Midwest Algebra and Group Theory Conference

DePaul University, November 3-4, 2001

ABSTRACTS

Yiftach Barnea, University of Wisconsin-Madison

Maximal graded subalgebras of some $\Z^n$-graded Lie algebras

Abstract: Let $\g$ be a finite dimensional simple Lie algebra over a field $\F$. Let

$L_n(\g)=\g\otimes_{\F}\F[x_1,x_1^{-1},x_2,x_2^{-1},\ldots,x_n,x_n^{-1}]$, i.e. Laurent polynomial in $n$ variables over $\g$. Notice that $L_n(\g)$ has a structure of $\Z^n$-graded Lie algebra. We study maximal graded subalgebras of $L_n(\g)$. Our main tool is multilinear central polynomials. The case $n=1$ is a joint work with A. Shalev and E. Zelmanov.

Jeffrey Bergen, DePaul University

``Gradings and Skew Derivations of Algebras’’

Abstract: The gradings of associative algebras by groups has proven to be an extremely useful tool in examining the invariants of derivations and automorphisms.

We review some of the results in this area and then branch off in several other directions.

  1. We examine the invariants of $\sigma$-skew derivations $\delta$, where $\sigma$ is an automorphism. If $\delta$ is a $q$-skew derivation, then results on group gradings can be used. However, for more general skew derivations, results on group gradings are not applicable and we discuss some of the combinatorial arguments needed to take their place.
  2. We also examine a class of nonassociative algebras we call $(\alpha,\beta, \gamma)$-algebras. Many of the results for gradings of associative algebras do not hold in this situation and we discuss some results on gradings which do hold in this situation and can be used to study the invariants of derivations and automorphisms.
The work discussed in this talk is joint with Piotr Grzeszczuk.

Harvey I. Blau, Northern Illinois University

``Sylow Theory for Table Algebras’’

Abstract: Table algebras are finite dimensional algebras over the complex numbers with a distinguished basis whose properties generalize those of group algebras, centers of group algebras, double coset algebras, character rings, and the adjacency algebras derived from association schemes. Generalizations
of the Sylow theorems are proved for a broad class of table algebras.  This is joint work with Paul-Hermann Zieschang.

Everett C. Dade, University of Illinois at Urbana-Champaign

``Generalized Glauberman Correspondences''

Abstract: Suppose that G is a normal subgroup of a finite group E, with a cyclic factor group E/G.  Let E' be the set of all elements e in E such that the coset eG generates the cyclic group E/G.  Look at the additive group Ch(E|E') of all virtual complex characters of E vanishing outside E'. The usual inner product <X,Y>_E of virtual characters of E is positive definite on Ch(E|E'). It turns out that there is a one to one correspondence between all irreducible characters of G fixed by E and certain configurations of characters in Ch(E|E') describable using only the inner product <X,Y>_E. So the additive group Ch(E|E') and the inner product <X,Y>_E on it determine in some sense the set Irr^E(G) of all E-invariant irreducible characters of G.
     Now we add a subgroup F of E such that FG = E, The above conditions are satisfied with F and its intersection H with G in place of E and G, respectively. In this situation the role of E' is played by its intersection F' with F, while the inner product <X,Y>_E is replaced by <X,Y>_F. If F' is a trivial intersection subset of E with F as its normalizer, then induction of characters is a bijection of Ch(F|F') onto Ch(E|E'), preserving inner products. In view of the above correspondence, this bijection determines a bijection of Irr^F(H) onto Irr^E(G).
        Among the bijections which can be constructed in this manner are the Glauberman correspondence for cyclic acting groups A, a relative Glauberman correspondence, and, for a few groups of Lie type, what looks suspiciously like Shintani Descent.

George Glauberman, University of Chicago

"A Conjecture of Oliver about Finite p-groups"

Abstract:  Suppose p is an odd  prime.  In unpublished work, Bob Oliver has proved the Martino-Priddy Conjecture (a condition for two p-completed classifying spaces to be homotopy equivalent) for p.  However, he needed to assume the Classification of Finite Simple Groups.  He observed that one could remove this assumption and possibly obtain a further application of his methods to the representation theory of finite groups (particularly block theory)
if one could prove a conjecture about the Thompson subgroup of a finite p-group.  I plan to discuss the conjecture and some partial results, in the hope of eliciting further interest in the conjecture.

