The complexity of the $q$-analog of the $n$-cube

TL;DR

This paper provides a positive combinatorial formula for counting spanning trees in the q-analog of the n-cube and offers explicit block diagonalization related to the symmetry group actions.

Contribution

It introduces a new combinatorial formula for the complexity of the q-analog of the n-cube and details the block diagonalization of the associated symmetry algebra.

Findings

Derived a positive combinatorial formula for the number of spanning trees.

Explicitly diagonalized the commutant algebra of the group action.

Connected the formula to the representation theory of GL(n, F_q).

Abstract

We present a positive, combinatorial, good formula for the complexity (= number of spanning trees) of the $q$-analog of the $n$-cube. Our method also yields the explicit block diagonalization of the commutant of the $GL(n,F_q)$ action on the $q$-analog of the Boolean algebra.