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Equivariant multiplicities via representations of quantum affine algebras.

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Matroid psi classes.

Jeshu Dastidar1, Dustin Ross2

  • 1Department of Mathematics, University of California, Davis, USA.

Selecta Mathematica. New Series
|July 31, 2023
PubMed
Summary

We introduce psi classes in matroid Chow rings, generalizing properties from moduli spaces of curves. These new matroid psi classes provide novel proofs for matroid characteristic polynomials, volume polynomials, and Poincaré duality.

Area of Science:

  • Algebraic Geometry
  • Combinatorics
  • Algebraic Combinatorics

Background:

  • The intersection theory of moduli spaces of curves is a well-developed area.
  • Psi classes are fundamental objects in the study of moduli spaces.
  • Matroid theory has connections to combinatorial algebraic geometry.

Purpose of the Study:

  • To introduce and study psi classes within the framework of matroid Chow rings.
  • To generalize known properties of psi classes from moduli spaces of curves to the matroid setting.
  • To leverage these new matroid psi classes to provide new proofs for key results in matroid theory.

Main Methods:

  • Definition of psi classes in matroid Chow rings.
  • Development of properties for these matroid psi classes.

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  • Application of these properties to derive new proofs for existing theorems.

    • Main Results:

    • Introduction of psi classes in matroid Chow rings.
    • Proof of properties for matroid psi classes analogous to those in moduli spaces.
    • New proofs for a Chow-theoretic interpretation of reduced characteristic polynomial coefficients.
    • New explicit formulas for matroid volume polynomials.
    • New proofs of Poincaré duality for matroid Chow rings.

    Conclusions:

    • The introduction of psi classes enriches the study of matroid Chow rings.
    • Matroid psi classes offer a powerful tool for understanding combinatorial structures.
    • This work bridges concepts from algebraic geometry and matroid theory, opening avenues for future research.