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The addable-minus-removable node count is the level-one Cartan pairing

Statement

For every partition $λ$ , every residue $i \in I$ , and $N_{i} (λ) : = # {addable i -nodes of λ} - # {removable i -nodes of λ}$ one has

$N_{i} (λ) = δ_{i 0} - \sum_{j \in I} a_{i j} m_{j} (λ),$

where $m_{j} (λ)$ is the number of residue- $j$ cells of $λ$ and $A = (a_{i j})$ is the cyclic affine type $A$ Cartan matrix. Consequently $\sum_{i \in I} N_{i} (λ) = 1$ ; the toral operator acts by $K_{i} ∣ λ ⟩ = q^{N_{i} (λ)} ∣ λ ⟩ = q^{⟨ h_{i}, w t (∣ λ ⟩) ⟩}$ , the central product $K_{0} K_{1} \dots K_{e - 1}$ acts by $q$ , and $w t (∣ λ ⟩) = Λ_{0} - \sum_{j} m_{j} α_{j}$ has level $1$ . In particular $N_{i} (\emptyset) = δ_{i 0}$ , and adding a node of residue $k$ changes $N_{i}$ by $- a_{i k}$ , hence the weight by $- α_{k}$ .

Facts & Assumptions

Given: The partition Fock space with residues $c - r m o d e$ , weight $Λ_{0} - \sum_{j} m_{j} α_{j}$ , the node formulas, and the Cartan datum with row sums $\sum_{j} a_{i j} = 0$ .

[F1]

An addable node of $λ$ is $(r, λ_{r} + 1)$ with $r = 1$ or $λ_{r - 1} > λ_{r}$ , and a removable node is $(r, λ_{r})$ with $λ_{r} > λ_{r + 1}$ (Removable and addable nodes).

[F2]

The residues of the nodes around a corner are determined by the row and column indices as $c - r$ ; in particular the new bottom row $(k, 1)$ has residue $1 - k$ and the node $(r, λ_{r} + 1)$ has residue $λ_{r} + 1 - r$ (The level-one partition Fock space and node residues).

[F3]

$⟨ α_{j}, h_{i} ⟩ = a_{i j}$ , $⟨ c, Λ_{i} ⟩ = 1$ , and $\sum_{j} a_{i j} = 0$ (The cyclic affine type A Cartan datum and its residues).

Proof

technique · verify the identity on the empty partition and then induct on the number of cells by the local corner change

1.1F1F2givenalgebra

The empty partition. The empty partition has exactly one addable node, the cell $(1, 1)$ of residue $0$ , and no removable nodes; hence $N_{i} (\emptyset) = δ_{i 0}$ . The right-hand side is $δ_{i 0} - \sum_{j} a_{i j} \cdot 0 = δ_{i 0}$ .

2.1F1F2step 1.1algebra

The local step. Encode $λ$ by its occupied integer indices $S_{λ} = {λ_{r} - r + 1 : r \geq 1}$ , with trailing zero rows understood, and let $χ_{m}$ be the indicator of $m \in S_{λ}$ . A pair $(m, m + 1)$ of occupancy $(1, 0)$ is exactly an addable node of content $m$ ; occupancy $(0, 1)$ is exactly a removable node of content $m$ . Indeed increasing or decreasing the corresponding row length by one is admissible exactly when the neighbouring occupied index does not collide, by [F1]. Hence $N_{i} (λ) = \sum_{m \equiv i} (χ_{m} - χ_{m + 1})$ , a finite sum because all sufficiently negative indices are occupied and all sufficiently positive indices are empty. Adding a node of residue $k$ changes $χ_{m}$ from $1$ to $0$ and $χ_{m + 1}$ from $0$ to $1$ at an index $m \equiv k$ . Thus the count changes by $- 2 δ_{i k} + δ_{i, k - 1} + δ_{i, k + 1} = - a_{i k}$ ; at $e = 2$ the last two contributions coincide and give the double edge. Also $m_{j} (μ) = m_{j} (λ) + δ_{j k}$ , so the claimed right side changes by the same $- a_{i k}$ .

3.1step 2.1F3givenalgebra

The identity and its consequences. Induction on the number of cells from step 1.1 with the local step 2.1 gives $N_{i} (λ) = δ_{i 0} - \sum_{j} a_{i j} m_{j} (λ)$ for every partition. Summing over $i$ and using $\sum_{i} a_{i j} = 0$ and $\sum_{i} δ_{i 0} = 1$ gives $\sum_{i} N_{i} (λ) = 1$ . Therefore the toral action $K_{i} ∣ λ ⟩ = q^{N_{i} (λ)} ∣ λ ⟩$ equals $q^{δ_{i 0} - \sum_{j} a_{i j} m_{j}} = q^{⟨ h_{i}, Λ_{0} - \sum_{j} m_{j} α_{j} ⟩}$ , and the product over $i$ acts by $q^{\sum_{i} N_{i}} = q$ , so $w t (∣ λ ⟩) = Λ_{0} - \sum_{j} m_{j} α_{j}$ has $⟨ c, w t ⟩ = \sum_{i} N_{i} = 1$ and level one; adding a node of residue $k$ replaces $m_{k}$ by $m_{k} + 1$ , changing the weight by $- α_{k}$ and $N_{i}$ by $- a_{i k}$ .

4.1step 1.1step 2.1step 3.1∎

Conclusion. The addable-minus-removable count is the level-one Cartan pairing, the sum of the counts is $1$ , the central product acts by $q$ , and the declared toral action of the partition Fock space is the one required for a level-one module.

Depends on

Used by

Dependency tree · two levels

11 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)
Kailash C. Misra and Tetsuji Miwa, Crystal base for the basic representation of $U_q(\mathfrak{sl}(n))$, Comm. Math. Phys. 134 (1990), 79-88 (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)