The Algebra Seminar

FALL 2002

September 10 : Alex Feingold
Title: ``Bosons and Fermions: Some Applications of Representation Theory to Particle Physics''
Abstract: I will give some explicit representations of Lie algebras constructed from Clifford or Weyl algebras, which can be interpreted as fermionic or bosonic creation and annihilation operators.

September 24 : Ben Brewster
Title: ``Fit(S_3)''
Abstract: A Fitting class is a class of finite groups closed under the closure operations S_N, taking subnormal subgroups, and N_0, taking products of normal subgroups. It is amazingly an open question - can one give an alternative description of the groups in Fit(S_3), the smallest Fitting class containing S_3, the symmetric group of degree 3 ? The intent of the seminar is to give some positive and a negative result about groups in this Fitting class. Maybe some insight into the nature of the difficulty in classifying all such groups can be produced. There are several related and interesting questions about this Fitting class which can be discussed, time providing.

October 1 : Dikran B. Karagueuzian
Title: ``Kirillov Theory for Finite Groups''
Abstract: The Kirillov theory gives a one-to-one correspondence between the unitary representations of a nilpotent Lie group and the group's coadjoint orbits. "Coadjoint orbits" are the orbits of the Lie group acting by conjugation on the dual of its Lie algebra. I will discuss analogues of this correspondence for finite groups.

October 8 : Joshua Palmatier
Title: ``A Property of Identity Ceilings for Finite, Totally-Ordered M-Zeroids''
Abstract:

October 15 : Marcin Mazur
Title: ``Around a theorem of Jordan''
Abstract: This will be an expository talk about a well known theorem of Jordan and its relatively recent generalization, the only known proof of which relies on the classification of finite simple groups. We will describe arithmetic applications of these theorems in order to motivate a search for a "nicer" proof of the above mentioned generalization of Jordan's result.

October 22 : Alex Feingold
Title: ``Representation Theory of Lie Algebras: From Characters To Power Series Identities''
Abstract: This will be an introduction to representations of Lie algebras, their characters, and connections with combinatorics coming from the Weyl character formula. Both finite and infinite dimensional examples (Kac-Moody theory) will be discussed.

October 29 : Walter Carlip
Title: ``Prime Testing in Polynomial Time: The Agrawal-Kayal-Saxena Theorem''
Abstract: In August of this year, Manindra Agrawal, Neeraj Kayal, and Nitin Saxena of the Indian Institute of Technology in Kanpur, proved a long-standing conjecture concerning primality testing by providing a deterministic polynomial-time algorithm for deciding whether a given integer is prime. Their result is remarkable in that their algorithm is straight-forward and easily proved. In this expository talk I will describe the Agrawal-Kayal-Saxena algorithm, prove its correctness, and analyze its complexity.

November 5 : Gail Yamskulna
Title: ``Vertex operator algebras and associative algebras''
Abstract: Let V be a vertex operator algebra. We construct a sequence of associative algebras A_n(V) (n=0,1,2,3,...). We will show that there is a one to one correspondence between the isomorphism classes of simple objects in the category of A_n(V)-modules which cannot factor through A_{n-1}(V) and V-modules respectively.

November 12 : Alex Feingold
Title: ``Representation Theory of Lie Algebras: From Characters To Power Series Identities (Continued)''
Abstract: I will continue and complete the talk started on October 22.

November 19 : Joseph Petrillo
Title: ``Covering and Avoidance in a Direct Product''
Abstract: Let G be a group, H/K a chief factor of G, and U a subgroup of G. We say that U covers H/K if H <= UK, and we say that U avoids H/K if U (intersect) K <= K. If U covers or avoids each chief factor of G, then U is said to have the cover-avoidance property in G, and U is called a CAP-subgroup of G. In this talk, two types of necessary and sufficient conditions for U to be a CAP-subgroup of a direct product G = G1 x G2 will be discussed. The first depends upon the normal subgroups of the components G1 and G2, and the second has a fundamental connection with Goursat's theorem on the subgroup structure of a direct product.

November 26 : Joseph Smith
Title: ``Nonisomorphic Groups with Identical Subgroup Lattices''
Abstract: After a brief overview, we will discuss in detail the Rottlander group, which is an example of the above.

December 3 : Omar Saldarriaga
Title: ``Tensor Products of Lie Algebra Modules and Berenstein- Zelevinsky Triangles''
Abstract: Tensor products of irreducible modules for finite dimensional simple Lie algebras can be decomposed into the direct sum of irreducibles, with multiplicities. Techniques for computing these decompositions will be discussed, including the combinatorial objects known as BZ triangles.

December 10 : Lori Koban
Title: ``Supersolvable Geometric Lattices''
Abstract: A sufficient condition for the characteristic polynomial of a geometric lattice to have a complete integral factorization is that the lattice be supersolvable, which means it has a maximal chain of modular elements. (However, supersolvability is not a necessary condition for such a factorization.) Other than some basic lattice theory, I will not assume prior knowledge of the topics discussed during this talk. This talk is part I of Ms. Koban's candidacy exam. All are welcome to attend. The examining committee consists of Laura Anderson, Matthias Beck, and Thomas Zaslavsky. On Wednesday, December 11 part II of Ms. Koban's candidacy exam will be given in the Combinatorics and Number Theory Seminar, 4:40-5:40 in LN-2205, under the title: ``Supersolvable Matroids of Biased Graphs''. The abstract for that talk is: In his 1997 paper "A characterization of supersolvable signed graphs", Young-Jin Yoon presents necessary and sufficient conditions for the bias matroid of a signed graph to be supersolvable. In his 2001 paper "Supersolvable frame-matroid and graphic-lift lattices", Zaslavsky does the same for biased graphs, a generalization of signed graphs. I will discuss why the two results are not compatible and will prove parts of the correct theorem.