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Hall Algebra Modules, Fock Spaces, and Geometric Actions: Examples and Counterexamples

1 · Prerequisites

2 · Summary

These examples distinguish ordinary Hall norms from Jordan Green norms, exhibit distinct Hall-module characters, and compute the framing-one specialization of the proved arbitrary-framing correspondence action.

The A1 Nakajima correspondence in the fundamental sl2 representation: The A1 Nakajima correspondence in the fundamental sl2 representation.
Two distinct one-dimensional modules over the generic Jordan Hall algebra: Two distinct one-dimensional modules over the generic Jordan Hall algebra.
Degree-one normalization for the Jordan Hall Green pairing and Fock action: Degree-one normalization for the Jordan Hall Green pairing and Fock action.

Companion: hall-algebra-modules-and-geometric-actions.

3 · Logical flowchart

4 · Definitions, theorems and proofs

ExampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

The A1 Nakajima correspondence in the fundamental sl2 representation

Example

For the one-vertex, no-arrow quiver with framing dimension one, the two nonempty Nakajima fibres are points at $v = 0$ and $v = 1$ . On their fundamental classes $e_{0}, e_{1}$ , the Hecke correspondence gives $E = (\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}), F = (\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}), H = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}) .$

Facts & Assumptions

Given: The A1 quiver varieties and their point Hecke correspondence, with basis ordered $(e_{0}, e_{1})$ by dimension $v = 0, 1$ .

[F1]

The A1 Nakajima Hecke correspondence gives the fundamental sl2 action: $E e_{0} = 0$ , $E e_{1} = e_{0}$ , $F e_{0} = e_{1}$ , $F e_{1} = 0$ , $H e_{0} = e_{0}$ and $H e_{1} = - e_{1}$ .

Verification

1.1givenF1algebra

In the ordered basis $(e_{0}, e_{1})$ , the six actions in [F1] give the displayed matrices column by column.

2.1F1step 1.1algebra∎

Direct multiplication gives $E F = diag (1, 0)$ , $F E = diag (0, 1)$ , $H E - E H = 2 E$ and $H F - F H = - 2 F$ . Therefore $[E, F] = H$ , $[H, E] = 2 E$ and $[H, F] = - 2 F$ , the defining ${s l}_{2}$ brackets.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Two distinct one-dimensional modules over the generic Jordan Hall algebra

Example

Let $K = Q (t)$ and let $H_{J, K}$ be the generic Jordan Hall algebra. By The generic Jordan-quiver Hall algebra is the symmetric-function ring, $H_{J, K} = K [u_{1}, u_{2}, \dots]$ as an algebra, where $u_{r}$ is the generator corresponding to the class $u_{(1^{r})}$ . The assignments $ϵ_{0} (u_{r}) = 0 (r \geq 1), ϵ_{1} (u_{1}) = 1, ϵ_{1} (u_{r}) = 0 (r \geq 2)$ give two different one-dimensional Hall modules on the same $K$ -vector space $K$ , by $h \cdot z = ϵ_{i} (h) z$ .

Facts & Assumptions

Given: The generic Jordan Hall algebra over $K$ and the Hall-module definition.

[F1]

The generic Jordan-quiver Hall algebra is the symmetric-function ring: the generators $u_{r} = u_{(1^{r})}$ freely generate the generic Hall algebra as a polynomial algebra, so any assignment of their values in a commutative $K$ -algebra extends uniquely to a unital $K$ -algebra homomorphism.

[F2]

Modules over a Hall algebra are unital algebra representations: a unital representation $ρ : H \to {End}_{K} (M)$ defines a Hall-module action by $h \cdot m = ρ (h) m$ .

Verification

1.1givenF1construct

By [F1], assigning values to the free polynomial generators $u_{r}$ extends uniquely to a unital $K$ -algebra homomorphism for both $ϵ_{0}$ and $ϵ_{1}$ .

2.1givenF1F2step 1.1construct

For $i \in {0, 1}$ define $ρ_{i} (h) (z) = ϵ_{i} (h) z$ . Multiplicativity and unit preservation from step 1.1 make $ρ_{i}$ a unital representation, so [F2] gives a Hall-module action on $K$ .

