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Clément de Seguins Pazzis

Publications and source records attributed to Clément de Seguins Pazzis.

At least 37 records · Page 2Linked to original sources
arXiv2023

Products of two involutions in orthogonal and symplectic groups

An element of a group is called bireflectional when it is the product of two involutions of the group (i.e. elements of order 1 or 2). If an element is bireflectional then it is conjugated to its inverse. It is known that all elements of orthogonal groups of quadratic forms are bireflectional. F. Bünger has characterized the elements of unitary groups (over fields of characteristic not $2$) that are bireflectional. Yet in symplectic groups over fields with characteristic different from 2, in general there are elements that are conjugated to their inverse but are not bireflectional (however, over fields of characteristic 2, every element of a symplectic group is bireflectional). In this article, we characterize the bireflectional elements of symplectic groups in terms of Wall invariants, over fields of characteristic not 2: the result is cited without proof in Bünger's PhD thesis, and attributed to Klaus Nielsen. We also take advantage of our approach to give a simplified proof of Wonenburger's corresponding result for orthogonal groups and general linear groups.

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arXiv2022

Sums of two square-zero selfadjoint or skew-selfadjoint endomorphisms

Let $V$ be a finite-dimensional vector space over a field $\mathbb{F}$, equipped with a symmetric or alternating non-degenerate bilinear form $b$. When the characteristic of $\mathbb{F}$ is not $2$, we characterize the endomorphisms $u$ of $V$ that split into $u=a_1+a_2$ for some pair $(a_1,a_2)$ of $b$-selfadjoint (respectively, $b$-skew-selfadjoint) endomorphisms of $V$ such that $(a_1)^2=(a_2)^2=0$. In the characteristic $2$ case, we obtain a similar classification for the endomorphisms of $V$ that split into the sum of two square-zero $b$-alternating endomorphisms of $V$ when $b$ is alternating (an endomorphism $v$ is called $b$-alternating whenever $b(x,v(x))=0$ for all $x \in V$). Finally, if the field $\mathbb{F}$ is equipped with a non-identity involution, we characterize the pairs $(h,u)$ in which $h$ is a Hermitian form on a finite-dimensional space over $\mathbb{F}$, and $u$ is the sum of two square-zero $h$-selfadjoint endomorphisms.

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arXiv2022

Locally finite cycles of linear mappings in countable dimension

Let $n$ be a positive integer. An $n$-cycle of linear mappings is an $n$-tuple $(u_1,\dots,u_n)$ of linear maps $u_1 \in \mathrm{Hom}(U_1,U_2),u_2 \in \mathrm{Hom}(U_2,U_3),\dots,u_n \in \mathrm{Hom}(U_n,U_1)$, where $U_1,\dots,U_n$ are vector spaces over a field. We classify such cycles, up to equivalence, when the spaces $U_1,\dots,U_n$ have countable dimension and the composite $u_n\circ u_{n-1}\circ \cdots \circ u_1$ is locally finite. When $n=1$, this problem amounts to classifying the reduced locally nilpotent endomorphisms of a countable-dimensional vector space up to similarity, and the known solution involves the so-called Kaplansky invariants of $u$. Here, we extend Kaplansky's results to cycles of arbitrary length. As an application, we prove that if $u_n \circ \cdots \circ u_1$ is locally nilpotent and the $U_i$ spaces have countable dimension, then there are bases $\mathbf{B}_1,\dots,\mathbf{B}_n$ of $U_1,\dots,U_n$, respectively, such that, for every $i \in \{1,\dots,n\}$, $u_i$ maps every vector of $\mathbf{B}_i$ either to a vector of $\mathbf{B}_{i+1}$ or to the zero vector of $U_{i+1}$ (where we convene that $U_{n+1}=U_1$ and $\mathbf{B}_{n+1}=\mathbf{B}_1$).

