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A finite totally nonnegative Grassmannian

John Machacek

TL;DR

This work extends total positivity to Grassmannians over finite fields by defining nonnegativity via squares in $\mathbb{F}_q$ and studying $\mathrm{Gr}_{k,n}^{\ge 0}(\mathbb{F}_q)$ through Plücker coordinates. It establishes duality via orthogonal complements, builds subfield embeddings to construct nonnegative points, and provides explicit point counts for lines and certain $2$-planes, including generating functions and connections to Chebyshev polynomials and OEIS sequences for small fields ($q=3,5$). The paper also highlights where intuition from the real totally nonnegative Grassmannian fails in the finite-field setting, exploring sign variation, fixed points under cyclic shifts, and matroid notions, culminating in the concept of $\mathbb{F}_q$-positroids and their duals. Overall, it reveals rich combinatorial and algebraic structure underpinning positivity over finite fields with potential links to enumerative geometry and graph/poset combinatorics in finite settings.

Abstract

We introduce totally nonnegative Grassmannians over finite fields where an element of a finite field is nonnegative if it is a square of an element of the finite field. Explicit point counts are given in some special cases where we find new interpretations of sequences in the On-Line Encyclopedia of Integer Sequences (OEIS). We compare and contrast the theory of totally nonnegative Grassmannians over a finite field with the traditional case of the field of real numbers.

A finite totally nonnegative Grassmannian

TL;DR

This work extends total positivity to Grassmannians over finite fields by defining nonnegativity via squares in and studying through Plücker coordinates. It establishes duality via orthogonal complements, builds subfield embeddings to construct nonnegative points, and provides explicit point counts for lines and certain -planes, including generating functions and connections to Chebyshev polynomials and OEIS sequences for small fields (). The paper also highlights where intuition from the real totally nonnegative Grassmannian fails in the finite-field setting, exploring sign variation, fixed points under cyclic shifts, and matroid notions, culminating in the concept of -positroids and their duals. Overall, it reveals rich combinatorial and algebraic structure underpinning positivity over finite fields with potential links to enumerative geometry and graph/poset combinatorics in finite settings.

Abstract

We introduce totally nonnegative Grassmannians over finite fields where an element of a finite field is nonnegative if it is a square of an element of the finite field. Explicit point counts are given in some special cases where we find new interpretations of sequences in the On-Line Encyclopedia of Integer Sequences (OEIS). We compare and contrast the theory of totally nonnegative Grassmannians over a finite field with the traditional case of the field of real numbers.
Paper Structure (11 sections, 13 theorems, 40 equations, 1 figure, 2 tables)

This paper contains 11 sections, 13 theorems, 40 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Orthogonal complements
  3. Subfields
  4. Enumeration of positive and nonnegative lines
  5. Enumeration of nonnegative $2$-planes
  6. The field $\mathbb{F}_3$
  7. The field $\mathbb{F}_5$
  8. Examples and Counterexamples
  9. Sign variation
  10. Cyclic shifts and fixed points
  11. Matroids

Key Result

Lemma 2.1

Given any $0 \leq k \leq n$ and any field $\mathbb{F}$, as projective coordinates, $\Delta_{I}(\mathop{\mathrm{alt}}\nolimits(V^{\perp})) = \Delta_{[n] \setminus I}(V)$ for all $I \in \binom{[n]}{n-k}$ for any $V \in \mathop{\mathrm{Gr}}\nolimits_{k,n}(\mathbb{F})$. That is, if $V = [X]$ and $\matho

Figures (1)

  • Figure 1: The $f_5 = 5$ tilings in the figure demonstrate that $T_4(x) = 1 -8x^2 + 8x^4$.

Theorems & Definitions (31)

  • Remark 1.1
  • Lemma 2.1: Hoch see also karp
  • Theorem 2.2
  • proof
  • Conjecture 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 21 more