05/06/2017, 15:00 — 16:00 — Room 6.2.38, Faculty of Sciences of the Universidade de Lisboa
Alicia Roca, Universitat Politècnica de València
Lattices of invariant subspaces
Given $\mathbb{F}$ an arbitrary field and $A\in M_{n}(\mathbb{F})$, the set of $A$-invariant subspaces of $\mathbb{F}^{n}$ is a lattice with inclusion as order, intersection as meet and linear sum as join. We denote this lattice by $\operatorname{Inv}(A)$.
An $A$-invariant subspace $V\subseteq \mathbb{F}^{n}$ is $A$-hyperinvariant ($A$-characteristitc) if it is invariant for every matrix $T\in Z(A)$, i.e. commuting with $A$ ($T\in Z^*(A)$, i.e. commuting with $A$ and $T$ non singular). It is straightforward to see that the set of $A$-hyperinvariant ($A$-characteristic) subspaces is a sublattice of $\operatorname{Inv}(A)$. We denote this sublattice by $\operatorname{Hinv}(A)$ ($\operatorname{Chinv}(A)$). Obviously, \[\operatorname{Hinv}(A)\subseteq \operatorname{Chinv}(A)\subseteq \operatorname{Inv}(A).\] If the characteristic polynomial of $A$ splits over $\mathbb{F}$, the study of these lattices can be reduced to the nilpotent case. Let $J\in M_{n}(\mathbb{F})$ be a nilpotent Jordan matrix.
In this talk we recall the general properties of $\operatorname{Inv}(J)$ and analyze which of those properties are preserved in the sublattice $\operatorname{Hinv}(J)$ and, if $\mathbb{F}=GF(2)$, in $\operatorname{Chinv}(A)$ (the only case where $\operatorname{Hinv}(J)\neq \operatorname{Chinv}(J)$).
In addition we analyze the cardinality of $\operatorname{Hinv}(J)$ and $\operatorname{Chinv}(J)$.
Joint work with David Mingueza, M. Eulàlia Montoro.
