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Good and cogood nodes by residue-signature reduction

Definition

Fix a residue $i$ and a partition $λ$ , and write the signature of $λ$ at $i$ as the word in the symbols $+$ (addable $i$ -node) and $-$ (removable $i$ -node) whose letters are the addable and removable $i$ -nodes of $λ$ read in the fixed increasing-content order of The node order and the signed exponents $N_{i}^{l}$ and $N_{i}^{r}$ . Reduce the word by cancelling adjacent pairs $- +$ as long as such a pair remains. The surviving minus letters are called normal removable $i$ -nodes, and the first surviving minus is the good node of $λ$ for $e_{i}$ ; the surviving plus letters are called conormal addable $i$ -nodes, and the last surviving plus is the cogood node for $f_{i}$ . If no sign of the required type survives, the corresponding crystal operator is $0$ on $b_{λ} : = ∣ λ ⟩ + q F_{0}$ , where $F_{0}$ is the raw $A_{0}$ -partition lattice. Throughout, $e_{i}, f_{i}$ act on these residue classes; the normalized operators on field vectors need not send $∣ λ ⟩$ to a single standard vector or to zero. Equivalently, reading the signature in decreasing content with the same addable/removable symbols, one cancels $+ -$ and takes the good node as the last surviving $-$ and the cogood node as the first surviving $+$ .

Facts & Assumptions

Given: A residue $i$ , a partition $λ$ , its addable and removable $i$ -nodes in content order, and the cube lemma for the fixed residue.

[F1]

The fixed-residue orbit is a Boolean cube of two-vertex strings, the actual normalized operators preserve $F_{0}$ and induce the survivor rule on $F_{0} / q F_{0}$ , and both content orders give the same selection (Fixed-residue node cubes realize the actual Fock strings).

[F2]

The node formulas and the content order are those of The node order and the signed exponents $N_{i}^{l}$ and $N_{i}^{r}$ .

Verification

technique · show the reduction is confluent, identify the survivors with the combinatorial deletion/addition rules, and record two calibration examples

1.1givenalgebra

The reduction is well defined. Each deletion $- + \to \emptyset$ decreases the word length by two. Two distinct adjacent cancellable pairs cannot overlap, since an overlapping letter would have to be both $+$ and $-$ ; disjoint deletions commute. Induction on word length therefore gives the same surviving labelled letters for every deletion order. Equivalently a left-to-right stack pairs each plus with the latest unmatched preceding minus. Reversing the order of letters without exchanging their meanings turns $- +$ cancellation into $+ -$ cancellation and exchanges first and last positions.

2.1step 1.1F1F2algebra

The operator interpretation. By the cube lemma [F1] the induced crystal operators on $b_{λ} \in F_{0} / q F_{0}$ select exactly the survivors: $e_{i}$ removes the good (first surviving minus) node and $f_{i}$ adds the cogood (last surviving plus) node, with value $0$ when the respective sign is absent; the equivalent decreasing-content description cancels $+ -$ and takes the last surviving $-$ and the first surviving $+$ , which is the usual Fayers good-deletion/cogood-addition convention.

3.1step 1.1step 2.1F1F2algebra

Calibrations and the minimiser warning. At $e = 3$ , $λ = (2)$ and $i = 2$ , the addable nodes $(2, 1)$ and $(1, 3)$ have contents $- 1, 2$ , giving the increasing-content signature $+ +$ . Its last plus selects $(1, 3)$ , so $f_{2} (2) = (3)$ . At $λ = (2, 1)$ and $i = 2$ , the removable node $(2, 1)$ has content $- 1$ and the addable node $(1, 3)$ has content $2$ , giving $- +$ and hence $f_{2} (2, 1) = 0$ . Nevertheless the ordinary node formula gives $F_{2} ∣ 2, 1 ⟩ = ∣ 3, 1 ⟩$ . The normalized value is ${\tilde{f}}_{2} ∣ 2, 1 ⟩ = q (1 + q^{2})^{- 1} ∣ 3, 1 ⟩$ , whose residue is zero by the two-slot calculation of [F1]. Thus even a unique nonzero Chevalley term need not be a crystal arrow.

4.1step 1.1step 2.1step 3.1∎

Conclusion. The good and cogood nodes are defined by the confluent survivor reduction of the residue signature, they give the residue actions of the actual normalized operators by the cube lemma, and they are not a selection by minimum ordinary Chevalley exponent.

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

Matthew Fayers, Partition models for the crystal of the basic $U_q(\mathfrak{sl}_n)$-module, arXiv:0906.4129 (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, Fock space representations (author-hosted complete 38-page treatment) (standard reference, not scraped)