Algebra and Combinatorics Seminar
Fridays, Milner 317     3:00-3:50 PM		Cayley Graph of the Free Product Z3 * Z5
The Algebra and Combinatorics seminar is devoted to studying algebra, combinatorics, their interconnection, and their relations to mathematics and applications. The seminar's organizers are Frank Sottile and Luis Garcia.
This Week's Seminar   April 27  Laura Matusevich, Texas A&M University 3:00-3:50   Combinatorics of binomial primary decomposition

History of previous Alg/Comb seminars.

		January 18	Alvaro Rittatore, University of Uruguay
			The structure of algebraic monoids
		January 26	Fumei Lam, Brown University
			Polyhedral Study of Traveling Salesman Path Problems
		February 2	Volodymyr Kirichenko, Kyiv State University
			Quivers of rings
		February 9	Milena Hering, Institute of Mathematics and its Applications
			Syzygies of toric varieties
		February 16	Tien-Yien Li, Michigan State University
			Numerical Calculation of the Jordan Normal Form
		February 23	Josephine Yu, University of California, Berkeley
			The Newton Polytope of the Implicit Equation
		March 2	Tatyana Chmutova, University of Michigan
			Twisted symplectic reflection algebras
		March 9	James Ruffo, Texas A&M University
			A straightening law for the Drinfel'd Lagrangian Grassmannian
		March 23	Tobias Hagge, Indiana University
			Graphical calculus methods for fusion categories
		March 30	Luis David Garcia, Texas A&M University
			Linear precision for multi-sided toric patches
		April 6	Chris Hillar, Texas A&M University
			Algebraic Characterizations of Uniquely Colorable Graphs
		April 13	Michael Anshelevich, Texas A&M University
			Lattice paths and orthogonal polynomials: a case study
		April 20	Eric Rowell, Texas A&M University
			Braid group representations from twisted quantum doubles of finite groups
		April 27	Laura Matusevich, Texas A&M University
			Combinatorics of binomial primary decomposition

Abstract:
A classical theorem of Chevalley asserts that any connected algebraic group is an extension of an abelian variety by a connected affine group. In this talk we show an analogous result for normal algebraic monoids (joint work with M. Brion), namely that a normal irreducible algebraic monoid is obtained by the extension of an abelian variety by a normal affine algebraic monoid. We begin by describing some basic facts about the geometry of algebraic monoids (as embeddings of their unit groups).
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Abstract:
In the Traveling Salesman Path Problem, we are given a set of cities, traveling costs between city pairs and fixed source and destination cities. The objective is to find a minimum cost path from the source to destination visiting all cities exactly once. The problem is a generalization of the Traveling Salesman Problem with many important applications.

In this talk, we will discuss properties of the polytope corresponding to a linear programming relaxation for the traveling salesman path problem. We consider the set of traveling salesman path perfect graphs, graphs for which the convex hull of incidence vectors of traveling salesman paths can be described by linear inequalities. We show such graphs have a description by way of forbidden minors and also characterize them constructively. We extend this description to perfect graphs for the 2-edge connected subgraph problem and discuss approximation algorithms for these problems.
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Abstract:
The notion of a quiver for a finite dimensional algebra was introduced by Gabriel in 1972 in connection with problems of the representation theory of finite dimensional algebras. In 1975 the author carried over this notion to the case of semiperfect right Noetherian rings and applied it to the structural theory of rings, in particular, to the theory of serial rings. We consider the different types of quivers of rings: prime quivers, Pierce quivers, link graphs and others. Note the following results:

(1) a Noetherian semiperfect ring is indecomposable if and only if its quiver is connected;
(2) the quiver of a Noetherian semiperfect and semiprime ring is strongly connected;
(3) the quiver of a indecomposable right Artinian and hereditary ring is acyclic, i.e., it does not have oriented cycles.

