[Paper Review] Arrangements Of Minors In The Positive Grassmannian And a Triangulation of The Hypersimplex

Miriam Farber, Yelena Mandelshtam|arXiv (Cornell University)|Sep 8, 2015

Advanced Combinatorial Mathematics22 references3 citations

TL;DR

This paper investigates the combinatorial structure of equalities and inequalities among minors in the positive Grassmannian, linking arrangements of t-th largest minors to a triangulation of the hypersimplex via a novel cubical distance on the dual graph. The key contribution is a radius-bound on the location of t-th largest minors relative to the largest minors, showing they lie within a ball of radius $2^{t-1}$ in the cubical metric, with implications for the poset structure of minors and their geometric realization in the Lam-Postnikov triangulation.

ABSTRACT

The structure of zero and nonzero minors in the Grassmannian leads to rich combinatorics of matroids. In this paper, we investigate an even richer structure of possible equalities and inequalities between the minors in the positive Grassmannian. It was previously shown that arrangements of equal minors of largest value are in bijection with the simplices in a certain triangulation of the hypersimplex that was studied by Stanley, Sturmfels, Lam and Postnikov. Here we investigate the entire set of arrangements and its relations with this triangulation. First, we show that second largest minors correspond to the facets of the simplices. We then introduce the notion of cubical distance on the dual graph of the triangulation, and study its relations with the arrangement of t-th largest minors. Finally, we show that arrangements of largest minors induce a structure of partially ordered sets on the entire collection of minors. We use the Lam and Postnikov circuit triangulation of the hypersimplex to describe a 2-dimensional grid structure of this poset.

Motivation & Objective

• To understand the full combinatorial structure of equalities and inequalities among minors in the positive Grassmannian beyond the known cases of largest and smallest minors.
• To extend the known bijection between arrangements of largest minors and simplices in the Lam-Postnikov triangulation of the hypersimplex to arrangements of t-th largest minors.
• To define and analyze a cubical distance on the dual graph of the triangulation to characterize the spatial distribution of minors of decreasing magnitude.
• To establish a partially ordered set (poset) structure on all minors induced by arrangements of largest minors, revealing a 2D grid-like structure via the triangulation.

Proposed method

• Define a cubical distance $d_{ij}(x,y)$ on the dual graph of the Lam-Postnikov triangulation of the hypersimplex, based on separation by hyperplanes $H_{i,j,r}$ of the form $x_i + \cdots + x_j = r$.
• Use the notion of sorted sets and the $\epsilon_I$-coordinates of minors to formalize the distance and relate it to combinatorial properties of minors.
• Prove by induction that $t$-th largest minors lie within a ball of radius $2^{t-1}$ around the set of largest minors in the cubical metric, using Skandera’s inequalities and the sorting operation $\text{Sort}_1, \text{Sort}_2$.
• Leverage the Lam-Postnikov circuit triangulation of the hypersimplex to describe the poset of minors as a 2-dimensional grid structure.
• Apply results from [FP15] on sorted sets and weak separation to relate the geometry of the triangulation to the combinatorics of minor arrangements.
• Use the hyperplane separation condition to define when a minor lies within a certain distance from another, enabling the radius bound on $t$-th largest minors.

Experimental results

Research questions

• RQ1 How do arrangements of t-th largest minors in the positive Grassmannian relate to the triangulation of the hypersimplex?
• RQ2 What geometric or combinatorial structure governs the relative positions of minors of decreasing magnitude in the positive Grassmannian?
• RQ3 Can the cubical distance on the dual graph of the triangulation be used to bound the location of t-th largest minors relative to the largest minors?
• RQ4 How does the poset structure on all minors, induced by arrangements of largest minors, reflect the underlying 2D grid structure in the triangulation?
• RQ5 What is the precise radius in the cubical metric within which all t-th largest minors must lie relative to the set of largest minors?

Key findings

• The second largest minors correspond to the facets of the simplices in the Lam-Postnikov triangulation of the hypersimplex.
• All $t$-th largest minors lie within a ball of radius $2^{t-1}$ in the cubical distance metric around the set of largest minors.
• The poset of all minors, induced by arrangements of largest minors, admits a 2-dimensional grid structure realized via the Lam-Postnikov triangulation of the hypersimplex.
• The cubical distance $d_{ij}(I,J)$ is bounded by 1 if and only if the sets $I$ and $J$ are sorted, linking the metric to known combinatorial structures.
• The proof of the radius bound relies on inductive application of Skandera’s inequalities and the behavior of the sorting operations $\text{Sort}_1, \text{Sort}_2$ on minor coordinates.
• The result implies that the arrangement of $t$-th largest minors is constrained to a bounded region around the largest minors, with the bound growing exponentially in $t$.