Maximal generalized rank in graphical matrix spaces

Abstract

Let ${M_n}(\mathbb{F})$ be the space of n × n matrices over a field $\mathbb{F}$. For a subset ${\cal B} \subset {[n]^2}$ let ${M_{\cal B}}(\mathbb{F}) = \{ A \in {M_n}(\mathbb{F}):A(i,j) \notin {\cal B}\}$. Let ${\nu _b}({\cal B})$ denote the matching number of the n by n bipartite graph determined by ${\cal B}$. For $S \subset {M_n}(\mathbb{F})$ 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 ${M_{\cal B}}(\mathbb{F})$.

Theorem (Li, Qiao, Wigderson, Wigderson, Zhang): For any ${\cal B} \subset {[n]^2}$ (1) $\max \{ \dim \,W:W \le {M_{\cal B}}(\mathbb{F}),\rho (W) \le k\} = \max \{ |{\cal B}\prime |:{\cal B}\prime \subset {\cal B},{\nu _b}({\cal B}\prime ) \le k\}$.

The main results of this note are two extensions of (1). Let ${\mathbb{S}_n}$ denote the symmetric group on [n]. For $\omega :\coprod\nolimits_{n = 1}^\infty {{\mathbb{S}_n} \to } {\mathbb{F}^ * } =\mathbb{F} \backslash \{ 0\}$ define a function D_ω on each ${M_n}(\mathbb{F})$ by ${D_\omega }(A) = \sum\nolimits_{\sigma \in {\mathbb{S}_n}} {\omega (\sigma )\prod\nolimits_{i = 1}^n {A(i,\sigma (i))} }$. 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 $S \subset {M_n}(\mathbb{F})$ let ${\rho _\omega }(S) = \max \{ {\rm{r}}{{\rm{k}}_\omega }(A):A \in S\}$. The first extension of (1) concerns general weight functions.

Theorem: For any ω as above and ${\cal B} \subset {[n]^2}$

$$\max \{ \dim \,W:W \le {M_{\cal B}}(\mathbb{F}),{\rho _\omega }(W) \le k\} = \max \{ |{\cal B}\prime |:{\cal B}\prime \subset {\cal B},{\nu _b}({\cal B}\prime ) \le k\} $$

Let ${A_n}(\mathbb{F})$ denote the space of alternating matrices in ${M_n}(\mathbb{F})$. For a graph ${\cal G} \subset \left( {\matrix{{[n]} \cr 2 \cr } } \right)$ let ${A_{\cal G}}(\mathbb{F}) = \{ A \in {A_n}(\mathbb{F}):A(i,j) = 0\,{\rm{if}}\,\{ i,j\} \notin {\cal G}\}$. Let $\nu ({\cal G})$ denote the matching number of ${\cal G}$. The second extension of (1) concerns general graphs.

Theorem: For any ${\cal G} \subset \left( {\matrix{{[n]} \cr 2 \cr } } \right)$

$$\max \{ \dim \,U:U \le {A_{\cal G}}(\mathbb{F}),\rho (U) \le 2k\} = \max \{ |{\cal G}\prime |:{\cal G}\prime \subset {\cal G},\nu ({\cal G}\prime ) \le k\}.$$