Mathematics – Quantum Algebra
Scientific paper
1996-05-28
Algebra i Analiz 9 (1997), no. 2, 73--146 (Russian); English translation to appear in St. Petersburg Math. J. 9 (1998), no. 2
Mathematics
Quantum Algebra
65 pages, AMS-TeX
Scientific paper
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}.
Okounkov Andrei
Olshanski Grigori
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