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

TL;DR

This paper determines the diameter of the modular McKay graph for the special linear group over a finite field, revealing a precise formula based on the prime characteristic and the group's dimension.

Contribution

It provides an explicit formula for the diameter of the modular McKay graph of SL_n(p) with respect to its standard module, a new result in modular representation theory.

Findings

Diameter of the modular McKay graph is (p-1)(n^2-n)/2.

The graph is connected and its structure is explicitly characterized.

The result links group parameters to graph properties in modular representation theory.

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 $\frac{1}{2} (p-1)(n^2-n)$.