Corneliu Hoffman, Bowling Green State University

"Curtis-Phan-Tits theory" (joint with S Shpectorov, C. Bennett, R. Gramlich)

Abstract: In 1977 Kok-Wee Phan published a theorem (see \cite{Ph1}) on generation of the special unitary group $SU(n+1,q^2)$ by a system of its subgroups isomorphic to $SU(3,q^2)$. This theorem is similar in spirit to the famous Curtis-Tits theorem. In fact, both the Curtis-Tits theorem and Phan's theorem were used as principal identification tools in the classification of finite simple groups. The aim of this talk is to describe a geometric approach to the proofs of these theorems and some new results of the same type.

Martin Isaacs, University of Wisconsin-Madison

``Orbit sizes of actions on characters and classes''

Abstract: Let A be a group of automorphisms of a finite group G. Then there are natural actions of A on the set Cl(G) of conjugacy classes of G and on the set Irr(G) of irreducible characters of G. Of course, |Cl(G)| = |Irr(G)| and it is well known that there are equal numbers of A-orbits on these two sets. Although the orbit sizes in the two actions are usually different, we show that there are some previously unobserved connections between these orbit sizes.

Edward Letzter, Temple University

``Toward an Effective Representation Theory for Finitely Presented Noncommutative Algebras’’

Abstract: Let n be a positive integer, and let R be a finitely presented -- but not necessarily finite dimensional -- algebra over a field k. Consider the following questions:

In this talk I will present algorithms (over k) for answering these questions.

Mark Lewis, Kent State University

``Connectedness of Degree Graphs of Nonsolvable Groups.’’(joint work with Donald White)

Abstract: Let $G$ be a group, and write ${\rm cd} (G)$ for the character degrees of $G$. Take $\rho (G)$ to be the set of primes that divide degrees in ${\rm cd} (G)$. The degree graph of $G$, written $\Delta (G)$, is the graph with $\rho (G)$ as its vertex set and there is an edge between $p$ and $q$ if $pq$ divides some degree $a \in {\rm cd} (G)$. It is known that $\Delta (G)$ has at most three connected components.
    We prove that if $G$ is simple, then $G$ is disconnected if and only if $G \cong {\rm PSL}_2 (q)$ for some prime power $q \ge 4$. We prove that $\Delta (G)$ has three connected components if and only if $G \cong{\rm PSL}_2 (2^n) \times A$ where $n \ge 2$ is an integer and $A$ is an abelian group. Finally, we classify the nonsolvable groups $G$ where $\Delta (G)$ has two connected components.

Alexander Moreto, University of Wisconsin-Madison

"Character degrees and nilpotence class of finite $p$-groups"

Abstract: We say that a finite set $\SS$ of powers of a prime number $p$ containing 1 is class bounding if and only if the nilpotence class of any $p$-group whose set of irreducible character degrees is $\SS$ is bounded by some constant that depends on $\SS$. In 1968 Isaacs and Passman proved that if $|\SS|=2$ then $\SS$ is class bounding if and only if $p$ does not belong to $\SS$. These sets of cardinality two were the only known class bounding until 2001 when, in joint work with I. M. Isaacs, we found class bounding sets of arbitrarily large size. We also extended the result of Isaacs and Passman to sets of cardinality 3. All these results suggested that, perhaps, a set $\SS$ is class bounding if and only if $p$ does not belong to $\SS$. The ``only if" has recently been proved by Isaacs and Slattery. We will discuss a joint work with A. Jaikin-Zapirain where we show that the ``if" part is false and we find new class bounding sets. This work shows the relevance of pro-$p$ groups in this problem: we caracterize class bounding sets in terms of certain $p$-adic space groups.

Lance W. Small, University of California, San Diego

``Affine Rings---Good, Bad and Ugly’’

Abstract: We will discuss GK 2 rings and some recent examples. The constructions used will be then applied to other questions on affine rings relating to the nullstellensatz and endomorphism rings.