dorsal/arxiv
View SchemaShifted Schur Functions
| Authors | Andrei Okounkov, Grigori Olshanski |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9605042 |
| URL | https://arxiv.org/abs/q-alg/9605042 |
| Journal | Algebra i Analiz 9 (1997), no. 2, 73--146 (Russian); English translation to appear in St. Petersburg Math. J. 9 (1998), no. 2 |
Abstract
The classical algebra $\Lambda$ of symmetric functions has a remarkable deformation $\Lambda^*$, which we call the algebra of shifted symmetric functions. In the latter algebra, there is a distinguished basis formed by shifted Schur functions $s^*_\mu$, where $\mu$ ranges over the set of all partitions. The main significance of the shifted Schur functions is that they determine a natural basis in $Z(\frak{gl}(n))$, the center of the universal enveloping algebra $U(\frak{gl}(n))$, $n=1,2,\ldots$. The functions $s^*_\mu$ are closely related to the factorial Schur functions introduced by Biedenharn and Louck and further studied by Macdonald and other authors. A part of our results about the functions $s^*_\mu$ has natural classical analogues (combinatorial presentation, generating series, Jacobi--Trudi identity, Pieri formula). Other results are of different nature (connection with the binomial formula for characters of $GL(n)$, an explicit expression for the dimension of skew shapes $\lambda/\mu$, Capelli--type identities, a characterization of the functions $s^*_\mu$ by their vanishing properties, `coherence property', special symmetrization map $S(\frak{gl}(n))\to U(\frak{gl}(n))$. The main application that we have in mind is the asymptotic character theory for the unitary groups $U(n)$ and symmetric groups $S(n)$ as $n\to\infty$. The results of this paper were used in \cite{Ok1--3}.
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"abstract": "The classical algebra $\\Lambda$ of symmetric functions has a remarkable\ndeformation $\\Lambda^*$, which we call the algebra of shifted symmetric\nfunctions. In the latter algebra, there is a distinguished basis formed by\nshifted Schur functions $s^*_\\mu$, where $\\mu$ ranges over the set of all\npartitions. The main significance of the shifted Schur functions is that they\ndetermine a natural basis in $Z(\\frak{gl}(n))$, the center of the universal\nenveloping algebra $U(\\frak{gl}(n))$, $n=1,2,\\ldots$.\n The functions $s^*_\\mu$ are closely related to the factorial Schur functions\nintroduced by Biedenharn and Louck and further studied by Macdonald and other\nauthors.\n A part of our results about the functions $s^*_\\mu$ has natural classical\nanalogues (combinatorial presentation, generating series, Jacobi--Trudi\nidentity, Pieri formula). Other results are of different nature (connection\nwith the binomial formula for characters of $GL(n)$, an explicit expression for\nthe dimension of skew shapes $\\lambda/\\mu$, Capelli--type identities, a\ncharacterization of the functions $s^*_\\mu$ by their vanishing properties,\n`coherence property\u0027, special symmetrization map $S(\\frak{gl}(n))\\to\nU(\\frak{gl}(n))$.\n The main application that we have in mind is the asymptotic character theory\nfor the unitary groups $U(n)$ and symmetric groups $S(n)$ as $n\\to\\infty$.\n The results of this paper were used in \\cite{Ok1--3}.",
"arxiv_id": "q-alg/9605042",
"authors": [
"Andrei Okounkov",
"Grigori Olshanski"
],
"categories": [
"q-alg",
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],
"journal_ref": "Algebra i Analiz 9 (1997), no. 2, 73--146 (Russian); English\n translation to appear in St. Petersburg Math. J. 9 (1998), no. 2",
"title": "Shifted Schur Functions",
"url": "https://arxiv.org/abs/q-alg/9605042"
},
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