A remark on commutative subalgebras of Grassmann algebra

TL;DR

This paper investigates the structure of maximal commutative subalgebras within Grassmann algebras, demonstrating the existence of such subalgebras with dimensions below a specific bound for certain dimensions.

Contribution

It establishes the existence of maximal commutative subalgebras of smaller dimension than previously known bounds in Grassmann algebras when n=4k+1 and k≥4.

Findings

Existence of maximal commutative subalgebras with dimension less than 3·2^{n-2}

Results apply specifically for n=4k+1 with k≥4

Provides new bounds on the dimensions of these subalgebras

Abstract

Let $n=4k+1$ and $k\geq 4$. We show that there exists maximal commutative subalgebras (with respect to inclusion) of dimension less that $3\cdot 2^{n-2}$.