Shifted Schur functions II. Binomial formula for characters of classical groups and applications

Details Shifted Schur functions II. Binomial formula for characters of classical groups and applications Shifted Schur functions II. Binomial formula for characters of classical groups and applications

1996-12-17

arxiv.org/abs/q-alg/9612025v1

In: Kirillov's Seminar on Representation Theory. Amer. Math. Soc. Transl. 1998, pp. 245-271.

Mathematics

Quantum Algebra

28 pages, AMS-TeX, this paper is logically and chronologically preceding the paper q-alg/9611011

Scientific paper

Let G be any of the complex classical groups GL(n), SO(2n+1), Sp(2n), O(2n), let g denote the Lie algebra of G, and let Z(g) denote the subalgebra of G-invariants in the universal enveloping algebra U(g). We derive a Taylor-type expansion for finite-dimensional characters of G (binomial formula) and use it to specify a distinguished linear basis in Z(g). The eigenvalues of the basis elements in highest weight g-modules are certain shifted (or factorial) analogs of Schur functions. We also study an associated homogeneous basis in I(g), the subalgebra of G-invariants in the symmetric algebra S(g). Finally, we show that the both bases are related by a G-equivariant linear isomorphism \sigma: I(g)\to Z(g), called the special symmetrization.

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