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Word projectives need not give lower global basis vectors

Statement refuted

Once the isomorphism $[P r o j (R^{Λ})] ≅ V_{A} (Λ)$ of The cyclotomic projective $K_{0}$ is the integral highest-weight module is known, the classes of the word projectives $R^{Λ} (β) e (ν)$ are automatically lower global basis vectors.

Facts & Assumptions

Given: The rank-one datum $I = {i}$ , $(α_{i} ∣ α_{i}) = 2$ , $d_{i} = 1$ , $q_{i} = q$ , $Λ = 2 Λ_{1}$ , and the rank-two block $R^{Λ} (2 α_{i}) = N H_{2} / (x_{1}^{2})$ of Cyclotomic KLR quotients in ranks zero, one and two.

[F1]

At $λ_{i} = n = 2$ , the exact matrix calculation of Cyclotomic KLR quotients in ranks zero, one and two identifies $R^{Λ} (2 α_{i}) = {End}_{k} (S_{2})$ , where $S_{2} = C {- 1}$ has degrees $- 1, 1$ and is the normalized divided-power projective $P_{i, 2}$ . Thus its regular module is $P_{i, 2} {1} \oplus P_{i, 2} {- 1}$ . The divided-power construction is Nil-Hecke idempotents and divided-power KLR projectives.

[F2]

$F_{i}^{2} = [2]_{i} F_{i}^{(2)}$ with $[2]_{i} = q + q^{- 1}$ (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ). The module string operators are defined in The Kashiwara operators ${\tilde{e}}_{i}$ and ${\tilde{f}}_{i}$ of an integrable module. The lower global lift is characterized by the integral, zero and bar-chart intersection in The lower global basis of $U_{q}^{-}$ and of a highest weight module; the charts are defined in The Kashiwara lattices $L (λ)$ and $L (\infty)$ and their generating sets and The bar lattice and the divided-power integral lattice of a based module.

Counterexample

At root $2 α_{i}$ the regular word projective has the graded decomposition $R^{Λ} (2 α_{i}) ≅ P_{i, 2} {1} \oplus P_{i, 2} {- 1}, [R^{Λ} (2 α_{i})] = (q + q^{- 1}) [P_{i, 2}] .$

Under the vacuum-normalized identification the word class is $F_{i}^{2} v_{2 Λ_{1}}$ , whereas the lower global vector is $F_{i}^{(2)} v_{2 Λ_{1}}$ . Their ratio $q + q^{- 1}$ is a nonunit of $A = Z [q, q^{- 1}]$ . The normalized indecomposable divided-power projective gives the lower vector in this weight; the ordinary word projective does not.

1.1F1

The word class. By [F1] the word projective decomposes as $(q + q^{- 1})$ copies of $P_{i, 2}$ , so $[R^{Λ} (2 α_{i})] = (q + q^{- 1}) [P_{i, 2}]$ ; under the K_0 isomorphism of the integral projective theorem this is the class of the ordinary product $F_{i}^{2} v_{2 Λ_{1}}$ .

1.2F2algebra

In the rank-one module $V (2 Λ_{1})$ , the Kashiwara string is $v, F_{i} v, F_{i}^{(2)} v$ . These vectors generate its zero chart; they are bar fixed and generate the corresponding infinity chart. In weight $2 α_{i}$ the divided-power integral lattice is $A F_{i}^{(2)} v$ : any divided word of that weight is an $A$ -multiple of $F_{i}^{(2)} v$ , since $F_{i}^{2} v = [2]_{i} F_{i}^{(2)} v$ . Since an integral Laurent polynomial regular at both $0$ and infinity is an integer constant, the balanced intersection in that weight is $Z F_{i}^{(2)} v$ , and its positive string residue selects $F_{i}^{(2)} v$ as the lower global lift.

2.1step 1.1step 1.2F2algebra

By [F2], $F_{i}^{2} v = [2]_{i} F_{i}^{(2)} v$ . The coefficient $[2]_{i} = q + q^{- 1}$ is a nonunit: evaluation at $q = 1$ sends it to $2$ , while an invertible Laurent polynomial must evaluate to a unit of $Z$ . Hence the word class cannot be the lower global vector identified in step 1.2.

3.1step 1.1step 1.2step 2.1∎

This refutes the assertion that word-projective classes are automatically lower global vectors. Here the integral divided-power class $[P_{i, 2}]$ gives $F_{i}^{(2)} v$ directly, whereas the regular word class has the nonunit coefficient $q + q^{- 1}$ . The finite example makes no general impossibility claim about extracting a basis from additional categorical or quantum-module structure.

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Sources

Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups I, Representation Theory 13 (2009) 309-347, arXiv:0803.4121v2 (standard reference, not scraped)
Seok-Jin Kang and Masaki Kashiwara, Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras, arXiv:1102.4677v4 (standard reference, not scraped)