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Matrices of linear forms and curves. (English) Zbl 1107.15010

The author studies a mathematical problem originated from population genetics. The object is to understand the triple action of the projective general linear group PGL\(_{n+1}\) on the projective space of nonzero \((n+1)\times (n+1)\) matrices of linear forms on (the projective space) \(\mathbb P^n\). This is done by associating a quadratic rational map \(\phi : \mathbb P^n \to \mathbb P^n\) to any such matrix \(A\).
By iteration of \(\phi\) a dynamical system is obtained, and some of its properties (geometrical in nature) generate invariants and a canonical form of the orbit of \(A\).
This is applied to a family of matrices parametrized by \(\mathbb P^1\), whose associated geometry is given by the rational normal curve for each dimension \(n=2, 3, 4\). The analysis involves the osculating flags to the curves. Further, the stabilizers of the rational maps and matrices are calculated.

MSC:

15A30 Algebraic systems of matrices
20G15 Linear algebraic groups over arbitrary fields

Software:

Macaulay2
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Full Text: DOI

References:

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