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arXiv2021

Sums and products of two quadratic endomorphisms of a countable-dimensional vector space

Let $V$ be a vector space with countable dimension over a field, and let $u$ be an endomorphism of it which is locally finite, i.e. $(u^k(x))_{k \geq 0}$ is linearly dependent for all $x$ in $V$. We give several necessary and sufficient conditions for the decomposability of $u$ into the sum of two square-zero endomorphisms. Moreover, if $u$ is invertible, we give necessary and sufficient conditions for the decomposability of $u$ into the product of two involutions, as well as for the decomposability of $u$ into the product of two unipotent endomorphisms of index $2$. Our results essentially extend the ones that are known in the finite-dimensional setting. In particular, we obtain that every strictly upper-triangular infinite matrix with entries in a field is the sum of two square-zero infinite matrices (potentially non-triangular, though), and that every upper-triangular infinite matrix (with entries in a field) with only $\pm 1$ on the diagonal is the product of two involutory infinite matrices.

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arXiv2020

Products of involutions of an infinite-dimensional vector space

We prove that every automorphism of an infinite-dimensional vector space over a field is the product of four involutions, a result that is optimal in the general case. We also characterize the automorphisms that are the product of three involutions. More generally, we study decompositions of automorphisms into three or four factors with prescribed split annihilating polynomials of degree $2$.

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arXiv2019

The structured Gerstenhaber problem (III)

Let $b$ be a symmetric bilinear form on a finite-dimensional vector space over a field with characteristic $2$. Here, we determine the greatest possible dimension of a linear subspace of nilpotent $b$-symmetric or $b$-alternating endomorphisms of $V$, expressing it as a function of the dimension, the rank, the Witt index of $b$, and an additional invariant in a very special case.

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arXiv2018

On the finite topology of a vector space and the domination problem for families of norms

Let $V$ be a real or complex vector space. The finite topology of $V$ consists of all the subsets $U$ for which the intersection $U \cap F$ is closed in $F$ for every finite-dimensional linear subspace of $V$. It is known that if $V$ has countable dimension, then this topology coincides with the greatest vector space topology and with the greatest locally convex vector space topology. Here, we note that in this case the finite topology is simply the union of all the normed space topologies. In connection to this problem, we characterize the vector spaces on which every family of norms with given cardinality is dominated.

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arXiv2018

Products of involutions in the stable general linear group

In the stable general linear group over an arbitrary field, we prove that every element with determinant $\pm 1$ is the product of three involutions, and of no less in general. We also obtain several results of the same flavor, with applications to decompositions of automorphisms of an infinite-dimensional vector space that are scalar multiples of finite-rank perturbations of the identity.

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arXiv2018

On the linear preservers of Schur matrix functionals

Let $\mathbb{F}$ be a field and $f : \mathfrak{S}_n \rightarrow \mathbb{F} \setminus \{0\}$ be an arbitrary map. The Schur matrix functional associated to $f$ is defined as $M \in \text{M}_n(\mathbb{F}) \mapsto \widetilde{f}(M):=\sum_{σ\in \mathfrak{S}_n} f(σ) \prod_{j=1}^n m_{σ(j),j}$. Typical examples of such functionals are the determinant (where $f$ is the signature morphism) and the permanent (where $f$ is constant with value $1$). Given two such maps $f$ and $g$, we study the endomorphisms $U$ of the vector space $\text{M}_n(\mathbb{F})$ that satisfy $\widetilde{g}(U(M))=\widetilde{f}(M)$ for all $M \in \text{M}_n(\mathbb{F})$. In particular, we give a closed form for the linear preservers of the functional $\widetilde{f}$ when $f$ is central, and as a special case we extend to an arbitrary field Botta's characterization of the linear preservers of the permanent.

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arXiv2018

The structured Gerstenhaber problem (II)

Let $b$ be a non-degenerate symmetric (respectively, alternating) bilinear form on a finite-dimensional vector space $V$, over a field with characteristic different from $2$. In a previous work, we have determined the maximal possible dimension for a linear subspace of $b$-alternating (respectively, $b$-symmetric) nilpotent endomorphisms of $V$. Here, provided that the cardinality of the underlying field be large enough with respect to the Witt index of $b$, we classify the spaces that have the maximal possible dimension. Our proof is based on a new sufficient condition for the reducibility of a vector space of nilpotent linear operators. To illustrate the power of that new technique, we use it to give a short new proof of the classical Gerstenhaber theorem on large vector spaces of nilpotent matrices (provided, again, that the cardinality of the underlying field be large enough).

