TheoremStatement: 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 generic Jordan Hall regular module carries the normalized Heisenberg Fock action

Statement

Use the category of finite-dimensional nilpotent Jordan-quiver representations from Nilpotent representations of the Jordan quiver. Let $t$ be an indeterminate, $K = Q (t)$ , and $H_{J, K} : = H_{g e n} \otimes_{Z [t]} K$ be the generic Jordan-quiver Hall algebra extended to $K$ . Extend the exact isomorphism $Φ_{t}$ of The generic Jordan-quiver Hall algebra is the symmetric-function ring to $K$ ; its regular module is identified with $Λ_{K} : = K \otimes_{Z} Λ = K [p_{1}, p_{2}, \dots]$ . For a partition $λ$ , let $m_{r} (λ)$ be the multiplicity of $r$ , set $p_{λ} = \prod_{r \geq 1} p_{r}^{m_{r} (λ)}$ and $z_{λ} = \prod_{r \geq 1} r^{m_{r} (λ)} m_{r} (λ)!$ , and put $p_{\emptyset} = 1$ . Write $λ ∖ (n)$ for the partition obtained by removing one part $n$ when $m_{n} (λ) > 0$ ; define $m_{n} (λ) p_{λ ∖ (n)} = 0$ when $m_{n} (λ) = 0$ . Transport the Jordan Green pairing of Jordan Hall classes are Hall–Littlewood functions with the transported Green pairing to $Λ_{K}$ . On power sums it is $(p_{λ}, p_{μ})_{G} = δ_{λ μ} z_{λ} \prod_{i} (t^{λ_{i}} - 1)^{- 1},$ where $δ_{λ μ} = 1$ if $λ = μ$ and $0$ otherwise, and the product runs over the parts of $λ$ with multiplicity. For $n \geq 1$ , let $X_{n}$ be multiplication by $p_{n}$ and define $X_{n}^{†, G} p_{λ} = \frac{n m_{n} (λ)}{t^{n} - 1} p_{λ ∖ (n)}, P_{n} : = \frac{t^{n} - 1}{n} X_{n}^{†, G} .$ Then $X_{n}^{†, G}$ is the adjoint of $X_{n}$ for $(,)_{G}$ , $P_{n} = \partial / \partial p_{n}$ , and for each finite $N \geq 1$ the assignments $X_{i} \mapsto (p_{i} -multiplication)$ , $P_{i} \mapsto \partial / \partial p_{i}$ , and $Z \mapsto id$ give a representation of the finite Heisenberg Lie algebra $h_{N} (K)$ from Finite-dimensional Heisenberg Lie algebra. The constant $1$ is the vacuum: $P_{n} 1 = 0$ , and the vectors $\prod_{i = 1}^{N} X_{i}^{m_{i}} 1$ , with nonnegative integer exponents, span the subalgebra $K [p_{1}, \dots, p_{N}]$ ; as $N$ varies, they span $Λ_{K}$ .

Facts & Assumptions

Given: The generic Jordan Hall algebra over $K = Q (t)$ , the power-sum basis of $Λ_{K}$ , its transported Green pairing, and the finite Heisenberg Lie algebra.

[F1]

The generic Jordan-quiver Hall algebra is the symmetric-function ring: the generic Hall algebra is a free polynomial algebra and $Φ_{t} (u_{(1^{r})}) = t^{- r (r - 1) / 2} e_{r}$ extends to a graded algebra isomorphism with the symmetric-function ring.

[F2]

Jordan Hall classes are Hall–Littlewood functions with the transported Green pairing: on power sums the transported Green pairing is diagonal with $(p_{λ}, p_{μ})_{G} = δ_{λ μ} z_{λ} \prod_{i} (t^{λ_{i}} - 1)^{- 1}$ , and it is nondegenerate after extension to $K$ .

[F3]

Power sums form a rational but not integral stable basis: $Λ_{K} = K [p_{1}, p_{2}, \dots]$ and the monomials $p_{λ}$ form a basis.