Quivers and prime quivers of semiperfect and semidistributive (SPSD) rings are considered. We prove that for any semihereditary SPSD-ring A there exists the classical ring of fractions A' and the prime quiver of A coincides with the usual quiver of A'. This is false for Noetherian semiperfect piecewise domains. With any prime Noetherian SPSD-ring A with nonzero Jacobson radical we associate its exponent matrix E(A). The quiver of A is defined by E(A). Theory of non-negative matrices is used for the study of exponent matrices.
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Abstract:
Understanding the equations defining algebraic varieties and the relations, or syzygies, between them is a classical problem in algebraic geometry. Green showed that sufficient powers of ample line bundles induce a projectively normal embedding that is cut out by quadratic equations and whose first q syzygies are linear. In this talk I will present numerical criteria for line bundles on toric varieties to satisfy this property. I will also discuss criteria for the coordinate ring of such an embedding to be Koszul.
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Abstract:
Nontrivial Jordan Normal Forms are extremely difficult, if not impossible, to compute numerically mainly due to the fact that multiple eigenvalues are multiple roots of the characteristic polynomial of the underlying matrix. We solve this problem via a recently established robust algorithm that determines, with remarkable accuracy, the multiple roots and multiplicity structure of a polynomial, even under perturbations.
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Abstract:
We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope. This talk should be accessible to a general mathematical audience. No knowledge of tropical geometry will be assumed, and the talk will include an introduction to this subject. This is joint work with Bernd Sturmfels and Jenia Tevelev.
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Abstract:
Symplectic reflection algebras associated to a finite subgroup G of Sp(U) were introduced by P. Etingof and V. Ginzburg in 2000. These algebras and their representations turned out to be related to numerous other areas of mathematics, and hence they were studied extensively during the last several years.
We will consider two generalizations of symplectic reflection algebras. The first one is a notion of symplectic reflection algebra associated to a finite group G mapping, not necessarily injectively, to Sp(U). The second one is a twisted symplectic reflection algebra associated to a finite group G with its symplectic representation U and a two-cocycle $\psi$. We will see that these two generalizations are connected: the case of noninjective U can be reduced to the injective one, but in this reduction the cocycle $\psi$ might become nontrivial even if at the beginning it was trivial.
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Abstract:
We present a structured set of defining equations for the Drinfel'd compactification of the space of rational maps of a given degree in the Lagrangian Grassmannian. These equations give a straightening law on a certain ordered set, which allows the use of combinatorial arguments to establish useful geometric properties of this compactification. For instance, its coordinate ring is Cohen-Macaulay and Koszul.
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Abstract:
Fusion categories can be thought of as generalizations of the category of finite dimensional vector spaces, or as generalizations of representation categories of Hopf algebras. They have recently played important roles in representation theory, conformal field theory and low dimensional topology. No general classification is known, and it is not clear to what extent results known for vector spaces and Hopf algebra representations generalize to the fusion category context. One useful fact is that only finitely many fusion categories exist with a given tensor product structure. Finding these categories is possible in principle, but difficult in practice.

In this talk I will introduce fusion categories and explain some of their uses. I'll discuss some properties of the category of finite dimensional vector spaces which are conjectured hold for fusion categories, and the problem of classifying fusion categories with a given tensor structure. Categorical language for fusion categories can be replaced with elementary algebra; I will not assume substantial knowledge of category theory.
This talk is based on joint work with Seung-Moon Hong.
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Abstract:
In 2002, Krasauskas generalized the standard Bezier and tensor product patches of geometric modeling to multi-sided toric patches. While these offer the promise of greater design flexibility, it is not clear whether they possess the desirable properties of the standard patches. One such property is linear precision, which is the ability to replicate a linear function.

In this talk, I will discuss work with Luis Garcia on linear precision. Our geometric analysis separates the shape of a patch (given by a polytope and a system of weights) from its parametrization. We first show that for any system of weights, there is a unique parametrization by algebraic functions that has linear precision. The existence of a rational parametrization with linear precision becomes a geometric statement concerning the intersection of a toric variety with a particular linear subspace (which depends on the system of weights).
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Abstract:
The study of graph colorability from an algebraic perspective has introduced novel techniques and algorithms into the field. For instance, k-colorability of a graph can be characterized in terms of whether its graph polynomial is contained in a certain ideal. In this talk, we interpret unique colorability in an analogous manner and use Groebner basis techniques to prove an algebraic characterization for uniquely k-colorable graphs. Our result also gives algorithms for testing unique colorability. As an application, we verify a counterexample to a conjecture of Xu concerning uniquely 3-colorable graphs without triangles. (Joint with Troels Windfeldt).
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Abstract:
One can calculate moments of measures using lattice paths and orthogonal polynomials. I will explain this procedure for the specific class of so-called free Meixner polynomials. In this particular case, there are explicit combinatorial formulas not just for their moments but for their free cumulants (to be defined) as well. At the end, I will describe an alternative way to express these moments and cumulants, in terms of operators.

The material of the talk is related to the Linear Analysis talk following it, but the talks themselves will be independent.
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Abstract:
In joint work with Etingof and Witherspoon, we investigate the braid group representations arising from categories of representations of twisted quantum doubles of finite groups. For these categories, we show that the resulting braid group representations always factor through finite groups, in contrast to the categories associated with quantum groups at roots of unity. We also show that in the case of p-groups, the corresponding pure braid group representations factor through a finite p-group, which answers a question of Drinfeld.

If time permits, I will discuss some open questions and applications of these results. A preprint can be found at: http://arxiv.org/abs/math.QA/0703274
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Abstract:
Using examples, I will illustrate the main elements needed to explicitly describe the primary components of a binomial ideal, emphasizing the connections to combinatorics and (hypergeometric) differential equations. This is joint work with Alicia Dickenstein and Ezra Miller.
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