Grassmannians over a finite field

TL;DR

This paper characterizes Grassmannians over finite fields using row echelon forms and derives a polynomial formula for their cardinality, advancing combinatorial understanding of these geometric objects.

Contribution

It introduces a new characterization of Grassmannians over finite fields and provides an explicit polynomial formula for their size.

Findings

Characterization of Grassmannians via row echelon forms

Polynomial formula for the number of subspaces over finite fields

Enhanced combinatorial understanding of Grassmannians

Abstract

In this paper, we characterize the Grassmannian Gr$(d,n)$ in terms of the row echelon forms of rank $d$. Using this characterization, then in the case of finite field we give a polynomial-type formula for the cardinality of the Grassmannian.