The Algebra Seminar

Fall 2008

The seminar usually meets Tuesdays in room LN 2205 at 2:50 p.m. There will be refreshments in the Anderson Reading Room at 4:00.

Organizers: Alex Feingold and Adrian Vasiu

August 26 : Organizational meeting

September 2 : Alex Feingold
Title: An Introduction to Lie Algebras
Abstract: A Lie algebra is a non-associative algebraic structure with many applications to physics. I will present the basic definitions and examples, including some of the representation theory.

September 9 : Viji Thomas
Title: A universal construction for the nonabelian tensor product and some applications, Part I.
Abstract: R. Brown and J. L. Loday first introduced the non-abelian tensor product of two groups G and H in the context of applications to homotopy theory. Let G and H be groups which act on each other via automorphisms and which act on themselves via conjugation. The actions of G and H are said to be compatible if they satisfy certain technical conditions. The non-abelian tensor product of G and H can be defined provided G and H act compatibly. In that case the non-abelian tensor product is the group generated by the tensors of elements g in G and h in H, with certain relations. In their 1995 paper Computing Schur multipliers and tensor products of finite groups, Ellis and Leonard give a universal construction for the non-abelian tensor product. In my first talk, I will prove the main theorem of this paper which describes the non-abelian tensor product of G and H as a subgroup of a quotient of the free product G*H. In my second talk, I will address some applications of this construction. It turns out that this construction is better suited for computer calculations of the non-abelian tensor product than the one given by the definition by generators and relations. We will discuss some computer calculations for non- abelian tensor products and give an easy derivation of the universal construction in the case of non-abelian tensor squares. In a recent paper, Computing the non-abelian tensor squares of polycyclic groups, Blyth and Morse apply this construction to the tensor square of a polycyclic group, showing that the tensor square of a polycyclic group is polycyclic, hence finitely presented. This opens the door for an algorithm to determine the tensor square of infinite polycyclic groups.

September 16 : Viji Thomas
Title: A universal construction for the nonabelian tensor product and some applications, Part II.
Abstract: See the abstract for Part I.

September 16, 1:15 - 2:15, LN-2205 : Thomas Zaslavsky (Cross Listing from the Combinatorics Seminar)
Title: Quasigroups via graphs
Abstract: A quasigroup is essentially a group without associativity. An n-ary quasigroup is a generalization from 2 to n independent variables. I will explain how indecomposability of biased graphs explains decomposition of n-ary quasigroups, and vice versa. There are many open questions.

September 23 : Elizabeth Wilcox
Title: An introduction to wreath products
Abstract: What is a wreath product? What's the degree of a wreath product? Why are wreath products cool? These questions and others regarding wreath products will be answered at an introductory level. The talk will start at the definition of a wreath product, explore a few examples and useful properties, and then move on to give three reasons why wreath products are some of the neatest groups around (in my opinion!).

September 30 : No Meeting
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October 7 : Cristian Lenart, SUNY, Albany
Title: On Combinatorial Formulas for Macdonald Polynomials
Abstract: Macdonald polynomials are generalizations of Weyl characters depending on two parameters. Haglund, Haiman and Loehr exhibited a combinatorial formula for the type A Macdonald polynomials in terms of a pair of statistics on fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary Lie type in terms of the corresponding affine Weyl group. In this talk, I relate the above developments, by explaining how the Ram-Yip formula compresses to a new formula, which is similar to the Haglund-Haiman-Loehr one but contains considerably fewer terms; in this context, the statistics on Young diagrams mentioned above follow naturally from more general concepts. I also explain how this work extends to types B and C, where no analog of the Haglund-Haiman-Loehr formula exists. The talk is largely self-contained.

October 7, 1:15 - 2:15, LN-2205 : Thomas Zaslavsky (Cross Listing from the Combinatorics Seminar)
Title: Tutte Functions of Matroids
Abstract: A function defined on an arbitrary minor-closed class of matroids is a "Tutte function" if it satisfies the parametrized deletion-contraction law: F(M) = de F(M\e) + ce F(M/e), whenever e is a point of M that is neither a loop nor a coloop. F need not have any other Tutte-style properties like multiplicativity. Here de and ce are constants associated with e, independent of M but depending on the point e. Functions of this kind appear in statistical physics and knot theory. Joanna Ellis-Monaghan and I are studying the modules and algebras behind Tutte functions.

October 15 (Wednesday at 5 PM): Derek Robinson, University of Illinois at Urbana-Champaign
Title: The Number of Generators of an Operator Group
Abstract: After a review of groups with an operator domain, we will discuss inequalities involving the number of generators of an operator group and of its abelianization. Several theorems will be presented which assert that these inequalities are equalities under surprisingly weak assumptions.

October 21 : Martha Kilpak
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October 28 : Ryan McCulloch
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November 4 : Jinghao Li
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November 11 : Adam Perry
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November 18 : Xiao Xiao
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November 25 : Quincy Loney
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December 2 : Dandrielle Lewis
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December 9 : Ding Ding
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