发表机构
Jagiellonian University（雅盖隆大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对任意域上有限多个子空间，提出对称非递归格拉斯曼公式计算和的维数，证明维数亏格轮廓的单调性，并给出金斯拉克的几何分解及熵类比。
AI 中文摘要
我们研究任意域上有限多个子空间的维数亏格。我们推导出它们和的维数的一个对称、非递归的格拉斯曼型公式。该公式通过一个特定子空间与其余子空间的部分和的交集来表达总维数损失，系数由所涉及子空间的数量决定。我们还表明，该公式对所关联的可表示多拟阵具有沙普利值解释。按参与子空间的数量对修正项进行分组，可得到一个非负的维数亏格轮廓。我们证明该轮廓是单调的，并将其连续间隙识别为平均秩轮廓的离散曲率。这些间隙给出了两侧亏格界中的精确余项。当子空间在其公共交集的商空间中的像构成内直和时，两个界中的等式恰好成立。我们还给出了金斯拉克的一个精确几何分解，将其分解为非负商维数，并刻画了等式成立的条件。将这些斯拉克量在排列和收缩上取平均，可恢复除最后一个外的所有亏格曲率；额外的可表示性约束仍保留在单个有序斯拉克量中。加权和对偶公式伴随该展开，熵类比分别将亏格水平和曲率表示为互信息和条件互信息的平均值。
英文摘要
We study the dimension defect of finitely many subspaces over an arbitrary field. We derive a symmetric, nonrecursive Grassmann-type formula for the dimension of their sum. The formula expresses the total dimension loss through intersections of a distinguished subspace with partial sums of the remaining subspaces, with coefficients determined by the number of subspaces involved. We also show that the formula admits a Shapley-value interpretation for the associated representable polymatroid. Grouping the correction terms by the number of participating subspaces yields a nonnegative dimension-defect profile. We prove that this profile is monotone and identify its successive gaps with the discrete curvatures of the average-rank profile. These gaps give exact remainders in two-sided defect bounds. Equality in either bound holds precisely when the images of the subspaces in the quotient by their common intersection form an internal direct sum. We also give an exact geometric decomposition of the Kinser slack into nonnegative quotient dimensions and characterize equality. Averaging these slacks over permutations and contractions recovers every defect curvature except the final one; additional representability constraints remain in the individual ordered slacks. Weighted and dual formulas accompany the expansion, and entropy analogues express the defect levels and curvatures as averages of mutual and conditional mutual information, respectively.
Comments30 pages, 2 figures, comments are welcome
