有界秩交换子的矩阵空间的等号情形
Equality cases for matrix spaces with bounded-rank commutators
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中文总结 AI 辅助
本文证明了 Omladič、Radjavi 和 Šivic 关于有界秩交换子的矩阵空间维数锐界等号情形的猜想，给出了等号情形的完整分类，并确定了相应 Grassmann 流形上射影代数子集的不可约分支。
中文摘要 AI 辅助
设 $0\leq k < n$，且 $\mathcal V\subseteq M_n(\mathbb C)$ 是满足对所有 $S,T\in\mathcal V$ 有 $\operatorname{rank}[S,T]\leq k$ 的复线性子空间。Omladič、Radjavi 和 Šivic 证明了锐界 $\dim\mathcal V\leq nk+\left\lfloor (n-k)^2/4\right\rfloor+1$，并猜想等号情形的分类。我们证明了他们的猜想。若等号成立，则在相似变换和可能的转置之后，$\mathcal V$ 由所有分块上三角矩阵组成，其左上和右上块任意，而右下块位于 $M_{n-k}(\mathbb C)$ 的一个极大维交换子空间中。对于 $n-k\geq4$，这些交换子空间是 Schur 定理中的经典等号情形；在维数 $2$ 和 $3$ 时，还出现额外的等号情形。在等号维数下，秩条件定义了 Grassmann 流形的一个射影代数子集。对于 $2\leq k\leq n-2$，我们确定了它的所有不可约分支。若 $n-k\geq4$，当 $n-k$ 为偶数时恰有两个分支，当 $n-k$ 为奇数时恰有四个；若 $n-k\in\{2,3\}$，则恰有两个。当 $n-k\geq4$ 时，每个这样的空间处的 Zariski 切空间等于其共轭轨道的切空间。当 $n-k\in\{2,3\}$ 时，两个分支通过变化不变的 $k$ 维子空间和商上的极大维交换子空间及其转置得到。
英文摘要
Let $0\leq k < n$, and let $\mathcal V\subseteq M_n(\mathbb C)$ be a complex linear subspace satisfying $\operatorname{rank}[S,T]\leq k$ for all $S,T\in\mathcal V$. Omladič, Radjavi, and Šivic proved the sharp bound $\dim\mathcal V\leq nk+\left\lfloor (n-k)^2/4\right\rfloor+1$ and conjectured a classification of the equality cases. We prove their conjecture. If equality holds, then, after a similarity and possibly transposition, $\mathcal V$ consists of all block upper-triangular matrices with arbitrary upper-left and upper-right blocks and with lower-right block in a maximal-dimensional commuting subspace of $M_{n-k}(\mathbb C)$. For $n-k\geq4$, these commuting subspaces are the classical equality cases in Schur's theorem; in dimensions $2$ and $3$, the additional equality cases also occur. At the equality dimension, the rank condition defines a projective algebraic subset of a Grassmannian. For $2\leq k\leq n-2$, we determine all of its irreducible components. If $n-k\geq4$, there are exactly two components when $n-k$ is even and exactly four when $n-k$ is odd; if $n-k\in\{2,3\}$, there are exactly two. When $n-k\geq4$, the Zariski tangent space at each such space equals the tangent space to its conjugacy orbit. When $n-k\in\{2,3\}$, the two components are obtained by varying the invariant $k$-dimensional subspace and the maximal-dimensional commuting subspace on the quotient, together with their transposes.
