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

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.

arXiv comment12 pages. This paper is a revised and expanded version of part of the author's previous paper " Counting spanning trees of the hypercube and its $q$-analogs by explicit block diagonalization" (arXiv:1104.1481), to which a reference has been added