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Self-dual indecomposable cyclotomic projectives give the lower global basis

Statement

Assume the Axiom of Choice (The Axiom of Choice), inherited through the based-quotient, generic global-basis, projection and canonical crystal suppliers.

For a symmetric Cartan datum normalized by $d_{i} = 1$ , over any characteristic-zero field, and the SAME finite loopless quiver with arbitrary edge multiplicities and $Q_{i j} = (- 1)^{h_{i j}} (u - v)^{h_{i j} + h_{j i}}$ as the generic global theorem, use the STANDARD homogeneous first-strand quotient and RIGHT append. The vacuum-normalized isomorphism $ϕ : K_{0} (P r o j R^{Λ}) \to V_{A} (Λ)$ maps the classes of normalized $#$ -self-dual indecomposable cyclotomic projectives bijectively onto the lower global basis. The normalization is by projective HOM duality and normalized simple heads, not by finite vector duality or a half-trace shift. The projective with RIGHT crystal label $c \in B (Λ)$ maps to $G_{Λ}^{-} (c)$ . Every such projective is the surviving extension of a generic normalized cover, with the raw map $π_{Λ} \circ *$ and the FIRST/RIGHT label cancellation proved in the based-quotient lemma. The uniqueness is at the level of normalized indecomposable projective isomorphism classes; this is not an assertion that its generic lift is unique. No arbitrary-parameter based extension is asserted.

Facts & Assumptions

Given: Assume the Axiom of Choice. The symmetric normalized $d_{i} = 1$ , characteristic-zero, loopless-multiedge geometric-parameter scope, the standard first-strand quotient with right append, and the vacuum-normalized integral isomorphism $ϕ$ .

[F1]

The based-quotient lemma (Cyclotomic extension of scalars respects the based quotient): raw extension sends each surviving normalized generic cover to a single normalized cyclotomic cover with the map $π_{Λ} \circ *$ , distinct survivors do not merge, and every normalized cyclotomic cover is obtained.

[F2]

The generic global theorem (Self-dual indecomposable KLR projectives give the lower global basis (symmetric type)) and the quantum global projection (The based quotient: projection of the lower global basis to a highest weight module): normalized self-dual generic projectives give the negative-half lower global basis, and every lower global vector of $V_{A} (Λ)$ has a generic lift.

[F3]

The integral identification and the crystal labels (The cyclotomic projective $K_{0}$ is the integral highest-weight module, Simple and indecomposable-projective crystals agree with $B (Λ)$ , The lower global basis of $U_{q}^{-}$ and of a highest weight module, Cyclotomic diagram duality and graded module duals).

Proof

Proof technique: map each normalized cyclotomic cover to its based-quotient class, show injectivity by the free normalized basis, and show surjectivity by lifting lower global vectors through the generic theorem.

1.1F1

Cover classes land on lower global vectors. By [F1] all normalized cyclotomic covers are precisely the nonzero extensions of normalized generic covers, with distinct surviving simple heads; the based-quotient lemma sends each such class to a single positive lower global vector of $V_{A} (Λ)$ , with no grading ambiguity because Hom base change preserves the $#$ -normalization and the degree-zero head pairing.

2.1step 1.1F2

Surjectivity. Conversely the quantum global projection theorem [F2] supplies a generic global vector lifting every lower global vector of $V_{A} (Λ)$ , and the generic theorem supplies its normalized generic cover; mirroring if necessary to use the raw first/right convention, its extension has that nonzero class and is a normalized indecomposable cover. Hence the assignment is surjective onto the lower basis.

3.1step 2.1F1F2F3∎

Bijectivity and labels. The integral $K_{0}$ isomorphism and its free normalized projective basis make the assignment injective; the RIGHT crystal label identification of [F3] shows that the projective with RIGHT label $c \in B (Λ)$ maps to $G_{Λ}^{-} (c)$ , with the FIRST/RIGHT label cancellation proved in the based-quotient lemma. The uniqueness is at the level of normalized indecomposable projective isomorphism classes, not of generic lifts, and no arbitrary-parameter based extension is asserted. The Axiom of Choice is inherited through the generic global-basis and projection contracts of [F2], and the based-quotient and canonical crystal identifications of [F1] and [F3].

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Sources

Seok-Jin Kang and Masaki Kashiwara, Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras, arXiv:1102.4677v4 (standard reference, not scraped)
Owner direction, in-repo: research/frontier-44-special-representation-examples-28-owner-authoring-direction.md (standard reference, not scraped)