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The node order and the signed exponents $N_{i}^{l}$ and $N_{i}^{r}$

Definition

For a fixed residue $i$ list the addable and removable $i$ -nodes of a partition by increasing content $c - r$ ; larger content is read as further right or higher. For an addable $i$ -node $γ$ of $λ$ (so that $μ = λ + γ$ ) define

$N_{i}^{r} (λ, γ) : = # {addable i -nodes of λ with content > content (γ)} - # {removable i -nodes of λ with content > content (γ)},$

and for a removable $i$ -node $γ$ of $μ$ (so that $λ = μ - γ$ ) define $N_{i}^{l} (λ, μ)$ by the same formula inside the smaller diagram $λ = μ - γ$ with $>$ replaced by $<$ , always excluding $γ$ itself. The node formulas are then

$F_{i} ∣ λ ⟩ = \sum_{γ} q^{N_{i}^{r} (λ, γ)} ∣ λ + γ ⟩, E_{i} ∣ μ ⟩ = \sum_{γ} q^{- N_{i}^{l} (μ - γ, μ)} ∣ μ - γ ⟩,$

over the addable respectively removable $i$ -nodes. The node order is the content order just fixed, and it is used to define the finite signature selection: a Chevalley coefficient minimiser is not in general a Kashiwara arrow, and even a nonzero single Chevalley term can have zero crystal arrow.

Facts & Assumptions

Given: The partition Fock space with residues $c - r m o d e$ , the stable node operators and the signed exponents $N_{i}^{r}, N_{i}^{l}$ defined above.

[F1]

$F_{i}$ adds and $E_{i}$ removes a node of residue $i$ ; the sets of addable and removable $i$ -nodes are finite for every partition and every residue (The level-one partition Fock space and node residues, Removable and addable nodes).

[F2]

$N_{i} (λ) = # {addable}_{i} (λ) - # {removable}_{i} (λ) = δ_{i 0} - \sum_{j} a_{i j} m_{j} (λ)$ (The addable-minus-removable node count is the level-one Cartan pairing).

Verification

technique · split the boundary nodes at the chosen node and verify the two identities by the content order, then calibrate at $e=2$

1.1F1F2givenalgebra

The splitting identity. For an addable $i$ -node $γ$ of $λ$ , the addable $i$ -nodes of $λ$ are ${γ}$ , the nodes strictly above $γ$ in content and those strictly below, and likewise for the removable $i$ -nodes; hence, with the two signed counts of the Definition, $N_{i}^{r} (λ, γ) + N_{i}^{l} (λ, γ) = (# {addable}_{i} (λ) - 1) - # {removable}_{i} (λ) = N_{i} (λ) - 1$ , because $γ$ contributes to the addable count but to neither signed count and no removable node equals $γ$ . The analogous identity holds for a removable node with $μ$ and $λ$ exchanged.

1.2F1givenalgebra

The node formulas are the ones of the stable prefix action. By the content order, the finite-prefix coefficient of an added node is the product of the preceding $K$ -weights in decreasing-index order, which is exactly $q^{N_{i}^{r}}$ (each addable $i$ -node to the right contributes $+ 1$ , each removable $i$ -node to the right contributes $- 1$ ), and for a removed node the renormalised suffix coefficient is $q^{- N_{i}^{l}}$ with the smaller diagram $μ - γ$ ; these are the formulas (38)-(39) of the stable action, restated in the content order.

1.3F1givenalgebra

Calibration at $e = 2$ . Take $λ = (3, 1)$ . Its cells have contents $0, 1, 2$ in row $1$ and $- 1$ in row $2$ , so mod $2$ the residues are $0, 1, 0, 1$ . The addable $0$ -nodes are $(2, 2)$ , content $0$ , and $(3, 1)$ , content $- 2$ ; the removable $0$ -node is $(1, 3)$ , content $2$ . For $γ = (2, 2)$ : no addable $0$ -node lies above content $0$ and the removable node $(1, 3)$ does, so $N_{0}^{r} = - 1$ and the coefficient of $∣ 3, 2 ⟩$ is $q^{- 1}$ ; for $γ = (3, 1)$ : the addable $(2, 2)$ and the removable $(1, 3)$ both lie above content $- 2$ , so $N_{0}^{r} = 0$ and the coefficient of $∣ 3, 1, 1 ⟩$ is $1$ ; hence $F_{0} ∣ 3, 1 ⟩ = q^{- 1} ∣ 3, 2 ⟩ + ∣ 3, 1, 1 ⟩$ , which is Leclerc's Exercise 15 value. For $E_{0} ∣ 3, 1 ⟩$ : the only removable $0$ -node is $γ = (1, 3)$ , whose smaller diagram is $λ = (2, 1)$ ; inside $(2, 1)$ the addable $0$ -nodes below content $2$ are $(2, 2)$ (content $0$ ) and $(3, 1)$ (content $- 2$ ), while no removable $0$ -node lies below, so $N_{0}^{l} = 2$ and $E_{0} ∣ 3, 1 ⟩ = q^{- 2} ∣ 2, 1 ⟩$ , again matching the calibration.

2.1step 1.1step 1.2step 1.3∎

Conclusion. The node order is the increasing-content order on the boundary nodes, the signed exponents satisfy $N_{i}^{r} + N_{i}^{l} = N_{i} (λ) - 1$ at every node, the node formulas are the ones produced by the stable prefix action, and the $e = 2$ calibration reproduces Leclerc's Exercise 15; no tilde operator is obtained by setting $q = 0$ in these possibly singular coefficients.

Depends on

Used by

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

Bernard Leclerc, Fock space representations (author-hosted complete 38-page treatment) (standard reference, not scraped)
Bernard Leclerc and Jean-Yves Thibon, Canonical bases of q-deformed Fock spaces, arXiv:q-alg/9602025 (standard reference, not scraped)
Kashiwara-Miwa-Petersen-Yung, Perfect crystals and q-deformed Fock spaces (standard reference, not scraped)

The node order and the signed exponents NilNil​ and NirNir​