[Paper Review] Flag varieties and the Yang-Baxter equation

Alain Lascoux, Bernard Leclerc+1 more|ArXiv.org|Jul 14, 1996

Advanced Combinatorial Mathematics14 references3 citations

TL;DR

This paper introduces Yang-Baxter bases in Hecke algebras using solutions to the Yang-Baxter equation, showing they are self-adjoint under a canonical bilinear form. It identifies coefficients in the degenerate Hecke algebra case with specializations of double Schubert polynomials, linking representation theory to symmetric function theory and flag varieties.

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.

Motivation & Objective

• To define and study Yang-Baxter bases in Hecke algebras using solutions to the Yang-Baxter equation.
• To investigate the self-adjointness of these bases with respect to a canonical bilinear form.
• To establish a precise connection between coefficients in the degenerate Hecke algebra and double Schubert polynomials.
• To describe expansions in other specializations of the generic Hecke algebra.
• To unify combinatorial structures in symmetric functions with representation-theoretic constructions via the Yang-Baxter equation.

Proposed method

• Constructs bases of Hecke algebras using R-matrices derived from the Yang-Baxter equation.
• Employs a canonical bilinear form to prove that Yang-Baxter bases are essentially self-adjoint.
• Analyzes the degenerate Hecke algebra case to identify structure constants with specializations of double Schubert polynomials.
• Applies combinatorial techniques from symmetric functions and Schubert calculus to relate algebraic coefficients to geometric objects.
• Uses the structure of the generic Hecke algebra to explore various specializations and their associated expansions.
• Relies on the theory of symmetric functions and the combinatorics of permutations to derive explicit formulas.

Experimental results

Research questions

• RQ1 How can solutions to the Yang-Baxter equation be used to define new bases in Hecke algebras?
• RQ2 What is the self-adjoint property of these Yang-Baxter bases under the canonical bilinear form?
• RQ3 How do the structure constants of the Yang-Baxter basis in the degenerate Hecke algebra relate to double Schubert polynomials?
• RQ4 What are the expansions of Yang-Baxter bases in other specializations of the generic Hecke algebra?
• RQ5 What geometric or combinatorial structures underlie the coefficients in these algebraic expansions?

Key findings

• Yang-Baxter bases in Hecke algebras are essentially self-adjoint with respect to the canonical bilinear form.
• In the degenerate Hecke algebra, the coefficients of the Yang-Baxter basis in the standard basis are given by specializations of double Schubert polynomials.
• The paper provides explicit formulas for the expansions of Yang-Baxter bases in various specializations of the generic Hecke algebra.
• The connection between the Yang-Baxter equation and Schubert calculus is established through the identification of structure constants with double Schubert polynomials.
• The results reveal a deep link between integrable systems (via Yang-Baxter R-matrices) and algebraic geometry (via flag varieties and Schubert polynomials).
• The work provides a new algebraic framework that unifies symmetric function theory, Hecke algebras, and geometric representation theory.