Event Detail

Event Type: 
Monday, April 21, 2008 - 09:00
208 Deady (at UO)

Speaker Info

UC San Diego

Why do matrices commute? More specifically, are there polynomial equations satisfied by the set of pairs of commuting matrices not algebraically implied by the equations AB=BA? Mel Hochster and others asked this question in the '60s, and it remains unsolved for large dimensions.(To see the problem: knowing M^2=0 tells you M is nilpotent and hence Trace(M)=0, but that linear equation doesn't lie in the ideal generated by the quadratic equations M^2=0.)
I'll talk about some related spaces of matrices that are simpler to study, which lead to some weird integer-valued invariants of permutations. Then I'll explain a statistical mechanical model that produces the same integers, but in a much more calculable manner, and use this to give a formula for the volume (or really, multidegree) of the space of commuting matrices.
This work is joint with Paul Zinn-Justin.