Abstract

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Meeting ID: 951 8566 2780 Passcode: 040109

For G a finite group, F an algebraically closed field and W a faithful FG-module the McKay graph, MF (G, W), is a connected directed graph on the set of simple FG-modules. There is an edge in the graph from V1 to V2 if V2 occurs as a composition factor of V1 ⊗ W . These graphs famously come up in the McKay correspondence which says that such graphs for finite subgroups of SU2 will be affine Dynkin diagrams of type A, D or E.

In the case where the characteristic of F divides the order of G, finding the composition factors of tensor products is a very hard problem. However it might surprise you to know that taking G to be SLn(Fp), F the algebraic closure of Fp and W the standard n-dimensional FSLn(Fp)-module we can show

diam MF(G,W)= ½(p−1)(n2 −n).

In this talk I will describe these graphs in a bit more detail, give some background and explain how we are able to prove this neat formula for the diameter without explicitly constructing the graphs.