A representation for some groups, a geometric approach
Subject Areas : Group theory
1 - Department of Mathematics, Tafresh Universiy, 39518-79611 Tafresh, Iran
Keywords: Group representation, exponential matrix, integral curve, vector field,
Abstract :
In the present paper, we are going to use geometric and topological concepts, entities and properties of theintegral curves of linear vector fields, and the theory of differential equations,to establish a representation for some groups on $R^{n} (n\geq 1)$. Among other things, we investigate the surjectivity and faithfulness of the representation.At the end, we give some applications. .
[1] A. Arhangel'skii, M. Tkachenko, Topological groups and related structures, Atlantis press, Paris, 2008.
[2] S. Ball, Finite geometry and combinatorial applications, Cambridge University Press, London, 2015.
[3] W. Fulton, J. Harris, Representation theory: a rst course, Springer-Verlag, New York, 2004.
[4] B. C. Hall, Lie groups, Lie algebras, and representations: an elementary introduction, Springer-Verlag, New York, 2003.
[5] N. J. Higham, Functions of matrices, theory and computation, SIAM, Philadelphia, 2008.
[6] W. S. Hurewicz, Lectures on ordinary di erential equations, M.I.T. Press, Dover ed., Cambridge, 2014.
[7] C. Moler, C. V. Loan, Nineteen dubious ways to compute the exponential of a matrix, SIAM Rev. 20 (4) (1978), 801-836.
[8] C. Moler, C. V. Loan, Nineteen dubious ways to compute the exponential of a matrix, Twenty-Five Years Later, SIAM Rev. 45 (1) (2003), 3-49.
[9] A. N. Sengupta, Representing nite groups: a semisimple introduction, Springer-Verlage, USA, 2011.
[10] S. Shahshahani, An introductory course on di erentiable manifolds, Dover Publications Inc, USA, 2016.
[11] R. Steinberg, A geometric approach to the representations of the full linear group over a Galois eld, Trans. Amer. Math. Soc. 71 (2) (1951), 274-282.
[12] R. Steinberg, Collected papers, AMS, New York, 1997.