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The Cubic, the Quartic, and the Exceptional Group G2

  • Anthony van Groningen
  • Jeb F. WillenbringEmail author
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 38)

Abstract

We study an example first addressed in a 1949 paper of J. A. Todd, in which the author obtains a complete system of generators for the covariants in the polynomial functions on the eight-dimensional space of the double binary form of degree (3,1), under the action of SL2 ×SL2. We reconsider Todd’s result by examining the complexified Cartan complement corresponding to the maximal compact subgroup of simply connected split G 2. A result of this analysis involves a connection with the branching rule from the rank two complex symplectic Lie algebra to a principally embedded \(\mathfrak{s}\mathfrak{l}_{2}\)-subalgebra. Special cases of this branching rule are related to covariants for cubic and quartic binary forms.

Key words

Binary form Branching rule Double binary form G2 Harmonic polynomials Principal \(\mathfrak{s}\mathfrak{l}_{2}\) Symmetric space 

Mathematics Subject Classification (2010):

20G05 22E45 17B10. 

Notes

Acknowledgements

We thank Allen Bell, Nolan Wallach and Gregg Zuckerman for helpful conversations about the results presented here. The first author’s Ph.D. thesis [17] contains a more thorough treatment of the graded K-multiplicities associated with the symmetric pair (\(G_{2},\mathfrak{s}\mathfrak{l}_{2} \oplus \mathfrak{s}\mathfrak{l}_{2}\)), which was jointly directed by Allen Bell and the second author. The problem concerning the graded decomposition of \(\mathcal{H}_{\mathfrak{p}}\) as a K-representation was suggested by Nolan Wallach (see [19]), while the problem of studying the restriction of a finite-dimensional representation to a principally embedded \(\mathfrak{s}\mathfrak{l}_{2}\)-subalgebra was suggested by Gregg Zuckerman (see [20]).

Finally, we would like to thank the referee for many helpful suggestions, and corrections.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Milwaukee School of EngineeringMilwaukeeUSA
  2. 2.Department of Mathematical SciencesUniversity of Wisconsin-MilwaukeeMilwaukeeUSA

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