On the modular McKay graph of $SL_n(p)$ with respect to its standard representation

Abstract

Let $F$ be an algebraically closed field of prime characteristic $p$. The modular McKay graph of $G:=SL_n(p)$ 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 $V_1$ to $V_2$ if $V_2$ occurs as a composition factor of the tensor product $V_1 \otimes W$. We show that the diameter of this modular McKay graph is $\frac{1}{2} (p-1)(n^2-n)$.