论文标题
矩阵半不变的分离品种
The separating variety for matrix semi-invariants
论文作者
论文摘要
令$ g $为线性代数组，在矢量空间$ v $上线性作用，让$ k [v]^g $成为不变多项式函数的相应代数。分离集合$ s \ subseteq k [v]^g $是一组多项式，该属性对于所有$ v，w \ in v $ in v $ in v $，如果存在$ f \ in k [v]^g $分隔$ v $和$ w $，那么s $ f \ f \ in s $ f \ in s $ f \ in s $ f \ in s $ s $ sukate $ v $和$ v $。 在本文中，我们考虑了$ g = \ mathrm {sl} _2 \ times \ mathrm {sl} _2 $在$ \ mathbb {c} $ - vector Space $ m_ {2,2,2,2}^n $ n $ n $ n $ 2 $ 2 $ 2 $ 2 $ iTrices的$ 2 $ n $ the y和左右的乘以。最小生成集$ s_n $ of $ \ mathbb {c} [m_ {2,2}^n]^g $是已知的，$ | s_n | = \ frac {1} {24}（n^4-6n^3+23n^2+6n）$。在最近的工作中，多诺科斯（Domokos）表明，$ s_n $是通过包含的最小分离设置，即，$ s_n $的适当子集是一个分开集。我们的主要结果表明，$ \ mathbb {c} [m_ {2,2}^n]^g $的任何分离设置都有核心$ \ geq 5n-9 $。特别是，没有尺寸$ \ dim（\ mathbb {c} [m_2^n]^g）= 4n-6 $的分离集，对于$ n \ geq 4 $。我们还通过左乘法考虑了$ g = \ mathrm {sl} _l（\ mathbb {c}）$的动作。在这种情况下，不变的代数具有最小生成的尺寸$ \ binom {n} {l} $和dimension $ ln-l^2+1 $。我们表明，$ \ mathbb {c} [m_ {l，n}]^g $的分离集必须至少具有$（2L-2）N-2（l^2-l）$。尤其是，$ \ mathbb {c} [m_ {l，n}]^g $不包含一组尺寸$ \ dim（\ Mathbb {c} [m_ {l，n} g）$ for $ l \ geq 3 $和$ n \ geq l+2 $。我们包括根据Quivers表示结果的解释，并将猜想概括为Skowronski-Weyman定理。
Let $G$ be a linear algebraic group acting linearly on a vector space $V$, and let $k[V]^G$ be the corresponding algebra of invariant polynomial functions. A separating set $S \subseteq k[V]^G$ is a set of polynomials with the property that for all $v,w \in V$, if there exists $f \in k[V]^G$ separating $v$ and $w$, then there exists $f \in S$ separating $v$ and $w$. In this article we consider the action of $G = \mathrm{SL}_2 \times \mathrm{SL}_2$ on the $\mathbb{C}$-vector space $M_{2,2}^n$ of $n$-tuples of $2 \times 2$ matrices by multiplication on the left and the right. Minimal generating sets $S_n$ of $\mathbb{C}[M_{2,2}^n]^G$ are known, and $|S_n| = \frac{1}{24}(n^4-6n^3+23n^2+6n)$. In recent work, Domokos showed that $S_n$ is a minimal separating set by inclusion, i.e. that no proper subset of $S_n$ is a separating set. Our main result shows that any separating set for $\mathbb{C}[M_{2,2}^n]^G$ has cardinality $\geq 5n-9$. In particular, there is no separating set of size $\dim(\mathbb{C}[M_2^n]^G) = 4n-6$ for $n \geq 4$. We also consider the action of $G= \mathrm{SL}_l(\mathbb{C})$ on $M_{l,n}$ by left multiplication. In that case the algebra of invariants has a minimum generating set of size $\binom{n}{l}$ and dimension $ln-l^2+1$. We show that a separating set for $\mathbb{C}[M_{l,n}]^G$ must have size at least $(2l-2)n-2(l^2-l)$. In particular, $\mathbb{C}[M_{l,n}]^G$ does not contain a separating set of size $\dim(\mathbb{C}[M_{l,n}]^G)$ for $l \geq 3$ and $n \geq l+2$. We include an interpretation of our results in terms of representations of quivers, and make a conjecture generalising the Skowronski-Weyman theorem.