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arXiv2018

The structured Gerstenhaber problem (I)

Let $b$ be a symmetric or alternating bilinear form on a finite-dimensional vector space $V$. When the characteristic of the underlying field is not $2$, we determine the greatest dimension for a linear subspace of nilpotent $b$-symmetric or $b$-alternating endomorphisms of $V$, expressing it as a function of the dimension, the rank, and the Witt index of $b$. Similar results are obtained for subspaces of nilpotent $b$-Hermitian endomorphisms when $b$ is a Hermitian form with respect to a non-identity involution. In three situations ($b$-symmetric endomorphisms when $b$ is symmetric, $b$-alternating endomorphisms when $b$ is alternating, and $b$-Hermitian endomorphisms when $b$ is Hermitian and the underlying field has more than $2$ elements), we also characterize the linear subspaces with the maximal dimension. Our results are wide generalizations of results of Meshulam and Radwan, who tackled the case of a non-degenerate symmetric bilinear form over the field of complex numbers, and recent results of Bukovšek and Omladič, in which the spaces with maximal dimension were determined when the underlying field is the one of complex numbers, the bilinear form $b$ is symmetric and non-degenerate, and one considers $b$-symmetric endomorphisms.

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arXiv2017

The sum and the product of two quadratic matrices

Let $p$ and $q$ be polynomials with degree $2$ over an arbitrary field $\mathbb{F}$. In the first part of this article, we characterize the matrices that can be decomposed as $A+B$ for some pair $(A,B)$ of square matrices such that $p(A)=0$ and $q(B)=0$. The case when both polynomials $p$ and $q$ are split was already known. In the first half of this article, we complete the study by tackling the case when at least one of the polynomials $p$ and $q$ is irreducible over $\mathbb{F}$. In the second half of the article, we use a similar method to characterize, under the assumption that $p(0)q(0) \neq 0$, the matrices that can be decomposed as $AB$ for some pair $(A,B)$ of square matrices such that $p(A)=0$ and $q(B)=0$.

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arXiv2017

Sums of three quadratic endomorphisms of an infinite-dimensional vector space

Let $V$ be an infinite-dimensional vector space over a field. In a previous article, we have shown that every endomorphism of $V$ splits into the sum of four square-zero ones but also into the sum of four idempotent ones. Here, we study decompositions into sums of three endomorphisms with prescribed split annihilating polynomials with degree $2$. Except for endomorphisms that are the sum of a scalar multiple of the identity and of a finite-rank endomorphism, we achieve a simple characterization of such sums. In particular, we give a simple characterization of the endomorphisms that split into the sum of three square-zero ones, and we prove that every endomorphism of $V$ is a linear combination of three idempotents.

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arXiv2016

Large spaces of symmetric or alternating matrices with bounded rank

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. In a recent work, we have determined the maximal dimension for a linear subspace of $n$ by $n$ symmetric matrices with rank less than or equal to $r$, and we have classified the spaces having that maximal dimension. In this article, provided that $\mathbb{K}$ has more than two elements, we extend this classification to spaces whose dimension is close to the maximal one: this generalizes a result of Loewy. We also prove a similar result on spaces of alternating matrices with bounded rank, with no restriction on the cardinality of the underlying field.

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arXiv2016

A note on sums of three square-zero matrices

It is known that every complex trace-zero matrix is the sum of four square-zero matrices, but not necessarily of three such matrices. In this note, we prove that for every trace-zero matrix $A$ over an arbitrary field, there is a non-negative integer $p$ such that the extended matrix $A \oplus 0_p$ is the sum of three square-zero matrices (more precisely, one can simply take $p$ as the number of rows of $A$). Moreover, we demonstrate that if the underlying field has characteristic $2$ then every trace-zero matrix is the sum of three square-zero matrices. We also discuss a counterpart of the latter result for sums of idempotents.

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