Closed formula for Poincaré series in terms of adjacency matrix.

Question

Let $Q$ be a finite quiver with vertex set $I$. For each $n = 0, 1, 2, \dots,$ let $k^{(n)}Q \subset kQ$ be the $k$-linear span of all paths of length $n$, in particular, we have$$k^{(0)}Q = k\{I\}.$$Clearly, one has a vector space direct sum decomposition$$kQ = \bigoplus_{n \ge 0} k^{(n)}Q$$that gives $kQ$ a $\mathbb{Z}_{\ge 0}$-grading. What is a closed formula for the Poincaré series$$P_{kQ}(t) = \sum_{n \ge 0} \text{dim}_k (k^{(n)}Q) \cdot t^n$$in terms of the adjacency matrix$$A_Q := \|a_{i,j}\|_{i, j \in I}?$$Here, we write $a_{i, j}$ for the number of edges $j \to i$.

Answer 1

Say the quiver $Q$ has adjacency matrix $A$. Then the $ij$ entry of $A^n$ counts the number of paths of length $n$ from $i$ to $j$. So $\dim k^{(n)}Q$ is the sum of the entries of $A^n$. Therefore your generating function is the sum of the entries of the matrix $\sum_{n\ge0} A^nt^n=({\rm Id}-At)^{-1}$. Note one can write the entries of an inverse matrix $B^{-1}$ as rational functions of $B$'s entries using the formula for inverses in terms of adjugate matrices: $B^{-1}=\det(B)^{-1}{\rm adj}(B)$.