$G_2$ representations and semistandard tableaux

TL;DR

This work extends combinatorial realizations of irreducible polynomial representations to the exceptional group $G_2$ by using vector tableaux labeled by weights of the standard $7$-dimensional representation $V$. It introduces a finite, $G$-stable set of relations (alternating, exchange, orthogonal, exclusion, pairing, transposition) on $V_ olinebreak[4]_ olinebreak[4] to form $S_ olinebreak[4]_ olinebreak[4]$, and proves irreducibility and weight structure via a branching analysis to the subgroup $G' olinebreak[4] olinebreak[4] \, ext{A}_2$. The paper then defines $G_2$ tableaux as a basis for $S_ olinebreak[4]_ olinebreak[4]$ for two-row shapes, showing that every irreducible $G$-module arises this way and that the direct sum of all $S_ olinebreak[4]_ olinebreak[4]$ is the coordinate ring of the flag variety, with the relations generating its defining ideal. Overall, the results provide a concrete, weight-based combinatorial model for $G_2$-representations and an explicit description of the $G_2$ flag variety via tableaux.

Abstract

Continuing earlier work, we show how to realize irreducible finite-dimensional representations of the complex group of type $G_2$ via tableaux, along the way exhibiting explicit generators of the defining ideal of the flag variety

Theorems & Definitions (4)

• Lemma
• proof
• Theorem
• proof