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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.

Depends on

Used by

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Sources

Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2, §2.4, equation (2.9) (standard reference, not scraped)
I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Chapter I §4, equations (4.5)–(4.7) and Remark 8 (standard reference, not scraped)