The legacy of the Wall and Guralnick conjectures

Speaker: 

Len Scott

Affiliation: 

The University of Virginia

Date: 

Fri, 08/03/2013 - 2:30pm to 3:30pm

Venue: 

Chemistry Lecture Theatre 2, the University of Sydney

Abstract: 

In 1961 G.E. Wall conjectured that the number of maximal subgroups of a finite group is less than the order of the group. The conjecture holds for all finite solvable groups (proved by Wall himself in his original paper) and holds for almost all finite simple groups, possibly all of them (proved by Liebeck, Pyber and Shalev in 2007). it is now known to be false in general, at least as originally stated, with infinitely many negative composite group examples found through a combination of computational and theoretical techniques. (I cite in particular computer calculations of Frank Luebeck, as partly inspired and later confirmed by calculations of my undergraduate student, Tim Sprowl, with theoretical input from myself and Bob Guralnick.) Somewhat surprisingly, the Wall conjecture, through a related 1986 conjecture of Bob Guralnick, has had a tremendous impact on the development of cohomology theory of finite and algebraic groups, with many positive results proved regarding first 1-cohomology with irreducible coefficients and, most recently, higher degree cohomology. The negative examples that have been now found arise from combining deep considerations in algebraic group theory (such as rational and generic cohomology theory, cohomology theory related to the Lusztig conjecture, and the Lusztig conjecture itself) with computer calculations of Kazhdan-Lusztig polynomials. I will discuss all of these things, and, as time permits, a new algorithm my student and I have developed for calculating individual Kazhdan-Lusztig polynomials (which might have been guessed ahead of time to be especially interesting), or more accurately, individual Kazhdan-Lusztig basis elements in a Hecke algebra, with minimal input from lower degree calculations.

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