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
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August 26 : Organizational meeting
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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.
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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.
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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.
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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.
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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!).
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September 30 : No Meeting
Title:
Abstract:
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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.
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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.
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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.
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October 21 : Martha Kilpack
Title: Lattices, closure systems, and the lattice of closure
systems on a set
Abstract: We define and look at some properties of lattices and
closure systems, and how they are related. We then turn our
attention to the lattice K of closure systems on a set. The
talk is based on a paper by Nathalie Caspard and Bernard
Monjardet of Universite Paris. They show that K is atomic and
lower bounded.
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October 28 : Ryan McCulloch
Title: Recognition of Matrix Rings
Abstract: We present a criteria for a ring R to be an n by n matrix ring
and solve a problem concerning the integer quaternion ring. The
talk is based on a paper by J.C. Robson of University of Leeds.
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October 30, Thursday, 3:20 PM, LN - 2205: Bernard Badzioch (University at Buffalo, SUNY)
(Cross Listing from the
Geometry/Topology Seminar)
Title: Rigidification of homotopy structures
Abstract: Algebraic objects in general do not have good homotopy
properties: a space homotopy equivalent to e.g. a topological group
usually will have no group structure itself. On the other hand,
several classes of objects naturally occurring in algebraic topology
(loop spaces, Eilenberg-MacLane spaces, spectra) exhibit properties
closely resembling these known from algebra. These two facts inspired
an intensive study which showed that in special cases algebraic
structures can be described in homotopy meaningful terms. This
research led to the development of operads, PROPS, Gamma spaces, etc.
The talk will present an overview of these results. It will also
explain how they all admit a common generalization. As it turns out,
there is a very broad class of algebraic objects which have their
interesting homotopy theoretical analogs.
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November 4 : Jinghao Li
Title: An introduction to power series rings
Abstract: Basic facts about power series rings over a field will be stated. Proofs of Weierstrass
Division and Preparation Theorem will be sketched. The main purpose of this talk is to show that if K is a field,
then the power series ring Rm over K in m variables is a unique factorization domain (UFD). Some related topics
such as analytic functions in complex analysis will be mentioned as well.
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November 11 : Adam Perry
Title: Groups with distributive subgroup lattices
Abstract: I will be discussing the classification of groups with distributive
subgroup lattices. Amongst other things, applications to cyclic groups
will also be derived. If time permits, I will outline the proof of the
classification of locally cyclic groups.
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November 18 : Xiao Xiao
Title: Witt vectors
Abstract: I will define the ring of Witt vectors of any commutative ring with
identity. Basic properties will also be explained. Main theorem of the talk is to show that
over a perfect field $k$ of characteristic $p$, the ring of Witt vectors is the unique local
ring with maximal ideal $(p)$ and residue field $k$, plus it is $p$-adically complete and
characteristic zero. A well-known example is that the ring of Witt vectors of the finite
field with $p$ elements is the $p$-adic integers.
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November 25 : Quincy Loney
Title: (Simple) Lie Superalgebras
Abstract: Lie superalgebras arise naturally in mathematics and theoretical
physics as a tool for examining the supersymmetry of particles (bosons and
fermions). The theory of Lie superalgebras closely parallels that of Lie
algebras. This talk will introduce some of the definitions and theory
needed for V. Kac's classification of the finite dimensional (simple) Lie superalgebras.
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November 25, 1:15 - 2:15, LN-2205 : Thomas Zaslavsky (Cross Listing from the
Combinatorics Seminar)
Title: Graphic Matrices Over a Group
Abstract: Graphic matrices are such as the adjacency matrix, the incidence matrices, the
Kirkhhoff (or Laplacian) matrix, and the adjacency matrix of the line graph. Each of them has integral entries.
There are integral matrices of a signed graph (a graph with signed edges) that directly generalize the graphic matrices.
I will consider a different generalization: the entries of the matrices belong to the group ring of the sign group.
This type of matrix has interesting new properties and generalizes even further, to matrices of a gain graph, where
the edges are labelled (orientably) from any group G, with the elements of the matrix belonging to the group ring ZG
or, for another example, the complex group algebra CG.
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December 2 : Dandrielle Lewis
Title: The Order of the Automorphism Group of a Central Product
Abstract: Determining the order of the automorphism group of a finite p-group is an
important problem in group theory. It has been conjectured that if G is a non-
cyclic p-group of order p^n , where n >=3, then the order of G divides the order
of Aut(G). In this talk, I will prove that if a p-group G is the central product
of nontrivial subgroups H and A where A is abelian and the order of H divides
the order of Aut(H), then the order of G divides the order of Aut(G).
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December 2, 1:15 - 2:15, LN-2205 : Simon Joyce (Cross Listing from the
Combinatorics Seminar)
Title: The Symmetric Group and Non-Crossing Partitions
Abstract: I will define a poset relation on the symmetric group
Sn, which gives a natural order-preserving function from
Sn to the
lattice of partitions. If we restrict our attention to elements in
Sn under a particular n-cycle, we have a lattice which is
isomorphic to the lattice of non-crossing partitions. If time permits
I'll talk about some of the implications.
This work is based on a paper by Thomas Brady of Dublin City University.
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December 9 : Ilir Snopce
Title: A question of Iwasawa and the Golod-Shafarevich inequality
Abstract: For a finitely generated pro-p group G, let d(G) denote the minimal number of
topological generators of G. For a positive integer n, Iwasawa raised the question
of determining all pro-p groups G which satisfy the following condition:
d(H) - n = [G:H] (d(G)-n)
for all open subgroups H of G.
In the first part of this talk I will discuss this question. In the second part
of the talk, I will discuss the Golod-Shafarevich inequality, and show how we can use
it to answer the question of Iwasawa for finitely presented solvable pro-p groups.
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