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Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups

  Published:2016-01-27
 Printed: Apr 2016
  • Skip Garibaldi,
    Institute for Pure and Applied Mathematics, UCLA, 460 Portola Plaza, Box 957121, Los Angeles, California 90095-7121, USA
  • Daniel K. Nakano,
    Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA
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Abstract

The representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the question of whether a given complex representation is symplectic or orthogonal has been solved since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from 2 hold by using similar proofs. This paper considers analogues of these results for simple, induced and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.
Keywords: orthogonal representations, symmetric tensors, alternating forms, characteristic 2, split reductive groups orthogonal representations, symmetric tensors, alternating forms, characteristic 2, split reductive groups
MSC Classifications: 20G05, 11E39, 11E88, 15A63, 20G15 show english descriptions Representation theory
Bilinear and Hermitian forms
Quadratic spaces; Clifford algebras [See also 15A63, 15A66]
Quadratic and bilinear forms, inner products [See mainly 11Exx]
Linear algebraic groups over arbitrary fields
20G05 - Representation theory
11E39 - Bilinear and Hermitian forms
11E88 - Quadratic spaces; Clifford algebras [See also 15A63, 15A66]
15A63 - Quadratic and bilinear forms, inner products [See mainly 11Exx]
20G15 - Linear algebraic groups over arbitrary fields
 

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