论文标题
2x2矩阵不变的分离品种
The separating variety for 2x2 matrix invariants
论文作者
论文摘要
令$ g $为线性代数组，在$ g $ -variety $ \ mathcal {v} $上线性作用，让$ k [\ mathcal {v}]^g $是不变多项式函数的相应代数。一个分离集合$ s \ subseteq k [\ Mathcal {v}]^g $是一组多项式，该属性具有所有$ v，w \ in \ Mathcal {v} $，如果存在K [\ Mathcal {v}]^g $ v $和$ w $和$ f $，然后$ f \ $ f \ in \ in \ f \ in \ f \ in $ f \ in $ f \ $ f $ and $ f $，然后$ f $ f \ $ f \ $ f \ $ f \ $ f \ $ f \ $ f \ $ f \ $ f \。 在本文中，我们考虑了$ g = \ mathrm {gl} _2（\ mathbb {c}）$在品种上的$ \ mathcal {m} _2 _2^n $ of $ n $ - $ 2 \ $ 2 \ times times times 2 $矩阵的矩阵。最小生成集$ s_n $ of $ \ mathbb {c} [\ mathcal {m} _2^n]^g $是众所周知的，$ | s_n | = \ frac16（n^3+11n）$。在最近的工作中，Kaygorodov，Lopatin和Popov表明，对于所有$ n \ geq 1 $，$ s_n $是通过包含的最小分离设置，即$ s_n $的适当子集是一个分开集。这并不一定意味着$ s_n $在所有分离集合中的基数最小，用于$ \ mathbb {c} [\ mathcal {m} _2^n]^g $。我们的主要结果表明，任何$ \ mathbb {c} [\ Mathcal {m} _2^n]^g $的分离设置都具有心脏$ \ geq 5n-5 $。特别是，没有尺寸$ \ dim（\ Mathbb {c} [\ Mathcal {m} _2^n]）= 4n-3 $ for $ n \ geq 3 $。此外，$ s_3 $的确具有最低的基数作为分离集，但是对于$ n \ geq 4 $，可能存在的分离集可能比$ s_n $较小。我们表明，对于$ n \ geq 5 $，实际上确实存在一个比$ s_n $的较小的分离集。我们还证明了$ \ mathrm {sl} _2（\ Mathbb {c}）\ Times \ Times \ Mathrm {SL} _2（\ Mathbb {C}）$ on $ \ MATHCAL {M MATHCAL {M} _2^n $。
Let $G$ be a linear algebraic group acting linearly on a $G$-variety $\mathcal{V}$, and let $k[\mathcal{V}]^G$ be the corresponding algebra of invariant polynomial functions. A separating set $S \subseteq k[\mathcal{V}]^G$ is a set of polynomials with the property that for all $v,w \in \mathcal{V}$, if there exists $f \in k[\mathcal{V}]^G$ separating $v$ and $w$, then there exists $f \in S$ separating $v$ and $w$. In this article we consider the action of $G = \mathrm{GL}_2(\mathbb{C})$ on the variety $\mathcal{M}_2^n$ of $n$-tuples of $2 \times 2$ matrices by simultaneous conjugation. Minimal generating sets $S_n$ of $\mathbb{C}[\mathcal{M}_2^n]^G$ are well-known, and $|S_n| = \frac16(n^3+11n)$. In recent work, Kaygorodov, Lopatin and Popov showed that for all $n \geq 1$, $S_n$ is a minimal separating set by inclusion, i.e. that no proper subset of $S_n$ is a separating set. This does not necessarily mean that $S_n$ has minimum cardinality among all separating sets for $\mathbb{C}[\mathcal{M}_2^n]^G$. Our main result shows that any separating set for $\mathbb{C}[\mathcal{M}_2^n]^G$ has cardinality $\geq 5n-5$. In particular, there is no separating set of size $\dim(\mathbb{C}[\mathcal{M}_2^n]) = 4n-3$ for $n \geq 3$. Further, $S_3$ has indeed minimum cardinality as a separating set, but for $n \geq 4$ there may exist a smaller separating set than $S_n$. We show that for $n \geq 5$ there does, in fact, exist a smaller separating set than $S_n$. We also prove similar results for the left-right action of $\mathrm{SL}_2(\mathbb{C}) \times \mathrm{SL}_2(\mathbb{C})$ on $\mathcal{M}_2^n$.
