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Matching the canonical and lower global normalizations in rank one and $A_{2}$

Example

Rank one. In the rank-one half $U^{-}$ with generator $F$ , the lower global (and transported canonical) vectors are exactly the divided powers $G (b_{n}) = F^{(n)}$ : they are bar fixed and congruent to their normalized crystal classes, where $b_{n} = {\tilde{f}}^{n} 1 + q L (\infty)$ is represented by $F^{(n)}$ and balanced uniqueness identifies its lift. Their Kashiwara norms are $B_{K} (F^{(n)}, F^{(n)}) = 1 / [n]_{q^{2}}!$ , while the transported Lusztig norms are $B_{L} (F^{(n)}, F^{(n)}) = \prod_{r = 1}^{n} (1 - q^{2 r})^{- 1}$ ; the two differ by the full color factor $(1 - q^{2})^{- n} = C_{n α}$ . The ordinary power satisfies $F^{n} = [n]_{q}! F^{(n)}$ with symmetric factorial $[n]_{q}!$ , which is a nonunit for $n \geq 2$ , and the Drinfeld generator normalization contributes the scalar $(q - q^{- 1})^{- 1}$ since $D (E, F) = 1 / (q - q^{- 1})$ . The four conventions are thereby matched explicitly rather than conflated.

Type $A_{2}$ in degree $α_{1} + α_{2}$ . The two lower global (and canonical) vectors are exactly $F_{1} F_{2}$ and $F_{2} F_{1}$ , with no triangular correction: they are integral, bar fixed, congruent to their distinct crystal classes, and there is no Serre relation in this degree. Their Kashiwara Gram matrix is $(\begin{array}{cc} B_{K} (F_{1} F_{2}, F_{1} F_{2}) & B_{K} (F_{1} F_{2}, F_{2} F_{1}) \\ B_{K} (F_{2} F_{1}, F_{1} F_{2}) & B_{K} (F_{2} F_{1}, F_{2} F_{1}) \end{array}) = (\begin{array}{cc} 1 & q \\ q & 1 \end{array}),$ already the identity modulo $q$ ; the transported Lusztig Gram matrix is $(1 - q^{2})^{- 2}$ times this one. Inverting the Kashiwara Gram matrix over the field gives the genuine upper dual vectors $(1 - q^{2})^{- 1} (F_{1} F_{2} - q F_{2} F_{1}), (1 - q^{2})^{- 1} (F_{2} F_{1} - q F_{1} F_{2}),$ which are not the lower vectors; the Lusztig form-duals are further multiplied by $(1 - q^{2})^{2}$ . In degree $2 α_{1}$ the lower vector is the divided power $F_{1}^{(2)}$ with the rank-one norm above. These calculations exhibit the actual convention dictionary and make no Gram-unimodularity or unspecified-correction claim.

Facts & Assumptions

Given: The rank-one string formulas of Divided-power commutation and the simple $U_{q_{i}} ({s l}_{2})$ string modules and the divided powers of The Drinfeld-Jimbo quantized enveloping algebra by generators and relations; the skew derivations of Kashiwara's skew derivations $e_{i}^{'}$ and $e_{i}^{''}$ on the negative half; the lower global bases and uniqueness The lower global basis of $U_{q}^{-}$ and of a highest weight module, Bar invariance, congruence and uniqueness of the lower global basis; the upper dual construction The upper (form-dual) global basis and the required perfectness of the integral pairing; the normalization dictionary The Lusztig–Kashiwara form, parameter and coproduct dictionary; and the canonical/global agreement theorem Lusztig's canonical basis and Kashiwara's lower global basis agree in symmetric finite ADE type.

[L1]

Strings: in rank one the Kashiwara operators satisfy $\tilde{f} F^{(n)} = F^{(n + 1)}$ for $n \geq 0$ , $\tilde{e} F^{(n)} = F^{(n - 1)}$ for $n \geq 1$ , and $\tilde{e} F^{(0)} = 0$ , so $F^{(n)}$ is the unique lift of the $n$ -th crystal class; $F^{(n)}$ is bar fixed because $ψ (F) = F$ (The $i$ -string decomposition of the negative half, Bar invariance, congruence and uniqueness of the lower global basis).

