Journal of the Mathematical Society of Japan

Multiplicity-free branching rules for outer automorphisms of simple Lie algebras

Hidehisa ALIKAWA

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We find explicit multiplicity-free branching rules of some series of irreducible finite dimensional representations of simple Lie algebras 𝔤 to the fixed point subalgebras 𝔤 σ of outer automorphisms σ . The representations have highest weights which are scalar multiples of fundamental weights or linear combinations of two scalar ones. Our list of pairs of Lie algebras ( 𝔤 , 𝔤 σ ) includes an exceptional symmetric pair ( E 6 , F 4 ) and also a non-symmetric pair ( D 4 , G 2 ) as well as a number of classical symmetric pairs. Some of the branching rules were known and others are new, but all the rules in this paper are proved by a unified method. Our key lemma is a characterization of the ``middle'' cosets of the Weyl group of 𝔤 in terms of the subalgebras 𝔤 σ on one hand, and the length function on the other hand.

Article information

J. Math. Soc. Japan, Volume 59, Number 1 (2007), 151-177.

First available in Project Euclid: 25 May 2007

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Zentralblatt MATH identifier

Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 05E15: Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80]

branching rule multiplicity free outer automorphism graph automorphism semisimple Lie groups characters of Lie groups


ALIKAWA, Hidehisa. Multiplicity-free branching rules for outer automorphisms of simple Lie algebras. J. Math. Soc. Japan 59 (2007), no. 1, 151--177. doi:10.2969/jmsj/1180135505.

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