Gerstenhaber - Serezhkin theorem

Let $𝔽$ be an arbitrary field. Consider ${ℳ}_n(𝔽),$ the vector space of all $n\times n$ matrices over $𝔽.$ Define

• $𝒩=\{A\in {ℳ}_n(𝔽):A\text{is nilpotent}\},$

• $𝒢{ℒ}_n(𝔽)=\{A\in {ℳ}_n(𝔽):det(A)\ne 0\},$

• $𝒯=\{A\in {ℳ}_n(𝔽):A\text{is strictly upper triangular}\}.$

Notice that $𝒯$ is a linear subspace of ${ℳ}_n(𝔽).$ Moreover, $𝒯\subseteq 𝒩$ and $dim𝒯=n(n-1)/2.$

The Gerstenhaber – Serezhkin theorem on linear subspaces contained in the nilpotent cone [G, S] reads as follows.

Theorem 1

Let $\mathrm{L}$ be a linear subspace of ${\mathrm{M}}_n\mathit{}\mathrm{(}\mathrm{F}\mathrm{)}\mathrm{.}$ Assume that $\mathrm{L}\mathrm{\subseteq }\mathrm{N}\mathrm{.}$ Then
(i)$\dim \mathit{}\mathrm{L}\mathrm{\le }n\mathit{}\mathrm{(}n\mathrm{-}\mathrm{1}\mathrm{)}\mathrm{/}\mathrm{2}\mathrm{,}$(ii)$\dim \mathit{}\mathrm{L}\mathrm{=}n\mathit{}\mathrm{(}n\mathrm{-}\mathrm{1}\mathrm{)}\mathrm{/}\mathrm{2}$ if and only if there exists $U\mathrm{\in }\mathrm{G}\mathit{}{\mathrm{L}}_n\mathit{}\mathrm{(}\mathrm{F}\mathrm{)}$ such that $\mathrm{\{}U\mathit{}A\mathit{}{U}^{\mathrm{-}\mathrm{1}}\mathrm{:}A\mathrm{\in }\mathrm{L}\mathrm{\}}\mathrm{=}\mathrm{T}\mathrm{.}$

An alternative simple proof of inequality (i) can be found in [M].