LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The link-state Gram form is symmetric and invariant

Statement

Let $R$ , $δ$ , $n$ , $t$ and $W_{n, t} (R, δ)$ be as in Temperley–Lieb link-state standard modules by defect number. For link states $x, y$ , let $x^{*}$ denote reflection across a horizontal line. The diagram $y \circ x^{*} : t \to t$ is either $δ^{c} 1_{t}$ for some $c \geq 0$ or has fewer than $t$ propagating components. Define $ϕ_{n, t} (x, y) = δ^{c}$ in the first case and $0$ in the second, and extend $R$ -bilinearly. Then $ϕ_{n, t}$ is symmetric and satisfies

$ϕ_{n, t} (x \cdot a, y) = ϕ_{n, t} (x, y \cdot a^{*})$

for all $x, y \in W_{n, t} (R, δ)$ and $a \in {T L}_{n} (R, δ)$ . Here $a^{*}$ is reflection across a horizontal line. The matrix of $ϕ_{n, t}$ in the link-state basis is the Gram matrix obtained by reflecting and capping two link states, with one factor $δ$ for each closed loop.

Facts & Assumptions

Given: A commutative ring $R$ with unit, $δ \in R$ , $n \geq 0$ , an allowed $t$ , and link-state representatives in $M_{R} (n, t)$ .

[F1]

The link states are the basis cosets of diagrams with $t$ propagating components, and the right action is $x \cdot a = [x \circ a]$ (Temperley–Lieb link-state standard modules by defect number, The link-state quotient is a right Temperley–Lieb module).

[F2]

Diagram composition is associative and each closed component contributes one factor $δ$ (Temperley–Lieb planar diagram data and endomorphism modules, Stacking is associative and horizontal juxtaposition is strict monoidal).

Proof

technique · direct

1.1F1F2givenconstruct

For basis link states $x$ and $y$ , the composite $y \circ x^{*}$ is a diagram from $t$ to $t$ . If all $t$ boundary points propagate, planarity forces them to connect in their order, so after removing $c$ closed loops the result is $δ^{c} 1_{t}$ for a unique $c$ . Otherwise it has fewer than $t$ propagating components. Thus the stated value is determined by the matching and loop count, and it extends uniquely to an $R$ -bilinear form on the free link-state basis [F1, F2].

2.1F1F2step 1.1

Reflection reverses the order of stacking, fixes each scalar $δ$ , and sends $y \circ x^{*}$ to $x \circ y^{*}$ . Hence $y \circ x^{*} = δ^{c} 1_{t}$ exactly when $x \circ y^{*} = δ^{c} 1_{t}$ with the same $c$ ; the lower-propagating case is also preserved. Therefore $ϕ_{n, t} (x, y) = ϕ_{n, t} (y, x)$ on basis states and, by bilinearity, on all module elements.

3.1F1F2step 1.1step 2.1∎

For basis states $x, y$ and a basis diagram $a \in {T L}_{n}$ , reflection gives $(x \circ a)^{*} = a^{*} \circ x^{*}$ . Both form values are therefore the coefficient of $1_{t}$ in the same associative composite $y \circ (x \circ a)^{*} = y \circ a^{*} \circ x^{*} = (y \circ a^{*}) \circ x^{*}$ . They have the same exterior matching and number of removed loops, so their coefficients of $1_{t}$ agree. If an intermediate term has fewer than $t$ propagating components, it remains lower-propagating after further stacking and contributes zero on both sides. This proves the invariance identity for basis states and basis diagrams; bilinearity proves it for all $x, y, a$ .

Depends on

Used by

Dependency tree · two levels

13 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

David Ridout and Yvan Saint-Aubin, Standard Modules, Induction and the Structure of the Temperley-Lieb Algebra (standard reference, not scraped)
Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, complete prepublication manuscript (1998; expanded published edition: AMS University Lecture Series 15, 1999) (standard reference, not scraped)