Shifted Schur Functions

TL;DR

This paper introduces shifted Schur functions, a deformation of symmetric functions, which form a basis in the center of universal enveloping algebras and have applications in asymptotic character theory for unitary and symmetric groups.

Contribution

It develops the theory of shifted Schur functions, including classical analogues, identities, and their connection to representation theory and asymptotic analysis.

Findings

Established combinatorial and algebraic properties of shifted Schur functions

Derived explicit formulas for dimensions of skew shapes

Connected shifted Schur functions to asymptotic character theory

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}.