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Self-dual shift normalization for a cyclotomic projective

Example

For rank one $d_{i} = 1$ and $λ \geq 1$ , let $B = k [x] / x^{λ}$ and $P = B$ . The PROJECTIVE Hom dual satisfies $P^{#} ≅ P$ , so the canonical normalized projective is $P$ with no shift. Its finite vector dual is different: $D_{ψ} P ≅ P {- 2 (λ - 1)} .$ The unique vector-dual-centered shift of this regular module is $P {- (λ - 1)}$ . For $λ = 1$ both centers agree; for $λ = 2$ , projective Hom-self-duality centers $P$ at $0$ while vector-self-duality centers it at $- 1$ . If $λ = 0$ the block is zero and there is no indecomposable projective here. For any normalized cyclotomic projective the general relation is $D_{ψ} P ≅ P^{#} {- 2 {def}_{Λ} (β)}$ ; it does not redefine the lower-basis projective normalization.

Verification

Given: The rank-one cyclotomic quotient $B = k [x] / x^{λ}$ of The cyclotomic KLR quotient at a dominant integral weight with $d_{i} = 1$ , $λ \geq 1$ , and the two dualities of Cyclotomic diagram duality and graded module duals.

[F1] The vector dual $D_{ψ} M = {HOM}_{k} (M, k)^{ψ}$ reverses internal shifts and satisfies $D_{ψ}^{2} M ≅ M$ ; the projective dual $P^{#} = {HOM}_{B} (P, B)^{ψ}$ is a finite graded projective and the symmetric theorem gives $D_{ψ} P ≅ P^{#} {t_{β}}$ with $t_{β} = - 2 {def}_{Λ} (β)$ (Cyclotomic KLR blocks are finite-dimensional graded symmetric algebras).

1.1F1

Projective Hom self-duality. Evaluation ${HOM}_{B} (B, B) ≅ B$ gives $P^{#} = {HOM}_{B} (B, B)^{ψ} ≅ B = P$ directly, the generator-fixing anti-involution producing the same left module; hence the canonical normalized projective is $P$ with no shift.

2.1step 1.1F1

Vector duality and the shift. The functional $ϵ (x^{a}) = 0$ for $a < λ - 1$ and $ϵ (x^{λ - 1}) = 1$ is the coefficient-of- $x^{λ - 1}$ trace; it has degree $- 2 (λ - 1)$ and its anti-diagonal pairing $ϵ (x^{a} x^{b}) = δ_{a + b, λ - 1}$ is perfect over every field, so $D_{ψ} B ≅ B {- 2 (λ - 1)}$ . Since $D_{ψ} (P {s}) = D_{ψ} P {- s}$ , vector self-duality of $P {s}$ solves $- 2 (λ - 1) - s = s$ , namely $s = - (λ - 1)$ , while projective Hom self-duality solves $- s = s$ , namely $s = 0$ .

3.1step 2.1F1∎

The general defect shift and the boundary. For a general normalized cyclotomic projective the same computation with the symmetric theorem gives $D_{ψ} P ≅ P^{#} {- 2 {def}_{Λ} (β)}$ , which is a comparison of the two dualities and does not redefine the projective normalization. At $λ = 1$ the two centers agree; at $λ = 2$ they differ by $1$ as computed; at $λ = 0$ the defining relation is $1 = 0$ and the block vanishes, so there is no indecomposable projective here. The trace is never halved to pick the lower-basis class.

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Sources

Ben Webster, Knot Invariants and Higher Representation Theory I, arXiv:1001.2020v9 (standard reference, not scraped)
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)