Maximal generalized rank in graphical matrix spaces

Alexander Guterman, Roy Meshulam, Igor Spiridonov

Research output: Contribution to journalArticlepeer-review

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Let Mn(F) be the space of n × n matrices over a field F . For a subset ℬ⊂ [ n] 2 let M(F) = { A∈ Mn(F) : A(i, j) ∉ ℬ} . Let νb(ℬ) denote the matching number of the n by n bipartite graph determined by ℬ . For S⊂ Mn(F) let ρ(S) = max{rk(A): A ∈ S}. Li, Qiao, Wigderson, Wigderson and Zhang [6] have recently proved the following characterization of the maximal dimension of bounded rank subspaces of M(F) . Theorem (Li, Qiao, Wigderson, Wigderson, Zhang): For any ℬ⊂ [ n] 2 (1) max{ dimW: W≤ M(F) , ρ(W) ≤ k} = max{ ∣ ℬ′ ∣ : ℬ′ ⊂ ℬ, νb(ℬ′ ) ≤ k} . The main results of this note are two extensions of (1). Let Sn denote the symmetric group on [n]. For ω:∐n=1∞Sn→F∗=F\{0} define a function Dω on each Mn(F) by Dω(A)=∑σ∈Snω(σ)∏i=1nA(i,σ(i)) . Let rkω(A) be the maximal k such that there exists a k × k submatrix B of A with Dω(B)≠0. For S⊂ Mn(F) let ρω(S) = max{ rk ω(A) : A∈ S} . The first extension of (1) concerns general weight functions. Theorem: For any ω as above and ℬ⊂ [ n] 2max{ dimW: W≤ M(F) , ρω(W) ≤ k} = max{ ∣ ℬ′ ∣ : ℬ′ ⊂ ℬ, νb(ℬ′ ) ≤ k} Let An(F) denote the space of alternating matrices in Mn(F) . For a graph G⊂([n]2) let AG(F) = { A∈ An(F) : A(i, j) = 0 if { i, j} ∉ G} . Let ν(G) denote the matching number of G . The second extension of (1) concerns general graphs. Theorem: For any G⊂([n]2)max{ dimU: U≤ AG(F) , ρ(U) ≤ 2 k} = max{ ∣ G′ ∣ : G′ ⊂ G, ν(G′ ) ≤ k}.

Original languageEnglish
Pages (from-to)297-309
Number of pages13
JournalIsrael Journal of Mathematics
Issue number1
StatePublished - Sep 2023

Bibliographical note

Publisher Copyright:
© 2023, The Hebrew University of Jerusalem.


Supported by ISF grant 686/20.

FundersFunder number
Israel Science Foundation686/20


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