Symmetric Grassmann Formulas: Monotone Dimension-Defect Hierarchies

Abstract

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.