A finite totally nonnegative Grassmannian

TL;DR

This paper extends the concept of totally nonnegative Grassmannians to finite fields, providing explicit point counts and new OEIS sequence interpretations, and compares this with the classical real case.

Contribution

It introduces a finite field version of totally nonnegative Grassmannians and explores their properties and point counts, offering new combinatorial insights.

Findings

Explicit point counts in special cases

New interpretations of OEIS sequences

Comparison with real case theory

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.