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
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- by Murray R. Bremner and Jiaxiong Hu PDF
- Math. Comp. 82 (2013), 2421-2438 Request permission
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 $\mathfrak {sl}_3(\mathbb {C}) \oplus \mathfrak {sl}_3(\mathbb {C}) \oplus \mathfrak {sl}_3(\mathbb {C})$.References
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Additional Information
- Murray R. Bremner
- Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, Canada
- MR Author ID: 41310
- Email: bremner@math.usask.ca
- Jiaxiong Hu
- Affiliation: Department of Mathematics, Simon Fraser University, Canada
- Email: hujiaxiong@gmail.com
- 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.
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3073208