Shifted Schur Functions

Abstract

The classical algebra $\Lambda$ of symmetric functions has a remarkable deformation ${\Lambda }^{\ast }$, which we call the algebra of shifted symmetric functions. In the latter algebra, there is a distinguished basis formed by shifted Schur functions $s_{\mu }^{\ast }$, 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 $Z(\mathfrak{gl}(n))$, the center of the universal enveloping algebra $U(\mathfrak{gl}(n))$, $n=1,2,\dots$. The functions $s_{\mu }^{\ast }$ 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 $s_{\mu }^{\ast }$ 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 $s_{\mu }^{\ast }$ by their vanishing properties, `coherence property', special symmetrization map $S(\mathfrak{gl}(n))\to U(\mathfrak{gl}(n))$. 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}.

65 pages, AMS-TeX

References in corpus (1)

• Difference equations and symmetric polynomials defined by their zeros

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