Abstract In this paper we study rings of invariants arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form K [ U ] Γ where Γ is a product of general linear groups over a field K of characteristic zero, and U is a finite dimensional rational representation of Γ. We will calculate the Hilbert series of such rings using the representation theory of the symmetric groups and Schur-Weyl duality. We focus on the case where U = End ( W ⊕ k ) and Γ = GL ( W ) and on the case where U = End ( V ⊗ W ) and Γ = GL ( V ) × GL ( W ) , though the methods introduced here can also be applied in more general framework. For the two aforementioned cases we calculate the Hilbert function of the ring of invariants in terms of Littlewood-Richardson and Kronecker coefficients. When the vector spaces are of dimension 2 we also give an explicit calculation of this Hilbert series, using Mathematica (see the appendix by Dejan Govc).

中文翻译：

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通过对称群表示的某些不变量环的希尔伯特函数（Dejan Govc 的附录）

摘要 在本文中，我们研究了在有限维代数结构研究中出现的不变量环。我们遇到的环是 K [ U ] Γ 形式的分级环，其中 Γ 是特征为零的域 K 上的一般线性群的乘积，U 是 Γ 的有限维有理表示。我们将使用对称群的表示理论和 Schur-Weyl 对偶性来计算此类环的希尔伯特级数。我们专注于 U = End ( W ⊕ k ) 和 Γ = GL ( W ) 的情况以及 U = End ( V ⊗ W ) 和 Γ = GL ( V ) × GL ( W ) 的情况，尽管方法这里介绍的也可以应用在更通用的框架中。对于上述两种情况，我们根据 Littlewood-Richardson 和 Kronecker 系数计算不变量环的希尔伯特函数。