[Paper Review] Sums of binomial determinants, non-intersecting lattice paths and positivity of Chern-Schwartz-MacPherson classes

Leonardo C. Mihalcea|ArXiv.org|Feb 19, 2007

Advanced Combinatorial Mathematics2 references3 citations

TL;DR

This paper proves a positivity conjecture for sums of binomial determinants arising from non-intersecting lattice paths in Schubert calculus, establishing that the sum of determinants over a family of lattice point configurations remains positive for p=2 and p=3. The result is interpreted geometrically as the positivity of Chern-Schwartz-MacPherson class coefficients in Grassmannian Schubert homology, with a constructive combinatorial formula provided for p=3 using balanced path triples.

ABSTRACT

We give a combinatorial interpretation of a certain positivity conjecture of Chern-Schwartz-MacPherson classes, as stated by P. Aluffi and the author in a previous paper. It translates into a positivity property for a sum of p by p determinants consisting of binomial coefficients, generalizing the classical Theorem of Lindstrom-Gessel-Viennot et al. which computes these determinants in terms of non-intersecting lattice paths. We prove this conjecture for p=2,3.

Motivation & Objective

• To prove a conjecture on the positivity of sums of p×p binomial determinants arising from lattice path configurations.
• To provide a combinatorial interpretation of Chern-Schwartz-MacPherson class coefficients in Schubert homology of Grassmannians.
• To generalize the Lindström–Gessel–Viennot theorem by relaxing initial point constraints while preserving determinant sum positivity.
• To establish explicit positive combinatorial formulas for c(λ,μ) when p=2 and p=3.
• To connect algebraic geometry (CSM classes) with enumerative combinatorics (lattice paths and determinants).

Proposed method

• Define a family of 2p lattice points A₁(s), ..., Aₚ(s) and B₁, ..., Bₚ based on partitions λ, μ and triangular sequences s ∈ S(λ) with constraints on non-negative integers aᵢⱼ.
• Construct a matrix M(s) whose entries mᵢⱼ(s) count lattice paths from Aᵢ(s) to Bⱼ, with steps only right and down.
• Define the sum c(λ,μ) = ∑ₛ∈S(λ) det M(s), conjectured to be positive for all p.
• Use the Lindström–Gessel–Viennot theorem to interpret individual determinants as counts of non-intersecting path tuples.
• For p=2 and p=3, introduce the notion of 'balanced' path triples that avoid invalid transformations (horizontal or diagonal swaps), ensuring no overcounting.
• Prove positivity by contradiction: show that any path configuration leading to a negative contribution would require an impossible configuration of points, violating geometric constraints.

Experimental results

Research questions

• RQ1 Is the sum of p×p binomial determinants, indexed over a family of lattice point configurations, always non-negative for any p?
• RQ2 Can the positivity of Chern-Schwartz-MacPherson class coefficients in Grassmannian Schubert homology be combinatorially explained via lattice paths?
• RQ3 For p=2 and p=3, can a positive combinatorial formula be constructed for c(λ,μ) that counts only 'balanced' non-intersecting path triples?
• RQ4 What conditions on initial lattice points Aᵢ(s) ensure that the determinant sum remains positive despite individual determinants possibly being negative?
• RQ5 How do triangular constraints on parameters aᵢⱼ relate to the geometric realizability of non-intersecting path systems?

Key findings

• The sum c(λ,μ) is proven to be positive for p=2 and p=3, confirming the positivity conjecture in these cases.
• For p=3, a positive combinatorial formula is given: c(λ,μ) equals the number of balanced path triples (π₁,π₂,π₃) from A₁(s), A₂(s), A₃(s) to B₁, B₂, B₃, where no horizontal or diagonal swap of adjacent paths yields a valid triangular configuration.
• The proof for p=3 relies on contradiction: assuming a path system with a negative contribution leads to a point configuration violating geometric constraints like x and y coordinate inequalities.
• The determinant det M(s) counts non-intersecting lattice paths from Aᵢ(s) to Bⱼ only when the points are in NE–SW order; in the relaxed setup, individual determinants may be negative, but their sum remains positive.
• The sum c(λ,μ) corresponds to the coefficient of the fundamental class of a Schubert variety in the CSM class expansion of a Schubert cell in Gr(p,n), for large n.
• The construction of Aᵢ(s) involves recursive dependencies on partial sums Rⱼ and Cⱼ, and the triangular order condition ensures that the path systems remain geometrically consistent.