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The projective-simple Cartan pairing in a small cyclotomic block

Example

For rank one $d_{i} = 1$ , $λ = 2$ , let $B_{r} = R^{Λ} (r α_{i})$ . The normalized simples are $S_{0} = k$ , $S_{1} = k$ with $x = 0$ , and $S_{2} = C {- 1}$ of graded dimension $q^{- 1} + q$ , where $C = k [x_{1}, x_{2}] / (x_{1} + x_{2}, x_{1} x_{2})$ . Their normalized projective covers are $P_{0} = k$ , $P_{1} = k [x] / x^{2}$ , $P_{2} = S_{2}$ . Then $T (P_{r}, S_{s}) = δ_{r s}$ , with distinct roots paired by zero. Under the integral projective identification $P_{0}, P_{1}, P_{2}$ map to $v, F v, F^{(2)} v$ , respectively, while $[B_{2}] = (q + q^{- 1}) [P_{2}]$ maps to $F^{2} v$ . In $G_{0}$ the correct regular classes are $[B_{1}] = (1 + q^{2}) [S_{1}]$ and $[B_{2}] = (q + q^{- 1}) [S_{2}]$ . The canonical restricted-dual map sends $S_{r}$ to the form-dual of the corresponding divided lower vector; it does not identify $B_{1}$ with $S_{1}$ .

Verification

Given: The rank-one datum $I = {i}$ , $(α_{i} ∣ α_{i}) = 2$ , $d_{i} = 1$ , $q_{i} = q$ , $λ = 2$ , with the blocks $B_{r} = R^{Λ} (r α_{i})$ of Cyclotomic KLR quotients in ranks zero, one and two.

[F1] The rank-one blocks: $B_{0} = k$ , $B_{1} = k [x] / x^{2}$ , and $B_{2} = N H_{2} / (x_{1}^{2}) ≅ P_{2}^{\oplus (q + q^{- 1})}$ , where $P_{2} = B_{2} \otimes_{N H_{2}} P_{i, 2} = C {- 1}$ ; the centered polynomial column reproduces $C = k [x_{1}, x_{2}] / (x_{1} + x_{2}, x_{1} x_{2})$ with the shift $- d_{i}$ (Cyclotomic KLR quotients in ranks zero, one and two, Nil-Hecke idempotents and divided-power KLR projectives).

[F2] The quotient-cover pairing: $T (P, S) = gdim (P^{ψ} \otimes_{B} S) = δ_{P, S}$ in normalized cover/simple bases, and the normalized covers are the nonzero extensions of generic covers (Cyclotomic projective covers and dualities descend through the quotient).

[F3] The integral identifications: $ϕ (P_{0}) = v$ , $ϕ (P_{1}) = F v$ , $ϕ (P_{2}) = F^{(2)} v$ under the normalized projective basis, and the canonical restricted-dual map sends simples to form-dual lower vectors (The cyclotomic projective $K_{0}$ is the integral highest-weight module, The cyclotomic module $G_{0}$ is the restricted dual integral highest-weight module).

Verification.

1.1F1

The blocks and their composition structure. By [F1] the three nonzero blocks have graded dimensions $1$ , $1 + q^{2}$ and $(q + q^{- 1})^{2}$ ; the first regular module $B_{1}$ has two composition factors $S_{1}, S_{1} {2}$ with graded dimensions $1 + q^{2}$ , and the second block is the endomorphism matrix algebra of the graded column $S_{2}$ , whose regular left module is the column tensored with its graded vector dual of dimension $q + q^{- 1}$ .

2.1step 1.1F2F3

The Cartan pairing. The normalized cover-to-head Hom space is $k$ in degree zero and there are no other simple types in a block, so quotient-cover evaluation of [F2] identifies the tensor pairing with $T (P_{r}, S_{s}) = δ_{r s}$ , with distinct roots paired by zero. The integral identification of [F3] maps $P_{0}, P_{1}, P_{2}$ to $v, F v, F^{(2)} v$ and $[B_{2}] = (q + q^{- 1}) [P_{2}]$ to $F^{2} v$ , and in $G_{0}$ the regular classes are $[B_{1}] = (1 + q^{2}) [S_{1}]$ and $[B_{2}] = (q + q^{- 1}) [S_{2}]$ .

3.1step 2.1F3∎

The restricted dual. The canonical restricted-dual map sends $S_{r}$ to the form-dual of the corresponding divided lower vector, and since $T$ pairs $P_{r}$ with $S_{s}$ by $δ_{r s}$ it does not identify $B_{1}$ with $S_{1}$ ; the scalar $q + q^{- 1}$ is not removed by the pairing. This is a cover/simple pairing computation, not a claim that the Cartan matrix is all ones or that a trace removes a denominator, and it is a finite computation using no choice.

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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)
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)