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Chow classes of matroids and standard Young tableaux

Jon Pål Hamre, Benjamin Schröter, Lorenzo Vecchi, Emil Verkama

TL;DR

This work develops a purely combinatorial approach to Chow classes of matroids in Grassmannians, reducing computations to snake matroids and extending to all matroids via valuativity. A central achievement is identifying the Poincaré dual of the Chow class of a snake matroid with the ribbon Schur function, which yields explicit Schubert coefficients as counts of standard Young tableaux with prescribed descent sets and recovers known uniform-matroid formulas. The paper then extends these results to lattice-path matroids, derives determinant and LR-structure formulas, and derives impactful applications, including RS-K volume interpretations and Gessel–Viennot-type counting formulas, as well as new positivity results for paving matroids. The framework also provides practical computational tools (via the K-class viewpoint) and opens numerous open problems, notably in positroids, nonnegativity, and the relation between volume and the beta-invariant.

Abstract

We study the Chow classes of arbitrary matroids in the Grassmannian. We develop a new combinatorial approach for computing them, by first focusing on snake matroids and then extending our results via valuativity to any matroid. Our main contribution identifies the Poincaré dual of the Chow class of a snake matroid with a specific ribbon Schur function, providing an explicit formula for its coefficients in the Schubert basis as the number of standard Young tableaux of a given shape with a prescribed descent set. This agrees with a formula by Klyachko for the uniform matroid. As consequences, we recover and simplify classical results such as Gessel and Viennot's enumeration of permutations with fixed descent sets, and formulas for the volume of lattice path matroids. Furthermore, we demonstrate the power of our findings by proving that certain Schubert coefficients are positive for all connected paving matroids.

Chow classes of matroids and standard Young tableaux

TL;DR

This work develops a purely combinatorial approach to Chow classes of matroids in Grassmannians, reducing computations to snake matroids and extending to all matroids via valuativity. A central achievement is identifying the Poincaré dual of the Chow class of a snake matroid with the ribbon Schur function, which yields explicit Schubert coefficients as counts of standard Young tableaux with prescribed descent sets and recovers known uniform-matroid formulas. The paper then extends these results to lattice-path matroids, derives determinant and LR-structure formulas, and derives impactful applications, including RS-K volume interpretations and Gessel–Viennot-type counting formulas, as well as new positivity results for paving matroids. The framework also provides practical computational tools (via the K-class viewpoint) and opens numerous open problems, notably in positroids, nonnegativity, and the relation between volume and the beta-invariant.

Abstract

We study the Chow classes of arbitrary matroids in the Grassmannian. We develop a new combinatorial approach for computing them, by first focusing on snake matroids and then extending our results via valuativity to any matroid. Our main contribution identifies the Poincaré dual of the Chow class of a snake matroid with a specific ribbon Schur function, providing an explicit formula for its coefficients in the Schubert basis as the number of standard Young tableaux of a given shape with a prescribed descent set. This agrees with a formula by Klyachko for the uniform matroid. As consequences, we recover and simplify classical results such as Gessel and Viennot's enumeration of permutations with fixed descent sets, and formulas for the volume of lattice path matroids. Furthermore, we demonstrate the power of our findings by proving that certain Schubert coefficients are positive for all connected paving matroids.

Paper Structure

This paper contains 22 sections, 42 theorems, 152 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Partitions, diagrams and tableaux
  4. Matroids
  5. Lattice path, nested and snake matroids
  6. Valuations
  7. The Chow class of a matroid
  8. K-classes
  9. An identity for Schubert coefficients
  10. The Chow class of snake matroids
  11. A descent statistic interpretation
  12. Schubert coefficients of lattice path matroids
  13. Applications
  14. Robinson–Schensted–Knuth correspondence (RSK) and volume
  15. Support
  16. ...and 7 more sections

Key Result

Theorem 1

Let $\mathsf{S}$ be the snake matroid associated to the ribbon $\mathop{\mathrm{\rho}}\nolimits$. Then the Poincaré dual of its Chow class is equal to the ribbon Schur function $s_{\mathop{\mathrm{\rho}}\nolimits}$.

Figures (10)

  • Figure 1: The Young diagrams of the partitions $\lambda = [5,3,2^2] \vdash 12$ on the left, $\mu = [2,1^2] \vdash 4$ in the center and the ribbon $\mathop{\mathrm{\rho}}\nolimits(2,1,2,3)$ on the right.
  • Figure 2: The snake matroid $\mathsf{S}(2,1,2,3)$ from Example \ref{['ex:snake']}. We highlighted the upper path $\mathbf{U}= \mathbf{NENNENEEE}$ (red) and lower path $\mathbf{L}= \mathbf{EENNENEEN}$ (blue).
  • Figure 3: A lattice path matroid and the corresponding snake matroids.
  • Figure 4: An example on the left and nonexample on the right of a Littlewood--Richardson skew tableau with content $\eta = [4,3^2,1^2]$ and $b = k = 4$.
  • Figure 5: The snake matroid $\mathsf{S}(2,1,2,3)$ and the standard Young tableaux with its associated descent set.
  • ...and 5 more figures

Theorems & Definitions (99)

  • Conjecture : bergetfink_K
  • Theorem
  • Theorem
  • Theorem 2.1: Pieri's formula
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Example 2.7
  • ...and 89 more