Properties of sets of Subspaces with Constant Intersection Dimension

2019/04/25 by Lucas, Lisa Hernandez
#05B25 #51E20 #51E23 #94B60 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1904.11197

Abstract

A (k,k-t)-SCID (set of Subspaces with Constant Intersection Dimension) is a set of k-dimensional vector spaces that have pairwise intersections of dimension k-t. Let C=\π₁,…,π_n\ be a (k,k-t)-SCID. Define S:=⟨ π₁, …, π_n ⟩ and I:=⟨ π_i ∩ π_j | 1 ≤ i < j ≤ n ⟩. We establish several upper bounds for dim S + dim I in different situations. We give a spectrum result for the case (n-1)(k-t)≤ k and for the case n≤\fracq^t(n-η)-1q^t-1, giving examples of (k,k-t)-SCIDs reaching a large interval of values for dim S + dim I.

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