dorsal/arxiv
View SchemaShifted Schur functions II. Binomial formula for characters of classical groups and applications
| Authors | Andrei Okounkov, Grigori Olshanski |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9612025 |
| URL | https://arxiv.org/abs/q-alg/9612025 |
| Journal | In: Kirillov's Seminar on Representation Theory. Amer. Math. Soc. Transl. 1998, pp. 245-271. |
Abstract
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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"abstract": "Let G be any of the complex classical groups GL(n), SO(2n+1), Sp(2n), O(2n),\nlet g denote the Lie algebra of G, and let Z(g) denote the subalgebra of\nG-invariants in the universal enveloping algebra U(g). We derive a Taylor-type\nexpansion for finite-dimensional characters of G (binomial formula) and use it\nto specify a distinguished linear basis in Z(g). The eigenvalues of the basis\nelements in highest weight g-modules are certain shifted (or factorial) analogs\nof Schur functions. We also study an associated homogeneous basis in I(g), the\nsubalgebra of G-invariants in the symmetric algebra S(g). Finally, we show that\nthe both bases are related by a G-equivariant linear isomorphism \\sigma:\nI(g)\\to Z(g), called the special symmetrization.",
"arxiv_id": "q-alg/9612025",
"authors": [
"Andrei Okounkov",
"Grigori Olshanski"
],
"categories": [
"q-alg",
"math.QA"
],
"journal_ref": "In: Kirillov\u0027s Seminar on Representation Theory. Amer. Math. Soc.\n Transl. 1998, pp. 245-271.",
"title": "Shifted Schur functions II. Binomial formula for characters of classical groups and applications",
"url": "https://arxiv.org/abs/q-alg/9612025"
},
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