LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A symmetric Gaussian alternating sum vanishes in the Serre range

Statement

Let $m \geq 1$ , and let $d \in Z$ satisfy $1 - m \leq d \leq m - 1$ and $d \equiv m - 1 (m o d 2)$ . With the symmetric Gaussian binomial $C_{m, n} = {(\binom{m}{n})}_{ν}$ from Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ , $\sum_{n = 0}^{m} (- 1)^{n} ν^{- d n} C_{m, n} = 0$ for every real $ν > 1$ .

Facts & Assumptions

Given: The integer range and parity above, and the symmetric Gaussian coefficient convention.

[F1]

The Gauss product formula is a Laurent-polynomial identity and remains valid after substituting $ν^{- 1}$ for its parameter (The quantum Pascal recurrences, the Gauss product formula and Gaussian integrality).

Proof

technique · direct

1.1givenF1algebra

Substituting $ν^{- 1}$ for the parameter in [F1] gives $\prod_{j = 0}^{m - 1} (1 + ν^{- 2 j} z) = \sum_{n = 0}^{m} ν^{- n (m - 1)} C_{m, n} z^{n} .$ This identity is valid at the real number $ν > 1$ because both sides are Laurent polynomials. Set $z = - ν^{m - 1 - d}$ ; the right side becomes exactly $\sum_{n = 0}^{m} (- 1)^{n} ν^{- d n} C_{m, n}$ .

2.1givenstep 1.1algebra∎

The parity assumption makes $j = (m - 1 - d) / 2$ an integer, and the bounds on $d$ give $0 \leq j \leq m - 1$ . The factor indexed by this $j$ on the left side of step 1.1 is $1 - ν^{- 2 j} ν^{m - 1 - d} = 0$ . Therefore the right side, which is the sum in the Statement, is zero.

Depends on

Used by

The Ringel assignment is a bialgebra homomorphism onto the extended composition subalgebra Proposition

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Claus Michael Ringel, The Hall Algebra Approach to Quantum Groups, E.L.A.M. Lectures (1993) (standard reference, not scraped)
Claus Michael Ringel, Hall algebras and quantum groups, Inventiones Mathematicae 101 (1990), 583–591 (standard reference, not scraped)