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Lower and upper global bases in a small highest weight module

Example

For the rank-one highest weight module $V (m ω)$ use the divided string $w_{n} = F^{(n)} v_{m}$ , $0 \leq n \leq m$ , which is the lower global basis, and the lower integral lattice $M_{A} = ⨁_{n = 0}^{m} A w_{n}$ . Its normalized polarization is diagonal, $B (w_{n}, w_{s}) = δ_{n s} d_{m, n}, d_{m, n} = q^{n (m - n)} {(\binom{m}{n})}_{q},$ so the upper form-dual global basis is $w_{n}^{u p} = d_{m, n}^{- 1} w_{n}$ and the restricted integral dual is $M_{A}^{\lor} = ⨁_{n = 0}^{m} A d_{m, n}^{- 1} w_{n}$ . These two explicit $A$ -lattices are the objects compared; no finite-index or determinant-unit property is presumed. For $m = 1$ one has $d_{1, 0} = d_{1, 1} = 1$ and the two bases coincide. For $m = 2$ one has $d_{2, 0} = d_{2, 2} = 1$ and $d_{2, 1} = 1 + q^{2}$ , so $M_{A}^{\lor} = A w_{0} \oplus A (1 + q^{2})^{- 1} w_{1} \oplus A w_{2} ⊋ M_{A},$ with middle quotient $A / (1 + q^{2})$ . The $Z [q]$ -spans of the respective bases extend to the same $A_{0}$ crystal lattice, and both bases have the same three crystal residues. The upper operators are the weighted adjoints of the duality theorem: their $q = 0$ reductions give the same finite string, while exact lower adjunction differs by the displayed norm ratios, and the upper basis is fixed by the adjoint bar rather than the ordinary lower bar.

Facts & Assumptions

Given: The rank-one string module of Divided-power commutation and the simple $U_{q_{i}} ({s l}_{2})$ string modules; the lower global basis and its integral lattice The lower global basis of $U_{q}^{-}$ and of a highest weight module, Bar invariance, congruence and uniqueness of the lower global basis; the contravariant module form The contravariant form on an integrable highest weight module; the upper dual construction and mixed adjunctions The upper (form-dual) global basis and the required perfectness of the integral pairing, Upper and lower global bases are dual and crystal compatible.

[L1]

String and form recurrence: for $1 \leq n \leq m$ , with $E F^{(n)} v = [m - n + 1] F^{(n - 1)} v$ and $σ (F) = q^{- 1} K E$ , the normalized module form satisfies $d_{m, n} = q^{m - 2 n + 1} \frac{[m - n + 1]}{[n]} d_{m, n - 1}$ with $d_{m, 0} = 1$ ; distinct labels are orthogonal (Divided-power commutation and the simple $U_{q_{i}} ({s l}_{2})$ string modules, The contravariant form on an integrable highest weight module).

[L2]

Closed form: iterating the recurrence gives $d_{m, n} = q^{n (m - n)} {(\binom{m}{n})}_{q}$ (The quantum Pascal recurrences, the Gauss product formula and Gaussian integrality).

[L3]

Upper duality: the upper basis is the inverse-Gram field dual, the dual charts are spanned by the same vectors over $A_{0}$ and $A_{\infty}$ , and the exact mixed adjunctions hold with the displayed norm ratios (Upper and lower global bases are dual and crystal compatible).

[L4]

Residues: at $q = 0$ the coefficient rings give $A_{0}$ with $d_{m, n}$ a unit congruent to $1$ , so both the lower and upper basis vectors reduce to the same crystal class; the ordinary bar acts on coefficients by $q \mapsto q^{- 1}$ , while the adjoint bar is the one fixing the upper basis (The upper (form-dual) global basis and the required perfectness of the integral pairing).

Proof

technique · direct

1.1L1L2

The norms. The recurrence of [L1] with base $d_{m, 0} = 1$ gives the diagonal values; iterating it produces the closed form $d_{m, n} = q^{n (m - n)} {(\binom{m}{n})}_{q}$ of [L2], and orthogonality between distinct labels makes the Gram matrix diagonal.

1.2L3L4

Common crystal lattice and residues. Put $C_{0} = ⨁_{n} Z [q] w_{n}$ and $C_{0}^{\lor} = ⨁_{n} Z [q] w_{n}^{u p}$ . Since every $d_{m, n}$ is a unit of $A_{0}$ congruent to $1$ modulo $q$ , the inclusion maps identify $A_{0} \otimes_{Z [q]} C_{0}$ and $A_{0} \otimes_{Z [q]} C_{0}^{\lor}$ with $L_{0} = ⨁_{n} A_{0} w_{n} = ⨁_{n} A_{0} w_{n}^{u p}$ , and the residues of $w_{n}$ and $w_{n}^{u p}$ are the same crystal class $b_{n}$ ; the upper operators are the weighted adjoints whose $q = 0$ reductions reproduce the same partial-inverse string, while exact adjunction carries the ratios $d_{m, n} / d_{m, n \pm 1}$ given by the duality theorem and so the corresponding unweighted adjunction holds modulo $q$ and generally fails exactly over $Q (q)$ .

1.3L3L4

Bars for $m = 2$ . The ordinary bar fixes $w_{1}$ but sends its upper denominator to $1 + q^{- 2}$ , so it moves $w_{1}^{u p}$ ; the adjoint bar $ψ^{\lor}$ is characterized by $B (ψ u, ψ^{\lor} v) = \overline{B (u, v)}$ and, on the diagonal form, fixes each $w_{n}^{u p}$ . This is exactly the distinction between lower-bar and adjoint-bar fixedness asserted by the duality theorem, and no lower/upper basis equality is implied.

2.1L2step 1.1

The two lattices for $m = 2$ . Substituting $m = 2$ : $d_{2, 0} = 1$ , $d_{2, 1} = q {(\binom{2}{1})}_{q} = q (q + q^{- 1}) = 1 + q^{2}$ , $d_{2, 2} = 1$ . Hence $M_{A} = ⨁ A w_{n}$ and $M_{A}^{\lor} = A w_{0} \oplus A (1 + q^{2})^{- 1} w_{1} \oplus A w_{2}$ , whose middle quotient $(1 + q^{2})^{- 1} A / A ≅ A / (1 + q^{2})$ is nonzero: the map $a / (1 + q^{2}) + A \mapsto a + (1 + q^{2}) A$ is an isomorphism; in particular $M_{A}^{\lor}$ properly contains $M_{A}$ and the inclusion is not an isomorphism. For $m = 1$ , $d_{1, 0} = d_{1, 1} = 1$ and the two lattices coincide.

3.1step 1.1step 2.1step 1.2step 1.3∎

Conclusion. Steps 1.1–2.1 compute the rank-one norms, compare the two genuine $A$ -lattices, exhibit the common crystal lattice and residues with modulo- $q$ adjoint agreement and the stated exact weighted adjunction, and identify the correct bar. All calculations are finite and use no determinant-unit assumption.

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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)