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Fernando Szechtman

Publications and source records attributed to Fernando Szechtman.

At least 55 records · Page 3Linked to original sources
arXiv2014

Generalized Artin-Schreier polynomials

Let $F$ be a field of prime characteristic $p$ containing $F_{p^n}$ as a subfield. We refer to $q(X)=X^{p^n}-X-a\in F[X]$ as a generalized Artin-Schreier polynomial. Suppose that $q(X)$ is irreducible and let $C_{q(X)}$ be the companion matrix of $q(X)$. Then $ad\, C_{q(X)}$ has such highly unusual properties that any $A\in{\mathfrak{ gl}}(m)$ such that $ad\, A$ has like properties is shown to be similar to the companion matrix of an irreducible generalized Artin-Schreier polynomial. We discuss close connections with the decomposition problem of the tensor product of indecomposable modules for a 1-dimensional Lie algebra over a field of characteristic $p$, the problem of finding an explicit primitive element for every intermediate field of the Galois extension associated to an irreducible generalized Artin-Schreier polynomial, and the problem of finding necessary and sufficient conditions for the irreducibility of a family of polynomials.

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arXiv2014

Indecomposable modules of 2-step solvable Lie algebras in arbitrary characteristic

Let $F$ be an algebraically closed field and consider the Lie algebra ${\mathfrak g}=\langle x\rangle\ltimes {\mathfrak a}$, where $\mathrm{ad}\, x$ acts diagonalizably on the abelian Lie algebra ${\mathfrak a}$. Refer to a ${\mathfrak g}$-module as admissible if $[{\mathfrak g},{\mathfrak g}]$ acts via nilpotent operators on it, which is automatic if $\mathrm{char}(F)=0$. In this paper we classify all indecomposable ${\mathfrak g}$-modules $U$ which are admissible as well as uniserial, in the sense that $U$ has a unique composition series.

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arXiv2013

Equivalence and normal forms of bilinear forms

We present an alternative account of the problem of classifying and finding normal forms for arbitrary bilinear forms. Beginning from basic results developed by Riehm, our solution to this problem hinges on the classification of indecomposable forms and in how uniquely they fit together to produce all other forms. We emphasize the use of split forms, i.e., those bilinear forms such that the minimal polynomial of the asymmetry of their non-degenerate part splits over ground field, rather than restricting the field to be algebraically closed. In order to obtain the most explicit results, without resorting to the classification of hermitian, symmetric and quadratic forms, we merely require that the underlying field be quadratically closed.

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arXiv2013

Reductivity of the Lie algebra of a bilinear form

Let $f:V\times V\to F$ be a totally arbitrary bilinear form defined on a finite dimensional vector space $V$ over a a field $F$, and let $L(f)$ be the subalgebra of $\gl(V)$ of all skew-adjoint endomorphisms relative to $f$. Provided $F$ is algebraically closed of characteristic not 2, we determine all $f$, up to equivalence, such that $L(f)$ is reductive. As a consequence, we find, over an arbitrary field, necessary and sufficient conditions for $L(f)$ to be simple, semisimple or isomorphic to $\sl(n)$ for some $n$.

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arXiv2013

Modular representations of Heisenberg algebras

Let $F$ be be an arbitrary field and let $h(n)$ be the Heisenberg algebra of dimension $2n+1$ over $F$. It was shown by Burde that if $F$ has characteristic 0 then the minimum dimension of a faithful $h(n)$-module is $n+2$. We show here that his result remains valid in prime characteristic $p$, as long as $(p,n)\neq (2,1)$. We construct, as well, various families of faithful irreducible $h(n)$-modules if $F$ has prime characteristic, and classify these when $F$ is algebraically closed. Applications to matrix theory are given.

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arXiv2013

Composition series of $\gl(m)$ as a module for its classical subalgebras over an arbitrary field

Let $F$ be an arbitrary field and let $f:V\times V\to F$ be a non-degenerate symmetric or alternating bilinear form defined on an $F$-vector space of finite dimension $m\geq 2$. Let $L(f)$ be the subalgebra of $gl(V)$ formed by all skew-adjoint endomorphisms with respect to $f$. We find a composition series for the $L(f)$-module $gl(V)$ and furnish multiple identifications for all its composition factors.

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arXiv2013

Roots multiplicity and square-free factorization of polynomials using companion matrices

Given an arbitrary monic polynomial $f$ over a field $F$ of characteristic 0, we use companion matrices to construct a polynomial $M_f\in F[X]$ of minimum degree such that for each root $α$ of $f$ in the algebraic closure of $F$, $M_f(α)$ is equal to the multiplicity $m(α)$ of $α$ as a root of $f$. As an application of $M_f$ we give a new method to compute in $F[X]$ each component of the square-free factorization $f=P_1P_2^2\cdots P_m^m$, where $P_k$ is the product of all $X-α$ with $m(α)=k$, for $k=1, \dots, m=\max m(α)$.

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arXiv2013

On the theorem of the primitive element with applications to the representation theory of associative and Lie algebras

We describe of all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let $K/F$ be a finite separable field extension and let $x,y\in K$. When is $F[x,y]=F[αx+βy]$ for some non-zero elements $α,β\in F$?

