Shifted Schur Functions
Loading...
Date
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Description
The classical algebra $Λ$ of symmetric functions has a remarkable deformation $Λ^*$, which we call the algebra of shifted symmetric functions. In the latter algebra, there is a distinguished basis formed by shifted Schur functions $s^*_μ$, where $μ$ 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 $s^*_μ$ 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 $s^*_μ$ 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 $λ/μ$, Capelli--type identities, a characterization of the functions $s^*_μ$ 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}.
65 pages, AMS-TeX
65 pages, AMS-TeX