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Zbl 1178.37029
Kulkarni, Ravi S.
Dynamics of linear and affine maps.
(English)
[J] Asian J. Math. 12, No. 3, 321-344 (2008). ISSN 1093-6106

Author's summary: The well-known theory of rational canonical form of an operator" describes the invariant factors, or elementary divisors, as a complete set of invariants of a similarity class of an operator on a finite-dimensional vector space $\Bbb V$ over a given field $\Bbb F$. A finer part of the theory is the contribution by Frobenius dealing with the structure of the centralizer of an operator. The viewpoint is that of finitely generated modules over a PID. In this paper we approach the issue from a dynamic" viewpoint, as explained in the author's paper [J. Ramanujan Math. Soc. 22, No. 1, 35--56 (2007; Zbl 1181.22022)]. We also extend the theory to affine maps. The formulation is in terms of the action of the general linear group $\text{GL}(n)$, resp. the group of invertible affine maps $\text{GA}(n)$, on the semigroup of all linear, resp. affine, maps by conjugacy. The theory of rational canonical forms is connected with the orbits, and the Frobenius' theory with the orbit-classes, of the action of $\text{GL}(n)$ on the semigroup of linear maps. We describe a parametrization of orbits and orbit-classes of both $\text{GL}(n)$- and $\text{GA}(n)$-actions, and also provide a parametrization of all affine maps themselves, which is independent of the choices of linear or affine co-ordinate systems. An important ingredient in these parametrizations is a certain flag. For a linear map $T$ on $\Bbb V$, let $Z_L(T)$ denote its centralizer associative $\Bbb F$-algebra, and $Z_L(T)^*$the multiplicative group of invertible elements in $Z_L(T)$. In this situation, we associate a canonical, maximal, $Z_L(T)$-invariant flag, and precisely describe the orbits of $Z_L(T)^*$ on $\Bbb V$. Using this approach, we strengthen the classical theory in a number of ways.
[Olaf Ninnemann (Berlin)]
MSC 2000:
*37C99 Smooth dynamical systems
15A04 Linear transformations (linear algebra)
20G15 Linear algebraic groups over arbitrary fields

Keywords: parametrization of orbits and orbit-classes; action of the general linear group; group of invertible affine maps; parametrization of affine maps; canonical maximal invariant flag

Citations: Zbl 1181.22022

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