the problem gets extraordinarily more complicated. We will survey what is known for m=4 and discuss what is still left to be solved. Permutation Patterns
Dec. 02, 2008 2pm (MATH 350)	Lie Theory Rachel Krieger (CU) X Understanding molecular structure is a keystone concept in chemistry. Knowledge of structure precludes understanding molecular interactions, and therefore chemical reactions. Once the composition of a molecule is determined (i.e. how many and what type of atoms comprise a molecule), there are many possible configurations of the atoms. However, only one configuration provides an accurate model of the molecule. Different molecular structures have different groups of symmetries, and certain representations of this group correspond to physical, measurable phenomena. This talk will describe what kinds of representations are physically identifiable, and how such observations lead to an understanding of the symmetry structure of a molecule. Divination of molecular structure through an application of character theory
Dec. 09, 2008 1pm (Math 350)	PDE/Analysis James D. Meiss X Just as one can generate an exact, symplectic map by an implicit generating function, exact volume-preserving maps can be obtained by implicit generating forms. Hector Lomeli and I have shown recently how to generalize this idea, originally due to Carroll, to obtain generating forms of different "types" analogous to the four standard canonical generators. A first step in a theory of transport is computation of the flux though lobes of turnstiles. For the area preserving case, these fluxes are given by differences between the actions of the heteroclinic orbits that bound the lobes. We show how to generalize this to the exact volume-preserving case. In this case the generating forms give rise to an "action" given by an integral along heteroclinic curves. Generating Forms and Flux for Volume Preserving Maps
Dec. 09, 2008 2pm (MATH 350)	Lie Theory Josh Wiscons (CU) X The class of groups of finite Morley rank consists of groups that can be equipped with a (model theoretic) notion of dimension, called Morley rank, that applies to the definable subsets of the group in much the same way that the notion of dimension of an algebraic group applies to its constructible sets. In fact, much effort has been put into the Cherlin-Zil'ber conjecture which states that a group of finite Morley rank which is simple is isomorphic (as an abstract group) to an algebraic group over an algebraically closed field. We will discuss how BN-pairs, which have been very helpful in understanding simple algebraic groups, may be helpful in approaching this conjecture. BN-pairs and groups of finite Morley rank
Feb. 03, 2009 2pm (MATH 350)	Lie Theory Richard Green (CU) X Regular polytopes are the n-dimensional analogs of the Platonic solids. In this talk, I will sketch how the theory of Coxeter groups can be used to classify the regular polytopes. The classification includes higher dimensional analogs of the tetrahedron, the cube and the octahedron, and several exceptional cases. Coxeter groups and regular polytopes I
Feb. 10, 2009 2pm (MATH 350)	Lie Theory Richard Green (CU) X Regular polytopes are the n-dimensional analogs of the Platonic solids. In this talk, I will sketch how the theory of Coxeter groups can be used to classify the regular polytopes. The classification includes higher dimensional analogs of the tetrahedron, the cube and the octahedron, and several exceptional cases. Coxeter groups and regular polytopes II
Feb. 17, 2009 2pm (MATH 350)	Lie Theory Siye Wu (CU) X In this talk, we review Coxeter and dual Coxeter numbers and related concepts. We also establish a relations between these numbers of a simple Lie algebra and its Langlands dual. Coxeter and dual Coxeter numbers of simple Lie algebras
Feb. 24, 2009 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X Random walks on groups have come up in a surprising number of models. If they happen to also "respect" the group's representation theory, these walks become algebraically understandable. This talk explores some of the ideas of what makes an algebraically nice walk, and how one can exploit this knowledge. Nice random walks on groups
Mar. 03, 2009 2pm (MATH 350)	Lie Theory Ilia Mishev (CU) X We will explore Coxeter group actions on certain linear combinations of Saalschutzian 4F3(1) hypergeomteric series. We will show an invariance group and three-term relations. The invariance relations will be classified by a double coset decomposition and the three-term relations will be classified using the notion of "L-coherent sets". Joint research with Richard Green and Eric Stade. Coxeter group actions on linear combinations of Saalschutzian 4F3(1) hypergeometric series
Mar. 10, 2009 2pm (MATH 350)	Lie Theory Ryan Vinroot (William & Mary) X Let G = U(2m, Fq2) be the finite unitary group, with q the power of an odd prime p. We prove that the number of irreducible complex characters of G with degree not divisible by p and with Frobenius--Schur indicator -1 is qm-1. We also obtain a combinatorial formula for the value of any character of U(n, Fq2) at any central element, using the characteristic map of the finite unitary group. Semisimple symplectic characters of finite unitary groups
