---
tag: 807S91H4RY
kind: definition
name: Pure weights
msc: [11-XX]
defines: NW4VZRA5J1
sources:
  - arxiv: 2609.26477v1
    locator: Definition 2.1
    mention: ax://paper/2609.26477#0N9K
    licence: cc-by
    note: the paper that states the item (corpus arxiv AX9.2)
history:
  notes: The source is arXiv:2609.26477v1 (2026-09), by Tristan Ricoul. A paper also restates older results, so this is not the first statement until a person or a later pass confirms it.
inventory: [6DXYDTGN6R]
---

## Definition

Let $E$ be a [[quadratic field|7E23FBM9NQ]] that is real. Let $S_\infty$ be the set of its infinite [[places|BT5V3TSNY6]], that is, its embeddings into the real numbers. Let $G_E$ be the group $\mathrm{GL}_3$ over $E$, and let $\mu$ be a highest weight for $G_E$.

Such a weight is a family $\mu=(\mu_\tau)_{\tau \in S_\infty}$. Each $\mu_\tau=(m^+_\tau,m^-_\tau,v_\tau)$ is a dominant weight of the diagonal torus of $\mathrm{GL}_3$.

The weight $\mu$ is called **pure** when both clauses hold:

1. The entry $v_\tau$ is the same for every $\tau \in S_\infty$.
2. The equality $m^+_\tau=m^-_\tau$ holds for every $\tau \in S_\infty$.

The **pure weights** are the highest weights for $G_E$ that satisfy both clauses.