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The good-node crystal component of the empty partition for $e = 3$

Example

At $e = 3$ , charge $0$ , the empty component through size $3$ consists of the partitions $\emptyset$ (rank $0$ ); $1$ (rank $1$ ); $2, 11$ (rank $2$ ); $3, 21$ (rank $3$ ). Within this range the arrows are $\emptyset \overset{f_{0}}{\to} 1$ ; $1 \overset{f_{1}}{\to} 2$ and $1 \overset{f_{2}}{\to} 11$ ; $2 \overset{f_{2}}{\to} 3$ and $11 \overset{f_{1}}{\to} 21$ ; raising arrows are the inverses. All other additions from ranks $\leq 2$ vanish, and additions from rank $3$ leaving the displayed range are omitted. The shapes are exactly the $3$ -regular shapes at these sizes. The shape $111$ is outside the component and is a highest crystal vertex of weight $Λ_{0} - δ$ ; this does not say that $E_{i} ∣ 111 ⟩ = 0$ as a field vector. Every shape has the exact weight $Λ_{0} - \sum_{i} m_{i} α_{i}$ .

Facts & Assumptions

Given: The $e = 3$ partition Fock space with its node operators and signature reduction, and the empty regular component.

[F1]

Good/cogood selection is the survivor reduction of the residue signature, equivalently $f$ adds the last surviving plus and $e$ removes the first surviving minus in content order (Good and cogood nodes by residue-signature reduction).

[F2]

The empty component is exactly the set of $3$ -regular partitions and the crystal is the good-node graph on all partitions (The empty component is the highest weight crystal $B (Λ_{0})$ , The crystal of the level-one Fock space is the good-node partition graph).

[F3]

The ordinary node actions on the shapes of size $\leq 3$ are the $e = 3$ table of Node actions for partitions of size at most three, and the weights are $Λ_{0} - \sum_{i} m_{i} α_{i}$ ; the Axiom of Choice is inherited only through the decomposition theorem used by [F2] (The Axiom of Choice).

Verification

1.1F1F3algebra

The arrows. For each displayed shape and color, list the corners in decreasing content and cancel $+ -$ : at $11$ and color $1$ the word is $+ +$ , and the first surviving plus is the top addition $21$ ; at $2$ and color $2$ the word is $+ +$ and the top addition is $3$ , with deletion on $3$ returning $2$ ; at $21$ and color $2$ the word is $+ -$ and cancels, so no arrow; at $21$ and color $1$ the word is $- +$ , leaving $e_{1} \to 11$ and $f_{1} \to 211$ , outside the displayed range. At $1$ the color- $1$ and color- $2$ additions give $2$ and $11$ . This reproduces exactly the displayed arrows, and the raising arrows are their inverses.

2.1step 1.1F2F3algebra

Completeness and the boundary $111$ . The shapes reached are exactly the $3$ -regular shapes of size $\leq 3$ ; every other partition of size $\leq 3$ , namely $111$ , is not $3$ -regular and lies outside the empty component by [F2]. At $111$ and color $1$ the decreasing-content signature has the top addable cell $(1, 2)$ of content $1$ and the bottom removable cell $(3, 1)$ of content $- 2$ , giving $+ -$ which cancels, and the other colors have no surviving minus, so $111$ is a highest crystal vertex; its node residues are $0, 2, 1$ , so its weight is $Λ_{0} - α_{0} - α_{1} - α_{2} = Λ_{0} - δ$ . The statement that $111$ is highest as a crystal vertex is not the statement $E_{i} ∣ 111 ⟩ = 0$ for the ordinary field operators.

3.1step 1.1step 2.1∎

Conclusion. The displayed graph is the empty component through size $3$ , the shapes are the $3$ -regular ones, and $111$ is the first highest crystal vertex outside it with weight $Λ_{0} - δ$ , with all weights computed from the node residues.

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Sources

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

The good-node crystal component of the empty partition for e=3e=3