# Gerstenhaber - Serezhkin theorem

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

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$\mathcal{N}=\{A\in\mathcal{M}_{n}(\mathbb{F}):\,A\,\,\mbox{is nilpotent}\},$

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$\mathcal{GL}_{n}(\mathbb{F})=\{A\in\mathcal{M}_{n}(\mathbb{F}):\det(A)\neq 0\},$

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$\mathcal{T}=\{A\in\mathcal{M}_{n}(\mathbb{F}):\,A\,\,\mbox{is strictly upper % triangular}\}.$

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

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

###### Theorem 1

Let $\mathcal{L}$ be a linear subspace of $\mathcal{M}_{n}(\mathbb{F}).$ Assume that $\mathcal{L}\subseteq\mathcal{N}.$ Then
(i) $\dim\mathcal{L}\leq n(n-1)/2,$ (ii) $\dim\mathcal{L}=n(n-1)/2$ if and only if there exists $U\in\mathcal{GL}_{n}(\mathbb{F})$ such that $\{UAU^{-1}:\,A\in\mathcal{L}\}=\mathcal{T}.$

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

## References

• G M. Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices, I, Amer. J. Math. 80: 614β622 (1958).
• M B. Mathes, M. OmladiΔ, H. Radjavi, Linear Spaces of Nilpotent Matrices, Linear Algebra Appl. 149: 215β225 (1991).
• S V. N. Serezhkin, On linear transformations preserving nilpotency, Vests$\bar{\iota}$ Akad. Navuk BSSR Ser. F$\bar{\iota}$z.-Mat. Navuk 1985, no. 6: 46β50 (Russian).
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