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

Abstract

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