A $q$-analog of the adjacency matrix of the $n$-cube

Ghosh, Subhajit¹ ; Srinivasan, Murali K.²

¹ Bar-Ilan University Department of Mathematics Ramat-Gan 5290002 (Israel)
² Indian Institute of Technology, Bombay Department of Mathematics Powai, Mumbai 400076 (India)

Algebraic Combinatorics, Tome 6 (2023) no. 3, pp. 707-725

Résumé

Let $q$ be a prime power and define ${\left(n\right)}_q=1+q+q^2+\cdots +q^{n-1}$, for a nonnegative integer $n$. Let $B_q\left(n\right)$ denote the set of all subspaces of ${𝔽}_q^n$, the $n$-dimensional $𝔽q$-vector space of all column vectors with $n$ components.

Define a $B_q\left(n\right)\times B_q\left(n\right)$ complex matrix $M_q\left(n\right)$ with entries given by

$$M_q\left(n\right)(X,Y)=\left\{\begin{array}{cc}1\hfill & \text{if}Y\subseteq X,dim\left(Y\right)=dim\left(X\right)-1,\hfill \\ q^k\hfill & \text{if}X\subseteq Y,dim\left(Y\right)=k+1,dim\left(X\right)=k,\hfill \\ 0\hfill & \text{otherwise.}\hfill \end{array}\right.$$

We think of $M_q\left(n\right)$ as a $q$-analog of the adjacency matrix of the $n$-cube. We show that the eigenvalues of $M_q\left(n\right)$ are

$${(n-k)}_q-{\left(k\right)}_q\text{with}\text{multiplicity}{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_q,k=0,1,...,n,$$

and we write down an explicit canonical eigenbasis of $M_q\left(n\right)$. We give a weighted count of the number of rooted spanning trees in the $q$-analog of the $n$-cube.

Détail

Reçu le : 2022-04-29
Révisé le : 2022-11-15
Accepté le : 2022-12-03
Publié le : 2023-06-19

DOI : 10.5802/alco.282

Classification : 05E18, 05C81, 20C30
Keywords: $n$-cube, $q$-analog

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits