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

Pub Date : 2024-04-03 DOI:10.1007/s00031-024-09854-3

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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.

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Kac 多项式的广义化和 $${{\,\textrm{GL}\,}}_n(\mathbb {F}_q)$$ 的张量乘积表征

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 和 Rodriguez-Villegas (Adv. Math. 234:85-128, 2013, Theorem 1.4.1) 构建了一个星形四元组 $Q=(I,\Omega )$ 以及一个维向量 $\alpha \in \mathbb {N}^I$ ，他们证明了 0.0.1 $$（begin{aligned}）。\其中 $a_{Q,\alpha }(t)\in \mathbb {Z}[t]$ 是所谓的 Kac 多项式，即、它是有限域上维度为向量 $\alpha $的 Q 的绝对不可分解表示的同构类数的计数多项式。此外，Kac（1983）猜想并由 Hausel-Letellier-Villegas 证明（Ann. of Math. (2) 177(3):1147-1168, 2013），$a_{Q,\alpha }(t)$ 具有非负整数系数。根据上述公式和 Kac（1983）的结果，他们推导出：当且仅当 $\alpha $ 是 Q 的根时，$left\langle \mathcal {X}_1otimes \cdots \otimes \mathcal {X}_k,1\right\rangle \ne 0$ 是 Q 的根；此外，当（alpha）是实数根时，（left/langle /mathcal {X}_1/otimes /cdots /otimes /mathcal {X}_k,1/right/rangle =1）正好是实数根。在本文中，我们将他们的结果扩展到任何k-tuple $(\mathcal {X}_1,\dots ,\mathcal {X}_k)$ 的分裂半简单不可还原字符（不一定是泛型的）。为此，我们在$\textrm{GL}_n(\mathbb {F}_q)$的可分割半简单不可还原字符的k元组集合上引入一个由子集$V\subset \mathbb {N}^I$索引的分层。对应于 $V=\{alpha\}$ 的部分由通用 k 元组子集 $(\mathcal {X}_1,\dots ,\mathcal {X}_k)$ 组成。在对应于 $V\subset \mathbb {N}^I$ 的层中的 k 元组 $(\mathcal {X}_1,\dots ,\mathcal {X}_k)$ 称为 V 层的。如果((Q,\alpha )\)的不可分解分量的维向量属于 V，那么(\rho \otimes _\mathbb {F}_q}\overline\mathbb {F}_q}\)的表示(\rho \)被认为最多是 V 层的。给定 V 层的 k 元组 $(\mathcal {X}_1,\dots ,\mathcal {X}_k)$, 我们的主要定理是公式（0.0.1）的以下概括 $$\begin{aligned}\left\langle （mathcal {X}_1\otimes ）（cdots （otimes ）（mathcal {X}_k,1\right\rangle =M_{Q,\alpha ,V}(q) （end{aligned}}$$ 其中（M_{Q,\alpha 、V}(t)/in \mathbb {Z}[t]\) 是最多 V 层的$mathbb {F}_q$ 上的$(Q,\alpha )$表示的同构类数的计数多项式。此外，我们证明了一个用 Kac 多项式表达 $M_{Q,\alpha ,V}(t)$的公式，因此我们得到了一个用通项表达任意倍数 $\left\langle \mathcal {X}_1\otimes \cdots \otimes \mathcal {X}_k,1\right\rangle $的公式。作为另一个结果，我们证明了 $ \left\langle \mathcal {X}_1\otimes \cdots \otimes \mathcal {X}_k,1\right\rangle $是一个在 q 中具有非负整数系数的多项式，并且我们给出了它在 Q 的根系统中不相等的判据。

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