A Generalization of Kac Polynomials and Tensor Product of Representations of \({{\,\textrm{GL}\,}}_n(\mathbb {F}_q)\)

Given a generic k-tuple \((\mathcal {X}_1,\dots ,\mathcal {X}_k)\) $({\mathcal{X}}_1,\cdots ,{\mathcal{X}}_k)$ of split semisimple irreducible characters of \(\textrm{GL}_n(\mathbb {F}_q)\) ${\text{GL}}_n({\mathbb{F}}_q)$ , Hausel, Letellier and Rodriguez-Villegas (Adv. Math. 234:85–128, 2013, Theorem 1.4.1) constructed a star-shaped quiver \(Q=(I,\Omega )\) $Q=(I,\mathrm{\Omega })$ together with a dimension vector \(\alpha \in \mathbb {N}^I\) $\alpha \in {\mathbb{N}}^I$ and they proved that 0.0.1 \(\begin{aligned} \left\langle \mathcal {X}_1\otimes \cdots \otimes \mathcal {X}_k,1\right\rangle =a_{Q,\alpha }(q) \end{aligned}\) $$\begin{array}{c}\hfill 〈{\mathcal{X}}_1,\otimes ,\cdots ,\otimes ,{\mathcal{X}}_k,,,1〉=a_{Q,\alpha }(q)\end{array}$$ where \(a_{Q,\alpha }(t)\in \mathbb {Z}[t]\) $a_{Q,\alpha }(t)\in \mathbb{Z}[t]$ is the so-called Kac polynomial, i.e., it is the counting polynomial for the number of isomorphism classes of absolutely indecomposable representations of Q of dimension vector \(\alpha \) $\alpha$ over finite fields. Moreover, it was conjectured by Kac (1983) and proved by Hausel-Letellier-Villegas (Ann. of Math. (2) 177(3):1147–1168, 2013) that \(a_{Q,\alpha }(t)\) $a_{Q,\alpha }(t)$ has non-negative integer coefficients. From the above formula together with Kac’s (1983) results, they deduced that \(\left\langle \mathcal {X}_1\otimes \cdots \otimes \mathcal {X}_k,1\right\rangle \ne 0\) $〈{\mathcal{X}}_1,\otimes ,\cdots ,\otimes ,{\mathcal{X}}_k,,,1〉\ne 0$ if and only if \(\alpha \) $\alpha$ is a root of Q; moreover, \(\left\langle \mathcal {X}_1\otimes \cdots \otimes \mathcal {X}_k,1\right\rangle =1\) $〈{\mathcal{X}}_1,\otimes ,\cdots ,\otimes ,{\mathcal{X}}_k,,,1〉=1$ exactly when \(\alpha \) $\alpha$ is a real root. In this paper, we extend their result to any k-tuple \((\mathcal {X}_1,\dots ,\mathcal {X}_k)\) $({\mathcal{X}}_1,\cdots ,{\mathcal{X}}_k)$ of split semisimple irreducible characters (which are not necessarily generic). To do that, we introduce a stratification indexed by subsets \(V\subset \mathbb {N}^I\) $V\subset {\mathbb{N}}^I$ on the set of k-tuples of split semisimple irreducible characters of \(\textrm{GL}_n(\mathbb {F}_q)\) ${\text{GL}}_n({\mathbb{F}}_q)$ . The part corresponding to \(V=\{\alpha \}\) $V=\{\alpha \}$ consists of the subset of generic k-tuples \((\mathcal {X}_1,\dots ,\mathcal {X}_k)\) $({\mathcal{X}}_1,\cdots ,{\mathcal{X}}_k)$ . A k-tuple \((\mathcal {X}_1,\dots ,\mathcal {X}_k)\) $({\mathcal{X}}_1,\cdots ,{\mathcal{X}}_k)$ in the stratum corresponding to \(V\subset \mathbb {N}^I\) $V\subset {\mathbb{N}}^I$ is said to be of level V. A representation \(\rho \) $\rho$ of \((Q,\alpha )\) $(Q,\alpha )$ is said to be of level at most \(V\subset \mathbb {N}^I\) $V\subset {\mathbb{N}}^I$ if the dimension vectors of the indecomposable components of \(\rho \otimes _{\mathbb {F}_q}\overline{\mathbb {F}}_q\) $\rho {\otimes }_{{\mathbb{F}}_q}{\overline{\mathbb{F}}}_q$ belong to V. Given a k-tuple \((\mathcal {X}_1,\dots ,\mathcal {X}_k)\) $({\mathcal{X}}_1,\cdots ,{\mathcal{X}}_k)$ of level V, our main theorem is the following generalization of Formula (0.0.1) \(\begin{aligned} \left\langle \mathcal {X}_1\otimes \cdots \otimes \mathcal {X}_k,1\right\rangle =M_{Q,\alpha ,V}(q) \end{aligned}\) $$\begin{array}{c}\hfill 〈{\mathcal{X}}_1,\otimes ,\cdots ,\otimes ,{\mathcal{X}}_k,,,1〉=M_{Q,\alpha ,V}(q)\end{array}$$ where \(M_{Q,\alpha ,V}(t)\in \mathbb {Z}[t]\) $M_{Q,\alpha ,V}(t)\in \mathbb{Z}[t]$ is the counting polynomial for the number of isomorphism classes of representations of \((Q,\alpha )\) $(Q,\alpha )$ over \(\mathbb {F}_q\) ${\mathbb{F}}_q$ of level at most V. Moreover, we prove a formula expressing \(M_{Q,\alpha ,V}(t)\) $M_{Q,\alpha ,V}(t)$ in terms of Kac polynomials and so we get a formula expressing any multiplicity \(\left\langle \mathcal {X}_1\otimes \cdots \otimes \mathcal {X}_k,1\right\rangle \) $〈{\mathcal{X}}_1,\otimes ,\cdots ,\otimes ,{\mathcal{X}}_k,,,1〉$ in terms of generic ones. As another consequence, we prove that \( \left\langle \mathcal {X}_1\otimes \cdots \otimes \mathcal {X}_k,1\right\rangle \) $〈{\mathcal{X}}_1,\otimes ,\cdots ,\otimes ,{\mathcal{X}}_k,,,1〉$ is a polynomial in q with non-negative integer coefficients and we give a criterion for its non-vanishing in terms of the root system of Q.