Maximal generalized rank in graphical matrix spaces

Let M_n(F) be the space of n × n matrices over a field F . For a subset ℬ⊂ [ n] ² let M_ℬ(F) = { A∈ M_n(F) : A(i, j) ∉ ℬ} . Let ν_b(ℬ) denote the matching number of the n by n bipartite graph determined by ℬ . For S⊂ M_n(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_ℬ(F) . Theorem (Li, Qiao, Wigderson, Wigderson, Zhang): For any ℬ⊂ [ n] ² (1) max{ dimW: W≤ M_ℬ(F) , ρ(W) ≤ k} = max{ ∣ ℬ′ ∣ : ℬ′ ⊂ ℬ, ν_b(ℬ′ ) ≤ k} . The main results of this note are two extensions of (1). Let S_n denote the symmetric group on [n]. For ω:∐n=1∞Sn→F∗=F\{0} define a function D_ω on each M_n(F) by Dω(A)=∑σ∈Snω(σ)∏i=1nA(i,σ(i)) . Let rk_ω(A) be the maximal k such that there exists a k × k submatrix B of A with D_ω(B)≠0. For S⊂ M_n(F) let ρ_ω(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_ℬ(F) , ρ_ω(W) ≤ k} = max{ ∣ ℬ′ ∣ : ℬ′ ⊂ ℬ, ν_b(ℬ′ ) ≤ k} Let A_n(F) denote the space of alternating matrices in M_n(F) . For a graph G⊂([n]2) let A_G(F) = { A∈ A_n(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≤ A_G(F) , ρ(U) ≤ 2 k} = max{ ∣ G′ ∣ : G′ ⊂ G, ν(G′ ) ≤ k}.