发表机构
Faculty of Mathematics and Physics, University of Ljubljana; Institute of Mathematics, Physics, and Mechanics; Département de Mathématique, Université Libre de Bruxelles（卢布尔雅那大学数学物理学院; 数学、物理与力学研究所; 布鲁塞尔自由大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究紧群不可约表示的加性直径与覆盖复杂度，证明直径仅差对数因子，确定了$\mathrm{SL}_2(\mathbf{C})$等各类群表示的复杂度范围及相关性质，还对比了单项式直径与$\mathrm{SL}_3(\mathbf{C})$直径的优劣。
AI 中文摘要
设群$G$在有限维复向量空间$V$上线性作用。子空间$U \leq V$的群加性直径是指，满足其平移的和等于整个空间$V$所需的最少平移个数。通过计算维数可知，该直径至少为$\dim V / \dim U$。本文证明，当$G$为紧群且$V$为不可约表示时，每个非零子空间的直径至多为$\lceil (\dim V / \dim U) \ln \dim V \rceil$，即平凡下界仅差一个对数因子。我们通过覆盖复杂度$\mathsf{C}(V)$来衡量这种偏差，$\mathsf{C}(V)$定义为任意子空间的直径与其平凡下界的最大比值，满足$1 \leq \mathsf{C}(V) \leq 2 + \ln \dim V$，本文确定了各类表示在该范围内的位置。$\mathrm{SL}_2(\mathbf{C})$的每个不可约表示都满足$\mathsf{C}(V)=1$；$\mathrm{SL}_n(\mathbf{C})$的对称幂与外幂族（$n$变化）、有限海森堡群、小阶2-传递群（如$\mathrm{PSL}_2(\mathbf{F}_p)$）的复杂度中确实存在对数项；$\mathrm{SL}_n(\mathbf{C})$的共轭表示、$\mathrm{SL}_3(\mathbf{C})$的表示$\operatorname{Sym}^k \mathbf{C}^3$的复杂度有常数上界；另一方面，每个固定的连通约化群都存在一族不可约表示，其复杂度至少趋向旗簇的维数。最后，对于李代数$\mathfrak{sl}_3(\mathbf{C})$在$\operatorname{Sym}^k \mathbf{C}^3$上的作用，相对于平面$X \leq \mathbf{C}^3$的$\operatorname{Sym}^k X$的单项式直径是最优的，但对应的$\mathrm{SL}_3(\mathbf{C})$直径并非最优。
英文摘要
Let a group $G$ act linearly on a finite-dimensional complex vector space $V$. The group-additive diameter of a subspace $U \leq V$ is the least number of translates of $U$ whose sum is all of $V$. Counting dimensions, it is at least $\dim V / \dim U$. We show that when $G$ is compact and $V$ is irreducible, the diameter of every nonzero subspace is at most $\lceil (\dim V / \dim U) \ln \dim V \rceil$, so the trivial bound is correct up to a logarithmic factor. We measure the discrepancy by the covering complexity $\mathsf{C}(V)$, the largest ratio between the diameter of any subspace and its trivial lower bound, so that $1 \leq \mathsf{C}(V) \leq 2 + \ln \dim V$, and we determine where in this range various representations lie. Every irreducible representation of $\mathrm{SL}_2(\mathbf{C})$ has $\mathsf{C}(V) = 1$. The logarithm can be genuinely present along families of symmetric and exterior powers of $\mathrm{SL}_n(\mathbf{C})$ with $n$ varying, and it is present for finite Heisenberg groups and $2$-transitive groups of small order such as $\mathrm{PSL}_2(\mathbf{F}_p)$. The complexity is bounded above by a constant on the conjugation representations of $\mathrm{SL}_n(\mathbf{C})$ and on the representations $\operatorname{Sym}^k \mathbf{C}^3$ of $\mathrm{SL}_3(\mathbf{C})$. On the other hand, every fixed connected reductive group has a family of irreducible representations whose complexities tend to at least the dimension of the flag variety. Finally, for the Lie algebra $\mathfrak{sl}_3(\mathbf{C})$ acting on $\operatorname{Sym}^k \mathbf{C}^3$, the monomial diameter with respect to $\operatorname{Sym}^k X$ for a plane $X \leq \mathbf{C}^3$ is optimal, while the corresponding $\mathrm{SL}_3(\mathbf{C})$ diameter is not.
Comments33 pages
