Theta and eta polynomials in geometry, Lie theory, and combinatorics

TL;DR

This paper reviews the development of theta and eta polynomials, which generalize Schur polynomials to symplectic and orthogonal types, highlighting their geometric and combinatorial significance.

Contribution

It provides an overview of the known results and examples related to theta and eta polynomials in geometry, Lie theory, and combinatorics.

Findings

Theta and eta polynomials generalize Schur polynomials to other Lie types.

They represent Schubert classes in symplectic and orthogonal Grassmannians.

The paper summarizes current knowledge and examples of these polynomials.

Abstract

The classical Schur polynomials form a natural basis for the ring of symmetric polynomials, and have geometric significance since they represent the Schubert classes in the cohomology ring of Grassmannians. Moreover, these polynomials enjoy rich combinatorial properties. In the last decade, an exact analogue of this picture has emerged in the symplectic and orthogonal Lie types, with the Schur polynomials replaced by the theta and eta polynomials of Buch, Kresch, and the author. This expository paper gives an overview of what is known to date about this correspondence, with examples.