Explicit equations for exterior square of the general linear group

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 $\varpi_{2}$.

Figures (6)

• Figure 1: $(A_3,\varpi_2)$
• Figure 2: $(A_{3},\varpi_2)\times(A_{3},\varpi_2)$
• Figure 3: Paths incident to the vertex (15)
• Figure 4: Elementary square for the set $\{1,2,4,6\}$
• Figure 5: The signs for an elementary square

Theorems & Definitions (20)

• Theorem \oldthetheorem
• Lemma 1
• Proposition 1
• Remark
• Remark
• Theorem \oldthetheorem
• proof
• Theorem \oldthetheorem
• Theorem \oldthetheorem
• proof