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Weyl's construction and tensor power decomposition for $G_{2}$

Authors: Jing-Song Huang and Chen-Bo Zhu
Journal: Proc. Amer. Math. Soc. 127 (1999), 925-934
MSC (1991): Primary 22E46, 13A50
MathSciNet review: 1469412
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Abstract: Let $V$ be the 7-dimensional irreducible representations of $G_{2}$. We decompose the tensor power $V^{\otimes n}$ into irreducible representations of $G_{2}$ and obtain all irreducible representations of $G_{2}$ in the decomposition. This generalizes Weyl's work on the construction of irreducible representations and decomposition of tensor products for classical groups to the exceptional group $G_{2}$.

References [Enhancements On Off] (What's this?)

  • [H] Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1–182. MR 1321638,
  • [FH] William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249
  • [MP] W. G. McKay and J. Patera, Tables of dimensions, indices, and branching rules for representations of simple Lie algebras, Lecture Notes in Pure and Applied Mathematics, vol. 69, Marcel Dekker, Inc., New York, 1981. MR 604363
  • [S] Gerald W. Schwarz, Invariant theory of 𝐺₂ and 𝑆𝑝𝑖𝑛₇, Comment. Math. Helv. 63 (1988), no. 4, 624–663. MR 966953,
  • [W] H. Weyl, The classical groups, Princeton Mathematical Series, Princeton University Press, 1939; second edition, 1946; third edition, 1966. MR 1:42c
  • [Wo] Y.-K. Wong, The First Fundamental Theorem of Covariants for $G_{2}$ and $Spin_{7}$, Yale thesis, Yale University, 1995.

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Additional Information

Jing-Song Huang
Affiliation: Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

Chen-Bo Zhu
Affiliation: Department of Mathematics, National University of Singapore, Kent ridge, Singapore 0511

Keywords: Cayley numbers, invariant theory, tensor power decomposition
Received by editor(s): March 25, 1997
Received by editor(s) in revised form: July 7, 1997
Additional Notes: The first named author was partially supported by NSF Grant DMS 9306138 and RGC Competitive Earmarked Research Grant HKUST 588/94P
Communicated by: Roe Goodman
Article copyright: © Copyright 1999 American Mathematical Society