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  "body": "\n## Definition\n\nLet $E$ 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 $G_E$.\n\nSuch 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 $\\mathrm{GL}_3$.\n\nThe weight $\\mu$ is called **pure** when both clauses hold:\n\n1. The entry $v_\\tau$ is the same for every $\\tau \\in S_\\infty$.\n2. The equality $m^+_\\tau=m^-_\\tau$ holds for every $\\tau \\in S_\\infty$.\n\nThe **pure weights** are the highest weights for $G_E$ that satisfy both clauses.\n",
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      "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."
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        "arxiv": "2609.26477v1",
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        "locator": "Definition 2.1",
        "mention": "ax://paper/2609.26477#0N9K",
        "note": "the paper that states the item (corpus arxiv AX9.2)"
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  "markdown": "https://math2.tamnd.com/node/807S91H4RY.md",
  "msc": [
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  "name": "Pure weights",
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        "what": "edge instance_of in",
        "note": "one example",
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