3.1F1step 2.1algebra∎

The actions differ because $ρ_{0} (u_{1}) = 0$ while $ρ_{1} (u_{1}) = {id}_{K}$ . Thus the same one-dimensional vector space carries two different Hall-module structures.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

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

Example

For $p_{1}^{r}$ ( $r \geq 0$ ), the ordinary symmetric-function Hall form has norm $⟨ p_{1}^{r}, p_{1}^{r} ⟩_{H} = r!$ , while the transported Jordan Green pairing has norm $(p_{1}^{r}, p_{1}^{r})_{G} = r! (t - 1)^{- r}$ . If $X_{1}$ is multiplication by $p_{1}$ , its adjoints therefore satisfy $X_{1}^{†, H} p_{1}^{r} = r p_{1}^{r - 1}, X_{1}^{†, G} p_{1}^{r} = \frac{r}{t - 1} p_{1}^{r - 1} (r \geq 1),$ and the normalized Fock operator $P_{1} = (t - 1) X_{1}^{†, G}$ has the usual derivative action $P_{1} p_{1}^{r} = r p_{1}^{r - 1}$ ; both adjoints kill $1$ . Here $X_{1}^{†}$ denotes the adjoint characterized by $⟨ X_{1} f, g ⟩ = ⟨ f, X_{1}^{†} g ⟩$ for the indicated bilinear form; the diagonal nonzero power-sum norms make these adjoints well-defined degree by degree.

Facts & Assumptions

Given: The standard Hall form, the transported Green pairing, and the normalized positive mode from the A-page Fock representation.

[F1]

The Hall inner product on symmetric functions and Power sums are orthogonal for the Hall form: $⟨ p_{λ}, p_{μ} ⟩_{H} = δ_{λ μ} z_{λ}$ for all partitions $λ, μ$ , so $⟨ p_{1}^{r}, p_{1}^{r} ⟩_{H} = r!$ .

[F2]

Jordan Hall classes are Hall–Littlewood functions with the transported Green pairing: the transported Green pairing is diagonal, $(p_{λ}, p_{μ})_{G} = δ_{λ μ} z_{λ} \prod_{i} (t^{λ_{i}} - 1)^{- 1}$ , so $(p_{1}^{r}, p_{1}^{r})_{G} = r! (t - 1)^{- r}$ .

[F3]

The generic Jordan Hall regular module carries the normalized Heisenberg Fock action: $P_{1} = (t - 1) X_{1}^{†, G} = \partial / \partial p_{1}$ .

Verification

1.1F1F2algebra

The Hall norm formula in [F1] gives $⟨ p_{1}^{r}, p_{1}^{r} ⟩_{H} = r!$ because $z_{(1^{r})} = r!$ ; the Green norm in [F2] gives $(p_{1}^{r}, p_{1}^{r})_{G} = r! (t - 1)^{- r}$ . At $r = 0$ both are $1$ by the empty-product convention.

1.2F1F2algebra

For either form and any power-sum monomial $p_{μ}$ of degree $r - 1$ , orthogonality makes the pairing of $p_{1} p_{μ}$ with $p_{1}^{r}$ vanish unless $μ = (1^{r - 1})$ . Nondegenerate diagonal norms then force each adjoint image of $p_{1}^{r}$ to be a multiple of $p_{1}^{r - 1}$ . Comparing against that monomial gives $X_{1}^{†, H} p_{1}^{r} = r p_{1}^{r - 1}$ and $X_{1}^{†, G} p_{1}^{r} = \frac{r}{t - 1} p_{1}^{r - 1}$ .

2.1F2F3step 1.2algebra∎

By [F3], $P_{1} = (t - 1) X_{1}^{†, G}$ , so $P_{1} p_{1}^{r} = r p_{1}^{r - 1}$ , the same as the standard Hall adjoint. At $r = 0$ the degree-lowering adjoints send $1$ to zero.

5 · Examples, counterexamples and false statements

None yet.

Hall Algebra Modules, Fock Spaces, and Geometric Actions: Examples and Counterexamples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

The A1 Nakajima correspondence in the fundamental sl2 representation

Example

Facts & Assumptions

Verification

Two distinct one-dimensional modules over the generic Jordan Hall algebra

Example

Facts & Assumptions

Verification

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

Example

Facts & Assumptions

Verification

5 · Examples, counterexamples and false statements

Sources