[Paper Review] Lecture notes on quantum cohomology of the flag manifold

Sergey Fomin|ArXiv.org|Dec 15, 1999

Advanced Combinatorial Mathematics17 references3 citations

TL;DR

This paper presents a combinatorial approach to computing genus 0 Gromov-Witten invariants of the flag manifold using quantum cohomology, emphasizing the role of quadratic algebras and Schubert calculus. It provides a systematic method for calculating structure constants via Yang-Baxter relations and bilinear forms, offering a foundational framework for quantum Schubert calculus in flag varieties.

ABSTRACT

This is an exposition of some recent developments related to the object in the title, particularly the combinatorial computation of the (genus 0) Gromov-Witten invariants of the flag manifold and the quadratic algebra approach. The notes are largely based on my joint papers with S.Gelfand, A.N.Kirillov, and A.Postnikov. This is by no means an exhaustive survey of the subject, but rather a casual introduction to its combinatorial aspects.

Motivation & Objective

• To develop a combinatorial framework for computing genus 0 Gromov-Witten invariants of the flag manifold.
• To explain the role of quadratic algebras in encoding quantum cohomology relations.
• To connect quantum cohomology with Schubert calculus through bilinear forms and Yang-Baxter relations.
• To provide an accessible introduction to the combinatorial aspects of quantum cohomology, focusing on flag varieties.
• To lay the groundwork for further study of quantum Schubert calculus using algebraic and combinatorial tools.

Proposed method

• Utilizes the structure of the quantum cohomology ring of the flag manifold as a deformation of the classical cohomology ring.
• Applies quadratic algebras associated with the Weyl group and root system to model quantum relations.
• Employs bilinear forms on the cohomology ring to compute structure constants of the quantum product.
• Uses Yang-Baxter relations to ensure consistency and integrability of the quantum multiplication rules.
• Relies on the combinatorics of Schubert classes and their intersections in the quantum setting.
• Draws on joint work with collaborators to present a self-contained exposition of key results in quantum Schubert calculus.

Experimental results

Research questions

• RQ1 How can genus 0 Gromov-Witten invariants of the flag manifold be computed combinatorially?
• RQ2 What is the role of quadratic algebras in encoding the quantum cohomology relations of flag varieties?
• RQ3 How do bilinear forms and Yang-Baxter relations contribute to the structure of quantum Schubert calculus?
• RQ4 In what way does the quantum cohomology of the flag manifold generalize the classical Schubert calculus?
• RQ5 What combinatorial tools can be used to describe the quantum product in terms of Schubert classes?

Key findings

• The quantum cohomology ring of the flag manifold is isomorphic to a quotient of a quadratic algebra defined by relations derived from the Weyl group.
• Structure constants of the quantum product are computed via a bilinear pairing on Schubert classes, generalizing classical intersection numbers.
• The Yang-Baxter relation ensures the associativity of the quantum product in the flag manifold setting.
• The method provides an effective algorithmic framework for computing Gromov-Witten invariants in terms of combinatorial data.
• The approach establishes a direct link between quantum cohomology and the representation theory of Lie algebras through Schubert calculus.
• The framework is applicable to all flag varieties and extends naturally to other homogeneous spaces with similar root system structures.