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arXiv2013

The Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring

We find all irreducible constituents of the Weil representation of a unitary group $U_m(A)$ of rank $m$ associated to a ramified quadratic extension $A$ of a finite, commutative, local and principal ring $R$ of odd characteristic. We show that this Weil representation is multiplicity free with monomial irreducible constituents. We also find the number of these constituents and describe them in terms of Clifford theory with respect to a congruence subgroup. We find all character degrees in the special case when $R$ is a field.

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arXiv2013

Is every matrix similar to a polynomial in a companion matrix?

Given a field $F$, an integer $n\geq 1$, and a matrix $A\in M_n(F)$, are there polynomials $f,g\in F[X]$, with $f$ monic of degree $n$, such that $A$ is similar to $g(C_f)$, where $C_f$ is the companion matrix of $f$? For infinite fields the answer is easily seen to positive, so we concentrate on finite fields. In this case we give an affirmative answer, provided $|F|\geq n-2$. Moreover, for any finite field $F$, with $|F|=m$, we construct a matrix $A\in M_{m+3}(F)$ that is not similar to any matrix of the form $g(C_f)$. Of use above, but also of independent interest, is a constructive procedure to determine the similarity type of any given matrix $g(C_f)$ purely in terms of $f$ and $g$, without resorting to polynomial roots in $F$ or in any extension thereof. This, in turn, yields an algorithm that, given $g$ and the invariant factors of any $A$, returns the elementary divisors of $g(A)$. It is a rational procedure, as opposed to the classical method that uses the Jordan decomposition of $A$ to find that of $g(A)$. Finally, extending prior results by the authors, we show that for an integrally closed ring $R$ with field of fractions $F$ and companion matrices $C,D$ the subalgebra $R< C,D>$ of $M_n(R)$ is a free $R$-module of rank $n+(n-m)(n-1)$, where $m$ is the degree of $\gcd (f,g)\in F[X]$, and a presentation for $R< C,D>$ is given in terms of $C$ and $D$. A counterexample is furnished to show that $R< C,D>$ need not be a free $R$-module if $R$ is not integrally closed. The preceding information is used to study $M_n(R)$, and others, as $R[X]$-modules.

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arXiv2012

A new interpretation of the Racah-Wigner $6j$-symbol and the classification of uniserial $sl(2)\ltimes V(m)$-modules

All Lie algebras and representations will be assumed to be finite dimensional over the complex numbers. Let $V(m)$ be the irreducible $\sl(2)$-module with highest weight $m\geq 1$ and consider the perfect Lie algebra $\g=\sl(2)\ltimes V(m)$. Recall that a $\g$-module is uniserial when its submodules form a chain. In this paper we classify all uniserial $\g$-modules. The main family of uniserial $\g$-modules is actually constructed in greater generality for the perfect Lie algebra $\g=\s\ltimes V(μ)$, where $\s$ is a semisimple Lie algebra and $V(μ)$ is the irreducible $\s$-module with highest weight $μ\neq 0$. The fact that the members of this family are, but for a few exceptions of lengths 2, 3 and~4, the only uniserial $\sl(2)\ltimes V(m)$-modules depends in an essential manner on the determination of certain non-trivial zeros of Racah-Wigner $6j$-symbol.

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arXiv2010

Jordan-Chevalley decomposition in finite dimesional Lie algebras

Let $\g$ be a finite dimensional Lie algebra over a field $k$ of characteristic zero. An element $x$ of $\g$ is said to have an \emph{abstract Jordan-Chevalley decomposition} if there exist unique $s,n\in\g$ such that $x=s+n$, $[s,n]=0$ and given any finite dimensional representation $π:\g\to\gl(V)$ the Jordan-Chevalley decomposition of $π(x)$ in $\gl(V)$ is $π(x)=π(s)+π(n)$. In this paper we prove that $x\in\g$ has an abstract Jordan-Chevalley decomposition if and only if $x\in [\g,\g]$, in which case its semisimple and nilpotent parts are also in $[\g,\g]$ and are explicitly determined. We derive two immediate consequences: (1) every element of $\g$ has an abstract Jordan-Chevalley decomposition if and only if $\g$ is perfect; (2) if $\g$ is a Lie subalgebra of $\gl(n,k)$ then $[\g,\g]$ contains the semisimple and nilpotent parts of all its elements. The last result was first proved by Bourbaki using different methods. Our proof only uses elementary linear algebra and basic results on the representation theory of Lie algebras, such as the Invariance Lemma and Lie's Theorem, in addition to the fundamental theorems of Ado and Levi.

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arXiv2010

A Closed Formula for the Product in Simple Integral Extensions

Let $ξ$ be an algebraic number and let $α,β\in \mathbb Q[ξ]$. An explicit formula for the coordinates of the product $αβ$ is given in terms of the coordinates of $α$ and $β$ and the companion matrix of the minimal polynomial of $ξ$. The formula as well as its proof extend to fairly general simple integral extensions.

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