Vargas, J. A. Matrices of linear forms and curves. (English) Zbl 1107.15010 Linear Algebra Appl. 418, No. 2-3, 363-379 (2006). 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. Reviewer: A. Arvanitoyeorgos (Rion) Cited in 2 Documents MSC: 15A30 Algebraic systems of matrices 20G15 Linear algebraic groups over arbitrary fields Keywords:trilinear algebra; matrix of linear forms; rational map; dynamical system; rational normal quartic; osculating flag; polynomial identity; population genetics; invariants; canonical form Software:Macaulay2 PDFBibTeX XMLCite \textit{J. A. Vargas}, Linear Algebra Appl. 418, No. 2--3, 363--379 (2006; Zbl 1107.15010) Full Text: DOI References: [1] Borel, A., Linear Algebraic Groups. Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 126 (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0726.20030 [2] Eisenbud, D., Commutative Algebra with a view toward Algebraic Geometry. Commutative Algebra with a view toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150 (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0819.13001 [3] Evans, L. C., A survey of entropy methods for partial differential equations, Bull. Amer. Math. Soc., 41, 409-438 (2004) · Zbl 1053.35004 [4] D.R. Grayson, M.E. Stillman, Macaulay2, a software system for research in algebraic geometry. Available from: <http://www.math.uiuc.edu/Macaulay2>; D.R. Grayson, M.E. Stillman, Macaulay2, a software system for research in algebraic geometry. Available from: <http://www.math.uiuc.edu/Macaulay2> [5] Hartshorne, R., Algebraic Geometry. Algebraic Geometry, Graduate Texts in Mathematics, vol. 52 (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0367.14001 [6] Harris, J., Algebraic Geometry. Algebraic Geometry, Graduate Texts in Mathematics, vol. 133 (1992), Springer-Verlag: Springer-Verlag New York · Zbl 0932.14001 [7] Humphreys, J. E., Linear Algebraic Groups. Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 21 (1981), Springer-Verlag: Springer-Verlag New York · Zbl 0471.20029 [8] Springer, T. A., Linear Algebraic Groups (1998), Birkäuser: Birkäuser Boston · Zbl 0927.20024 [9] Vargas, J. A.; del Castillo, R. F., Inbreeding depression in a zygotic algebra, Comm. Alg., 27, 4425-4432 (1999) · Zbl 1015.17030 [10] Vargas, J. A., Hardy-Weinberg theory for tetraploidy with mixed mating, Adv. Appl. Math., 24, 369-383 (2000) · Zbl 0980.92024 [11] Vargas, J. A.; del Castillo, R. F., Genetic associations under mixed-mating systems: the Bennett-Binet effect, IMA J. Math. Appl. Med. Biol., 18, 327-341 (2001) · Zbl 1013.92028 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.