Frobenius-Schur functions: summary of results

TL;DR

This paper introduces Frobenius-Schur functions, a new family of symmetric functions related to Schur functions, providing explicit formulas for skew diagram dimensions and exploring their properties and generalizations.

Contribution

It presents the first explicit determinantal expression for Frobenius-Schur functions and introduces multiparameter Schur functions interpolating between known symmetric functions.

Findings

Determinantal formula for Frobenius-Schur functions in terms of Schur functions

Explicit expression for skew Young diagram dimensions

Introduction of multiparameter Schur functions

Abstract

We introduce and study a family of inhomogeneous symmetric functions which we call the Frobenius-Schur functions. These functions are indexed by partitions and differ from the conventional Schur functions in lower terms only. Our interest in these new functions comes from the fact that they provide an explicit expression for the dimension of a skew Young diagram in terms of the Frobenius coordinates. This is important for the asymptotic theory of the characters of the symmetric groups. Our main result is a surprisingly simple determinantal expression of the Frobenius-Schur functions in terms of the conventional Schur functions. Other results include certain generating series, the Giambelli formula, vanishing properties and interpolation, a combinatorial formula (representation in terms of tableaux), and a Sergeev-Pragacz-type formula. Actually, we deal with a wider class of inhomogeneous symmetric functions which we call multiparameter Schur functions. These functions depend on an arbitrary doubly infinite sequence of parameters and interpolate between the Frobenius--Schur functions and the conventional Schur functions. This paper contains the statements of the results and the main formulas. Proofs will be given in an expanded version of the paper which will be posted in the arXiv.