SIMPLE GROUPS STABILIZING POLYNOMIALS
SIMPLE GROUPS STABILIZING POLYNOMIALS
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简单群稳定多项式
DOI：
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发表时间：
2013
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通讯作者：
R. Guralnick
中科院分区：
文献类型：
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作者：
S. Garibaldi;R. Guralnick
本文研究了向量空间V上多项式函数f的线性变换g的确定问题，使得f =f。当$f$在不可约地作用于$V$的单代数群$G$下不变时，我们注意到$的子群 ext{GL}（V）稳定f的单位分支G，给出了将各种群，包括最大例外群E_{8}，实现为多项式和代数的自同构群的应用.我们表明，从一个简单的群$G$和一个不可约表示$V$，人们几乎总是可以找到一个$f$的稳定剂有单位分量$G$，并没有这样的$f$存在于短列表中的排除情况。这依赖于我们的核心技术结果，夹杂物$G ext{SL}（V）$使得$V/H$具有与$V/G$相同的维数。本文的主要结果是新的，即使在特殊情况下，$k$是复数。
We study the problem of determining, for a polynomial function $f$ on a vector space $V$, the linear transformations $g$ of $V$ such that $fcirc g=f$. When $f$ is invariant under a simple algebraic group $G$ acting irreducibly on $V$, we note that the subgroup of $ ext{GL}(V)$ stabilizing $f$ often has identity component $G$, and we give applications realizing various groups, including the largest exceptional group $E_{8}$, as automorphism groups of polynomials and algebras. We show that, starting with a simple group $G$ and an irreducible representation $V$, one can almost always find an $f$ whose stabilizer has identity component $G$, and that no such $f$ exists in the short list of excluded cases. This relies on our core technical result, the enumeration of inclusions $G<Hleqslant ext{SL}(V)$ such that $V/H$ has the same dimension as $V/G$. The main results of this paper are new even in the special case where $k$ is the complex numbers.