University of Cambridge > > Junior Algebra and Number Theory seminar > The diameter of the modular McKay graph of SLn(Fp).

The diameter of the modular McKay graph of SLn(Fp).

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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.

This talk is part of the Junior Algebra and Number Theory seminar series.

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