[L2]

Norms: $B_{K} (F^{(n)}, F^{(n)}) = 1 / [n]_{q^{2}}!$ with $[n]_{q^{2}}! = \prod_{r = 1}^{n} (1 + q^{2} + \dots + q^{2 (r - 1)})$ , and $B_{L} = C_{n α} B_{K}$ with $C_{n α} = (1 - q^{2})^{- n}$ (The Lusztig–Kashiwara form, parameter and coproduct dictionary, The negative-half bilinear form is symmetric, nondegenerate and integral).

[L3]

Degree $α_{1} + α_{2}$ : there is no Serre relation between $F_{1}$ and $F_{2}$ ; the skew derivations give $e_{1}^{'} (F_{2}) = 0 = e_{2}^{'} (F_{1})$ , and $e_{1}^{'} (F_{2} F_{1}) = q F_{2}$ (Kashiwara's skew derivations $e_{i}^{'}$ and $e_{i}^{''}$ on the negative half, Uniqueness of the $i$ -string decomposition of an integrable module).

[L4]

Balanced uniqueness: an integral bar-fixed vector lying in the zero chart is determined by its crystal residue (Bar invariance, congruence and uniqueness of the lower global basis).

[L5]

Upper duals: the upper basis is the inverse-Gram dual in the field, with no $A$ -unimodularity assumed (The upper (form-dual) global basis and the required perfectness of the integral pairing).

Proof

technique · direct

1.1L1L2

Rank one. By [L1] the lower global vector of the $n$ -th class is $F^{(n)}$ , bar fixed and congruent to it; by [L2] its Kashiwara norm is $1 / [n]_{q^{2}}!$ and its Lusztig norm is the color-factor multiple $\prod_{r = 1}^{n} (1 - q^{2 r})^{- 1} = (1 - q^{2})^{- n} / [n]_{q^{2}}!$ . Since $F^{n} = [n]_{q}! F^{(n)}$ and $[n]_{q}!$ is not a unit of $A$ for $n \geq 2$ , the ordinary power differs by a nonunit, and the Drinfeld generator pairing value $1 / (q - q^{- 1})$ records the remaining scalar.

1.2L3L4

The two mixed $A_{2}$ vectors are lifts. By [L3] the classes ${\tilde{f}}_{1} {\tilde{f}}_{2} 1$ and ${\tilde{f}}_{2} {\tilde{f}}_{1} 1$ equal $F_{1} F_{2}$ and $F_{2} F_{1}$ ; both are integral (products of generators), bar fixed ( $ψ$ fixes each $F_{i}$ and reverses no letters here since $ψ$ is an algebra automorphism fixing $F_{i}$ ), and congruent to their distinct crystal residues, so by [L4] they are exactly the lower global lifts with no correction.

2.1L2L3step 1.2

The Gram matrices. The diagonal pairings are $1$ by the normalization of the half form; the cross pairing $⟨ F_{1} F_{2}, F_{2} F_{1} ⟩$ is computed by the adjunction with $e_{1}^{'}$ , using $e_{1}^{'} (F_{2} F_{1}) = q F_{2}$ from [L3], giving $q$ . Hence the Kashiwara Gram matrix is $(\begin{array}{cc} 1 & q \\ q & 1 \end{array})$ with determinant $1 - q^{2}$ , and the transported Lusztig Gram matrix is its color-factor multiple $(1 - q^{2})^{- 2}$ times it.

3.1L2L5step 2.1

Upper duals and the last degree. Inverting the field Gram matrix gives the two displayed upper dual vectors, which are not the lower basis vectors; the reciprocal color factor multiplies the Lusztig form-duals by $(1 - q^{2})^{2}$ . In degree $2 α_{1}$ the only class is that of $F_{1}^{(2)}$ , whose norm is the rank-one value of step 1.1.

4.1step 1.1step 3.1∎

Conclusion. Steps 1.1–3.1 exhibit the exact normalizations in rank one and in degree $α_{1} + α_{2}$ of $A_{2}$ , the absence of triangular corrections in that degree, and the explicitly non-unimodular inverse-Gram duals. All displayed matrices and scalars are finite computations, and no false Gram-unimodularity is asserted.

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Sources

Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Mathematical Journal 63 (1991), 465–516 (standard reference, not scraped)
George Lusztig, Introduction to Quantum Groups (Progress in Mathematics 110, Birkhauser, 1993; second printing 1994; author-hosted scan) (standard reference, not scraped)
Jonathan Brundan, Dual Canonical Bases and Kazhdan–Lusztig Polynomials, arXiv:math/0509700v2 (Transform. Groups 14 (2009), 535–558) (standard reference, not scraped)

Matching the canonical and lower global normalizations in rank one and A2A2​