Let $A$ be any $k\times k$ matrix. Also, let $\otimes$ denote the Kronecker product and define $A^{\otimes n},$ the $n$th Kronecker power of $A$, by $A^{\otimes 1}:=A$ and $A^{\otimes n}:=A\otimes A^{\otimes (n-1)}.$
Next, let $[k]=\{1,2,\dots, k\}$ be an alphabet and any $w\in [k]^n$ be a $k$-ary word of length $n$. We write $w:=w_1w_2\cdots w_n$. Since $A^{\otimes n}$ is a $k^n \times k^n$ matrix and there are $k^n$ total $k$-ary words of length $n$, we can label rows and columns of $A^{\otimes n}$ using elements of $[k]^n$. We use lexicographic ordering so that the case $n=1$ corresponds to the usual indexing of $A$.
Normally, we are not interested in the exact entries of Kronecker products. With such labeling above, however, we obtain a simple description of entries of $A^{\otimes n}.$ For $u,v\in [k]^n$, let $A^{\otimes n}(u,v)$ denote the entry of $A^{\otimes n}$ corresponding to row $u$ and column $v$. Then, one can easily show by induction that $$A^{\otimes n}(u,v)=\prod_{i=1}^n A(u_i,v_i).$$ If $A$ is a diagonal matrix, we further get the combinatorial expression $$A^{\otimes n}(w,w)=\prod_{i=1}^k A(i,i)^{c_i(w)}$$ where $c_i(w)$ counts the number of $i$'s in $w$.
Question: I was wondering if there are any references/results that further exploit the idea of labeling entries of Kronecker powers using $k$-ary words.