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A weight basis for representations of even orthogonal Lie algebras. (English) Zbl 1008.17003

Koike, Kazuhiko (ed.) et al., Combinatorial methods in representation theory. Papers of the conference on combinatorial methods in representation theory, July 21-July 31, 1998 and the conference on interaction of combinatorics and representation theory, October 26-November 6, 1998, Kyoto, Japan. Tokyo: Kinokuniya Company Ltd. Adv. Stud. Pure Math. 28, 221-240 (2000).
The author constructs a basis for any finite dimensional representation of the Lie algebra \(\mathfrak o(2n)\). The basis differs from that of Gelfand and Tsetlin in the following ways: It is consistent with the chain of type D subalgebras \(\mathfrak o(2)\subset\mathfrak o(4)\subset\dots\subset\mathfrak o(2n-2) \subset\mathfrak o(2n)\) rather than with \(\mathfrak o(2)\subset\mathfrak o(3)\subset\dots \subset\mathfrak o(2n-1)\subset\mathfrak o(2n)\), so that the basis vectors can be chosen to be simultaneous weight vectors. However, the restriction from \(\mathfrak o(2n)\) to \(\mathfrak o(2n-2)\) is not multiplicity free. To separate the multiplicities, the author uses the irreducible action of the twisted Yangian \(Y^+(2)\) on the space \(V(\lambda)_{\mu}^+\) of highest weight vectors for \(\mathfrak o(2n-2)\) of weight \(\mu\) in the \(\mathfrak o(2n)\)-module \(V(\lambda)\) of highest weight \(\lambda\). In addition, he computes the action of a generating set of matrix elements on this basis, which is parametrized by Gelfand-Tsetlin patterns.
This paper follows a previous one [Commun. Math. Phys. 201, No. 3, 591-618 (1999; Zbl 0931.17005)], in which the author constructed a similar basis for the finite dimensional representations of the symplectic Lie algebras.
For the entire collection see [Zbl 0963.00024].

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B20 Simple, semisimple, reductive (super)algebras

Citations:

Zbl 0931.17005
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