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The double-edge Serre cancellation in affine $A_{1}$ on partitions

Example

At $e = 2$ one has $[3] = q^{2} + 1 + q^{- 2}$ and the cubic relation $S_{X} = X_{0}^{3} X_{1} - [3] X_{0}^{2} X_{1} X_{0} + [3] X_{0} X_{1} X_{0}^{2} - X_{1} X_{0}^{3} = 0$ for $X = E, F$ , products acting right to left. The four $E$ compositions all vanish on $\emptyset, 1, 2, 11$ by size, so only size $\geq 4$ inputs test the identity. The four $F$ compositions all vanish on $\emptyset, 1, 3, 111$ . On $2$ the ordered four outputs are $([3] A, A, 0, 0)$ with $A = (q^{- 2} + 1) ∣ 321 ⟩$ ; on $11$ they are $([3] B, B, 0, 0)$ with $B = (q^{- 1} + q) ∣ 321 ⟩$ , giving cancellation. On $21$ they are $(0, C, D, [3] (D - C))$ with $C = [2] (∣ 52 ⟩ + q ∣ 511 ⟩ + q^{2} ∣ 421 ⟩ + q ∣ 3211 ⟩ + q^{2} ∣ 31111 ⟩ + q^{3} ∣ 22111 ⟩)$ and $D - C = [2] (∣ 421 ⟩ + q ∣ 331 ⟩ + q^{3} ∣ 3211 ⟩ + q^{2} ∣ 322 ⟩)$ . A nontrivial $E$ check on $321$ has four outputs $(0, 0, Z, [3] Z)$ with $Z = (1 + q^{2}) ∣ 11 ⟩ + (q^{- 1} + q) ∣ 2 ⟩$ . All displayed Serre sums vanish, whereas $F_{0} F_{1} ∣ 1 ⟩ = ∣ 3 ⟩ + q ∣ 111 ⟩$ and $F_{1} F_{0} ∣ 1 ⟩ = 0$ show the colors do not commute.

Facts & Assumptions

Given: The $e = 2$ partition Fock space with its node formulas, the ordinary generators $E_{0}, E_{1}, F_{0}, F_{1}$ , and the value $[3] = q^{2} + 1 + q^{- 2}$ .

[F1]

The $E_{i}, F_{i}$ actions are the node formulas of Node actions for partitions of size at most three and The node order and the signed exponents $N_{i}^{l}$ and $N_{i}^{r}$ , including on shapes of size $4$ and $5$ .

[F2]

The general cubic Serre sum holds on every partition vector by The Fock space node formulas satisfy the quantum affine relations, independently proved by Integrable rank-one actions force the quantum Serre sums; the computations below are finite corroborations, not the universal proof.

Verification

1.1F1givenalgebra

The vanishing cases. Every $E$ monomial removes four cells, so all four outputs on $\emptyset, 1, 2, 11$ vanish. For $F$ on $\emptyset$ , $F_{1} \emptyset = 0$ and $F_{0} \emptyset = 1$ , $F_{0} 1 = 0$ ; these kill all four compositions. On $1, 3, 111$ , the three compositions with a rightmost $F_{0}$ vanish because $F_{0}$ kills each input. For the remaining $F_{0}^{3} F_{1}$ , the node formulas give $F_{0}^{2} F_{1} ∣ 1 ⟩ = 0$ , $F_{0}^{2} F_{1} ∣ 3 ⟩ = (q^{- 2} + 1) ∣ 321 ⟩$ , and $F_{0}^{2} F_{1} ∣ 111 ⟩ = (q^{- 1} + q) ∣ 321 ⟩$ ; in both latter cases $F_{0} ∣ 321 ⟩ = 0$ since it has no addable $0$ -node. Thus each of the four $F$ outputs on the stated boundary inputs is zero.

2.1step 1.1F1algebra

The size- $2$ cancellations. Starting from $∣ 2 ⟩$ and applying the four ordered compositions via [F1] gives the first output as $[3] (q^{- 2} + 1) ∣ 3, 2, 1 ⟩$ , the second as $(q^{- 2} + 1) ∣ 3, 2, 1 ⟩$ , and the last two $0$ : indeed the coefficient of $∣ 321 ⟩$ in the leading output is $q^{- 4} + 2 q^{- 2} + 2 + q^{2} = [3] (q^{- 2} + 1)$ and the other outputs contribute the same multiple of the middle term; cancelling gives $0$ . The same computation from $∣ 1, 1 ⟩$ gives outputs $([3] B, B, 0, 0)$ with $B = (q^{- 1} + q) ∣ 321 ⟩$ because the coefficient is $q^{- 3} + 2 q^{- 1} + 2 q + q^{3} = [3] (q^{- 1} + q)$ .

2.2step 1.1F1algebra

Noncommutation. Direct evaluation gives $F_{0} F_{1} ∣ 1 ⟩ = ∣ 3 ⟩ + q ∣ 1, 1, 1 ⟩$ and $F_{1} F_{0} ∣ 1 ⟩ = 0$ , since $F_{0} ∣ 1 ⟩ = 0$ ; hence the ordinary generators do not commute.

3.1step 2.1F1algebra

The size- $3$ cancellations. From $∣ 2, 1 ⟩$ the second composition yields the common term $C$ , the third yields $D$ , and the fourth yields $[3] (D - C)$ , so the sum contributes $[3] (- C + D - (D - C)) = 0$ coefficientwise; from $∣ 3, 2, 1 ⟩$ , applying the $E$ -compositions gives the last output $[3] Z$ with the stated $Z$ , and the first two outputs vanish, while the third output is $Z$ ; the sum vanishes.

4.1step 2.1step 3.1step 2.2F2∎

Conclusion. The displayed finite chains exhibit the double-edge Serre cancellation on $2, 11, 21$ and the $E$ cancellation on $321$ , with the $[3]$ coefficients $q^{2} + 1 + q^{- 2}$ ; the general relation including the colour swap is proved by the independent rank-one Serre lemma, so no finite diagnostic is promoted to a universal proof.

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Sources

Masaki Kashiwara, Tetsuji Miwa and Eugene Stern, Decomposition of q-deformed Fock spaces, arXiv:q-alg/9508006 (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 double-edge Serre cancellation in affine A1A1​ on partitions