Mar. 17, 2009 2pm (MATH 350)	Lie Theory Geir Helleloid (UT Austin) X In 2000, Hua gave a (complicated) formula for the number of isomorphism classes of absolutely indecomposable representations of a quiver over the finite field Fq with a fixed dimension vector. This number is a polynomial in q with integer coefficients (the coefficients are conjecturally non-negative). After introducing quiver representations and Hua's formula, I will discuss joint work with Fernando Rodriguez-Villegas in which we extract information from Hua's formula. In this work, we see connections to the enumeration of certain graphs and derivatives of q-binomial coefficients. Counting Quiver Representations over Finite Fields
Mar. 31, 2009 2pm (MATH 350)	Lie Theory Stephen Lewis (CU) X I will begin with a review of character theory and supercharacter theory, the role of the latter in Un(F2), discuss algorithms for computation of restriction and the role it may play in theoretical proofs, and finally address when the coefficient of the trivial character in a restriction is nonzero. Throughout this paper, I will recall the similarities and differences of Sn to Un(Fq), and character theory to supercharacter theory. Restriction of Supercharacters in U_n(F_2)
Apr. 09, 2009 2pm (MATH 350)	Lie Theory Mitya Boyarchenko (U. Chicago) X Let F_q denote a finite field with q elements, and let G be a subgroup of the general linear group GL_n(F_q) that is defined as the zero set of a certain collection of polynomial equations with coefficients in F_q (the variables are the entries of an n-by-n matrix). In particular, G is a finite group. Some typical examples of such groups are: GL_n(F_q), SL_n(F_q) (the special linear group), Sp_{2n}(F_q) (the symplectic group of matrices of size 2n), UL_n(F_q) (the group of upper-triangular n-by-n matrices with 1's along the main diagonal). The same collection of equations defines an algebraic group G' over the algebraic closure of F_q. One can ask how the geometry of G' (as an algebraic variety) is related to (complex) representations of the finite group G. This question was classically studied by many mathematicians, notably George Lusztig, in the case where G' is a reductive group. My talk is devoted to the "opposite" case, where G' is a unipotent group, so that G is a finite p-group (where p=char(F_q)). I will present several nontrivial results relating the geometry of G' to complex irreducible representations of G. Among these results is a vast generalization of a 1966 conjecture of J. Thompson stating that the dimension of every irreducible representation of UL_n(F_q) is a power of q. (The conjecture is now a theorem of I.M. Isaacs.) The talk is based on joint work with Vladimir Drinfeld. Geometric methods in representation theory of finite p-groups
Apr. 14, 2009 2pm (MATH 350)	Lie Theory Rachel Krieger (CU) X A supercharacter theory of the finite group of unipotent uppertriangular matrices Un(Fq) has been developed based on the existence of an underlying nilpotent associative algebra. In this talk, we consider a family of maps between the nilpotent associative algebra and Un(Fq). We first determine a method by which the image of any element in the algebra can be computed. We then show that the particular map chosen does not affect the location of the two-sided orbit, so that the superclasses defined in this way are independent of the map between the underlying algebra and the group. Hence, the superclasses arising in this particular supercharacter theory may have another natural definition for Chevalley groups. Modified Exponential Maps and the Finite Group of Unipotent Uppertriangular Matrices
Apr. 21, 2009 2pm (MATH 350)	Lie Theory Hugh Denoncourt (CU) X Many combinatorial questions that arise in the context of Coxeter groups rely on properties of reduced expressions for a Coxeter group element. We present a discrete geometric model for arbitrary Coxeter group elements of arbitrary Coxeter groups and show that the model can be used to answer questions about reduced expressions. In particular, we can ascertain the effect of deleting a generator from a reduced expression upon the length of a Coxeter group element. Additionally, we construct correspondences between reduced expressions of an element and tournaments placed "on top" of the element's discrete geometric model. Word combinatorics for Coxeter groups I
Apr. 28, 2009 2pm (MATH 350)	Lie Theory Hugh Denoncourt (CU) X Many combinatorial questions that arise in the context of Coxeter groups rely on properties of reduced expressions for a Coxeter group element. We present a discrete geometric model for arbitrary Coxeter group elements of arbitrary Coxeter groups and show that the model can be used to answer questions about reduced expressions. In particular, we can ascertain the effect of deleting a generator from a reduced expression upon the length of a Coxeter group element. Additionally, we construct correspondences between reduced expressions of an element and tournaments placed "on top" of the element's discrete geometric model. Word combinatorics for Coxeter groups II

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