False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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The Weyl quotient requires cancellation or extension

Statement

In the Weyl character formula the numerator divided by the denominator is an ordinary pointwise quotient on the whole torus before any cancellation or continuous extension is justified.

Facts & Assumptions

Given: The ${s l}_{2}$ -weight data: a root $α$ , the Weyl vector $ρ = α / 2$ of the positive system ${α}$ (The Weyl vector, Positive systems and simple roots, The special linear Lie algebra sl_2), the variable $z$ ranging over $C^{\times}$ , and, for an integer $n \geq 0$ , the Laurent polynomials $χ_{n} (z) = z^{n} + z^{n - 2} + \dots + z^{- n}, N (z) = z^{n + 1} - z^{- (n + 1)}, D (z) = z - z^{- 1} .$ The functions $N$ and $D$ are the numerator and denominator of the Weyl character formula in this rank-one instance, with $A_{λ + ρ} = N$ and $A_{ρ} = D$ after fixing the trivial Weyl alternant normalisation.

[L1]

Multiplication of the finite geometric sum gives the telescoping identity $χ_{n} (z) D (z) = N (z)$ , hence $χ_{n} (z) = N (z) / D (z)$ for every $z \neq 0$ with $D (z) \neq 0$ , that is, for $z \neq \pm 1$ , by cancellation of the common factor $z - z^{- 1}$ in the Laurent polynomial ring.

[L2]

At $z = 1$ one has $N (1) = 1 - 1 = 0$ and $D (1) = 1 - 1 = 0$ , while $χ_{n} (1) = n + 1$ .

Refutation

technique · direct

1.1

The identity of [L1] is an identity of Laurent polynomials, and it required multiplying the finite sum by $z - z^{- 1}$ and cancelling the common factor; on the set where $D (z) \neq 0$ the quotient equals $χ_{n}$ .

1.2

At the torus point $z = 1$ the displayed quotient $N (1) / D (1)$ is $0 / 0$ by [L2], so the formula gives no value there, whereas the character has the well-defined value $χ_{n} (1) = n + 1$ ; the value at $z = 1$ can be recovered only after the algebraic cancellation or a continuous extension of the quotient.

2.1

Hence the Weyl quotient is not an ordinary pointwise quotient on the whole torus: it is undefined at the identity point until cancellation or extension is justified, so the claim of the Statement section is false.

step 1.1step 1.2∎

Depends on

Used by

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Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)
Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)