On the modular Mckay graph of SLn(p) with respect to its standard representation

Abstract

Let F be an algebraically closed field of prime characteristic p. The modular McKay graph of G:=SLn(p) with respect to its standard FG-module W is the connected, directed graph whose vertices are the irreducible FG-modules and for which there is an edge from a vertex V1 to V2 if V2 occurs as a composition factor of the tensor product V1⊗W. We show that the diameter of this modular McKay graph is 12(p−1)(n2−n).