Let G be a linearly reductive group acting on a vector space V, and f a semi-invariant polynomial on V. In this paper we study systematically decompositions of the Bernstein–Sato polynomial of f in parallel with some representation-theoretic properties of the action of G on V. We provide a technique based on a multiplicity one property, that we use to compute the Bernstein–Sato polynomials of several classical invariants in an elementary fashion. Furthermore, we derive a “slice method” which shows that the decomposition of V as a representation of G can induce a decomposition of the Bernstein–Sato polynomial of f into a product of two Bernstein–Sato polynomials – that of an ideal and that of a semi-invariant of smaller degree. Using the slice method, we compute Bernstein–Sato polynomials for a large class of semi-invariants of quivers.

中文翻译：

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Bernstein-Sato多项式和切片的分解

让G ^是作用在矢量空间中的线性还原性基团V，和˚F在半不变多项式V。在本文中，我们系统地研究了f的Bernstein-Sato多项式的分解以及G对V的作用的某些表示理论性质。我们提供了一种基于多重性质的技术，该技术用于以基本方式计算几个经典不变式的Bernstein-Sato多项式。此外，我们推导了一种“切片方法”，该方法表明V的分解表示为G可以将f的Bernstein-Sato多项式分解为两个Bernstein-Sato多项式的乘积-理想乘积和较小次半不变式的乘积。使用切片法，我们为一类颤动的半不变量计算了Bernstein-Sato多项式。