Explicit equations for exterior square of the general linear group

TL;DR

This paper provides explicit algebraic equations defining the exterior square of the general linear group as an affine group scheme, using exterior numbers and weight diagrams for Lie groups of type A.

Contribution

It introduces a novel set of explicit equations for the exterior square of GL(n), connecting algebraic and Lie-theoretic concepts.

Findings

Explicit equations for the exterior square of GL(n)

Use of exterior numbers and weight diagrams

Connection to Lie groups of type A

Abstract

We present several explicit systems of equations defining exterior square of the general linear group as an affine group scheme. Algebraic ingredients of the equations, exterior numbers, are translated into the language of weight diagrams corresponding to Lie groups of type $A_{n-1}$ in representation with the highest weight $\varpi_{2}$.