Decomposable forms and automorphisms.
type: Item
instance of: scholarly article
title: Decomposable forms and automorphisms.
zbMATH Open document ID: 1042.11026
DOI: 10.1016/S0022-314X(02)00073-2
author: Leonhard Summerer
published in: Journal of Number Theory
publication date: 2003-05-25T00:00:00Z
review text: Let \(\mathcal{F}_{d}\) be the set of decomposable forms of degree \(d>0\) over an algebraically closed field \(K\) with variables in a \(K\)-vector space \(V\). Then \(\text{GL}(V) \) acts on \(\mathcal{F}_{d}\) via \(\gamma \cdot f=f\circ \gamma ^{-1}\), \(\gamma \in \text{GL}(V)\), \(f\in \mathcal{F}_{d}.\) For a given \(f\), \(\Aut(f) \) is defined to be the stabilizer of this action. This paper studies the group \( \Aut(f) \) for certain classes of forms. As \(f\) is decomposable we may write \(f=l_{1}l_{2}\cdots l_{d}\) for \(l_{i}\in V^{\ast }.\) Writing \(l=( l_{1},\ldots ,l_{d}) \) there is a map \( \widetilde{\rho}_{l}:\Aut(f) \to \Sigma _{d},\) where \(\Sigma _{d}\) is the symmetric group on \(d\) letters. While this map depends on the choice of decomposition \(l\), the author proves that, for \( m=( m_{1},\ldots ,m_{d}) \) another decomposition of \(f\), the maps \(\widetilde{\rho}_{l}\) and \(\widetilde{\rho}_{m}\) are conjugate in \(\Sigma _{d}.\) Let \(\Gamma _{l}( f) \) be the image of \(\Aut(f) \) under \(\widetilde{\rho}_{l}\). Then \(\Aut(f) \) is the semidirect product of \(\mu _{d}\) (the \(d\)th roots of unity) with \(\Gamma(f) \) where \(\Gamma(f) =\Gamma _{l}( f) \) for any decomposition \(l\). Thus the computation of \(\Aut(f) \) is equivalent to the computation of \(\Gamma(f) .\) In the second section, the author does some computations of \(\Gamma(f) \) in special cases. For a non-degenerate form \(f\) of degree \(d>0\), let \(k\) denote the number of variables. If \(k=2\) and \(d\geq 4\) then any nontrivial \(\tau \in \Gamma(f) \) is the product of \(( d-i) /t\) cycles of length \( t\) and \(i\) fixed points for some \(t\) and for \(0\leq i\leq 2.\) If \(k=3\) and \( d\geq 5\) then any nontrivial \(\tau \in \Gamma(f) \) is either the product of \(( d-i) /t\) \(t\)-cycles for \(0\leq i\leq 3\), or \((d-2-i)/t\) \(t\)-cycles, one \(2\)-cycle, and \(i\) fixed points for \( i=0\) or \(1.\) An element of \(\Aut(f) \) not in \(\mu _{d}\) is call…
reviewed by: Alan Koch
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 1915504
zbMATH Keywords: decomposable forms
zbMATH Keywords: automorphisms
zbMATH Keywords: generic forms
MaRDI profile type: Publication
cites work: Q5340151
cites work: Q4143433
cites work: Singularities. II: Automorphisms of forms
cites work: Algebraic geometry IV: linear algebraic groups, invariant theory. Transl. from the Russian by G.A. Kandall
cites work: Forms represented by linear forms
description: scientific article; zbMATH DE number 1915504