Flag varieties and the Yang-Baxter equation

TL;DR

This paper explores Yang-Baxter bases in Hecke algebras, revealing their self-adjoint properties and connections to double Schubert polynomials, advancing understanding of algebraic structures related to flag varieties.

Contribution

It introduces Yang-Baxter bases in Hecke algebras and links their coefficients to double Schubert polynomials, providing new insights into algebraic and geometric structures.

Findings

Yang-Baxter bases are self-adjoint with respect to a canonical bilinear form.

Coefficients in the basis expansion relate to specializations of double Schubert polynomials.

Descriptions of expansions for various specializations of the Hecke algebra.

Abstract

We investigate certain bases of Hecke algebras defined by means of the Yang-Baxter equation, which we call Yang-Baxter bases. These bases are essentially self-adjoint with respect to a canonical bilinear form. In the case of the degenerate Hecke algebra, we identify the coefficients in the expansion of the Yang-Baxter basis on the usual basis of the algebra with specializations of double Schubert polynomials. We also describe the expansions associated to other specializations of the generic Hecke algebra.