[F4]

Finite-dimensional Heisenberg Lie algebra and The Heisenberg bracket satisfies the Lie algebra axioms: $h_{N} (K)$ has generators $X_{i}, P_{i}, Z$ with $[P_{i}, X_{j}] = δ_{i j} Z$ and all same-type and central brackets zero, and these relations define a Lie algebra.

[F5]

A unital Hall algebra is a left module over itself: a unital associative Hall algebra acts on its underlying $R$ -module by left multiplication.

Proof

1.1F1F2F3F5algebra

By [F1] and [F5], scalar extension of the generic Hall algebra acts on its regular module by multiplication; extending the isomorphism [F1] from $Z [t, t^{- 1}]$ to $K$ identifies this action with multiplication by $p_{n}$ when the Hall element $Φ_{t}^{- 1} (p_{n})$ acts. Thus $X_{n}$ is the transported regular action. By [F2]–[F3], the power sums form an orthogonal basis with nonzero norms, since $z_{λ} \neq 0$ and $t^{r} - 1 \neq 0$ in $K$ for every part $r \geq 1$ . For partitions $α, β$ , both adjoint pairings vanish unless $β$ is obtained by adjoining one part $n$ to $α$ ; in that case, if $m = m_{n} (α)$ , then $z_{β} = n (m + 1) z_{α}$ and $(X_{n} p_{α}, p_{β})_{G} = (p_{α}, X_{n}^{†, G} p_{β})_{G} = n (m + 1) (t^{n} - 1)^{- 1} (p_{α}, p_{α})_{G}$ . Thus the displayed operator is the adjoint on every pair of basis elements, hence on all of $Λ_{K}$ .

1.2F3algebra

The formula in the Statement gives $P_{n} p_{λ} = m_{n} (λ) p_{λ ∖ (n)}$ , the formal derivative $\partial p_{λ} / \partial p_{n}$ by [F3], with value zero when $m_{n} (λ) = 0$ . Therefore the $P_{n}$ commute, the $X_{n}$ commute, and $P_{n} 1 = 0$ .

2.1F3F4step 1.2algebra

For any $m, n \geq 1$ and basis monomial $p_{λ}$ , the polynomial product rule gives $(P_{n} X_{m} - X_{m} P_{n}) p_{λ} = δ_{n m} p_{λ}$ . Since the $p_{λ}$ form a basis by [F3], $[P_{n}, X_{m}] = δ_{n m} id$ . Together with the commuting relations of step 1.2 these are precisely the generator brackets in [F4].

3.1F3F4step 1.2step 2.1∎

For $1 \leq i \leq N$ , send $X_{i}, P_{i}, Z$ in $h_{N} (K)$ to multiplication by $p_{i}$ , $\partial / \partial p_{i}$ , and $id$ , respectively. Step 2.1 and the commuting relations in step 1.2 preserve every defining bracket, so the assignments give a Lie algebra representation. For any nonnegative integers $m_{1}, \dots, m_{N}$ , the monomial $p_{1}^{m_{1}} \dots p_{N}^{m_{N}}$ is $\prod_{i = 1}^{N} X_{i}^{m_{i}} 1$ ; hence these vectors span $K [p_{1}, \dots, p_{N}]$ and, as $N$ varies, generate $Λ_{K}$ .

Remarks

The adjoint used here is the Jordan Green pairing from [F2]. It is not the ordinary Hall inner product defined in The Hall inner product on symmetric functions, whose power-sum norms are computed in Power sums are orthogonal for the Hall form. The scalar normalization of the positive modes accounts for the difference.

Depends on

Used by

Degree-one normalization for the Jordan Hall Green pairing and Fock action Example

Dependency tree · two levels

37 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

Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2, §§2.3–2.4 (standard reference, not scraped)
I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Chapter I §4, equation (4.5) and Remark 8 (standard reference, not scraped)