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Fundamental invariants for the action of $ SL_3(\mathbb{C}) \times SL_3(\mathbb{C}) \times SL_3(\mathbb{C})$ on $ 3 \times 3 \times 3$ arrays


Authors: Murray R. Bremner and Jiaxiong Hu
Journal: Math. Comp. 82 (2013), 2421-2438
MSC (2010): Primary 13A50; Secondary 15A72, 17B10
DOI: https://doi.org/10.1090/S0025-5718-2013-02706-8
Published electronically: May 2, 2013
MathSciNet review: 3073208
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Abstract: We determine the three fundamental invariants in the entries of a $ 3 \times 3 \times 3$ array over $ \mathbb{C}$ as explicit polynomials in the 27 variables $ x_{ijk}$ for $ 1 \le i, j, k \le 3$. By the work of Vinberg on $ \theta $-groups, it is known that these homogeneous polynomials have degrees 6, 9 and 12; they freely generate the algebra of invariants for the Lie group $ SL_3(\mathbb{C}) \times SL_3(\mathbb{C}) \times SL_3(\mathbb{C})$ acting irreducibly on its natural representation $ \mathbb{C}^3 \otimes \mathbb{C}^3 \otimes \mathbb{C}^3$. These generators have, respectively, 1152, 9216 and 209061 terms; we find compact expressions in terms of the orbits of the finite group $ ( S_3 \times S_3 \times S_3 ) \rtimes S_3$ acting on monomials of weight zero for the action of the Lie algebra $ \mathfrak{sl}_3(\mathbb{C}) \oplus \mathfrak{sl}_3(\mathbb{C}) \oplus \mathfrak{sl}_3(\mathbb{C})$.


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Additional Information

Murray R. Bremner
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, Canada
Email: bremner@math.usask.ca

Jiaxiong Hu
Affiliation: Department of Mathematics, Simon Fraser University, Canada
Email: hujiaxiong@gmail.com

DOI: https://doi.org/10.1090/S0025-5718-2013-02706-8
Keywords: Multidimensional arrays, polynomial invariants, semisimple Lie algebras, representation theory
Received by editor(s): December 13, 2011
Received by editor(s) in revised form: March 28, 2012
Published electronically: May 2, 2013
Additional Notes: The first author was partially supported by a Discovery Grant from NSERC. The authors thank the referees and Luke Oeding for helpful comments.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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