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Homogeneous dual-basis sums: basis independence and weightwise finiteness

Statement

Let $ν \in Q_{+}$ , let $(u_{a})$ be a field basis of $U_{q}^{+} [ν]$ and $(v^{a})$ its $D$ -dual basis in $U_{q}^{-} [- ν]$ with respect to the normalized Drinfeld pairing of The Drinfeld pairing is nondegenerate in each root degree and matches the shuffle pairing (so $D (u_{a}, v^{b}) = δ_{a}^{b}$ ), and set $Θ_{ν} : = \sum_{a} u_{a} \otimes v^{a} \in U_{q}^{+} [ν] \otimes U_{q}^{-} [- ν] \subseteq \hat{C},$ the ambient root-height completion of The root-height completion for weightwise quasi- $R$ intertwiners. Then $Θ_{ν}$ is independent of the chosen basis of $U_{q}^{+} [ν]$ , and $Θ_{0} = 1 \otimes 1$ . The family $(Θ_{ν})_{ν}$ has finite-dimensional homogeneous components and defines a vectorwise finite action under the exact finiteness hypotheses of the completion definition. Its degree- $ν$ action shifts a label pair $(μ, η)$ to $(μ + ν, η - ν)$ , preserving the total weight, and on finite-weight-space $O_{i n t}$ pairs every fixed total-weight space has a uniform finite tail bound beyond which all components act as $0$ . No integral algebra coefficients are asserted, and the positive-height components have no same-weight-pair diagonal term.

Facts & Assumptions

Given: The graded Serre halves $U_{q}^{\pm} = ⨁_{ν} U_{q}^{\pm} [\pm ν]$ , the perfect normalized pairing $D$ , and the completion $\hat{C} = \prod_{ν \in Q_{+}} U_{q}^{+} [ν] \otimes U_{q}^{-} [- ν]$ .

[F1]

Perfect graded pairing (The Drinfeld pairing is nondegenerate in each root degree and matches the shuffle pairing): $D$ is homogeneous and perfect on every finite root degree, so each finite-dimensional $U_{q}^{+} [ν]$ has a unique $D$ -dual basis in $U_{q}^{-} [- ν]$ and vice versa.

[F2]

Completion and finiteness (The root-height completion for weightwise quasi- $R$ intertwiners): $\hat{C}$ is the diagonal root-height completion with root-degree convolution multiplication $(A B)_{ν} = \sum_{β + γ = ν} A_{β} B_{γ}$ , each component sum being finite; a family acts on $m_{μ} \otimes n_{η}$ by its finite sum whenever $U_{q}^{+} [ν] m_{μ} \otimes U_{q}^{-} [- ν] n_{η} = 0$ for all sufficiently large $ν$ ; this holds for finite-union bounded-above first supports, for lowest second factors with bounded-below support, and uniformly on each fixed total-weight space of a pair of finite-weight-space $O_{i n t}$ modules.

[F3]

Sample weights: $U_{q}^{+} [ν]$ acts on weight $μ$ vectors by shifting to $μ + ν$ and $U_{q}^{-} [- ν]$ acts on weight $η$ vectors by shifting to $η - ν$ ; the tensor orientation is positive first, negative second.

Proof

technique · direct

1.1F1algebra

Basis independence. If $u_{a}^{'} = \sum_{b} u_{b} g_{b a}$ for an invertible matrix $g$ , then the dual basis is $v^{' a} = \sum_{b} (g^{- 1})_{a b} v^{b}$ , since $\sum_{b} g_{b c} (g^{- 1})_{a b} = δ_{a c}$ . Consequently $\sum_{a} u_{a}^{'} \otimes v^{' a} = \sum_{b, c} u_{b} \otimes v^{c} \sum_{a} g_{b a} (g^{- 1})_{a c} = \sum_{b} u_{b} \otimes v^{b}$ .

2.1F1F3step 1.1

Components and degree shift. By [F1] the pieces $U_{q}^{\pm} [\pm ν]$ are finite-dimensional, so each $Θ_{ν}$ lies in the finite-dimensional tensor product $U_{q}^{+} [ν] \otimes U_{q}^{-} [- ν]$ and $Θ_{0} = 1 \otimes 1$ ; by [F3] the degree- $ν$ action shifts $(μ, η)$ to $(μ + ν, η - ν)$ and preserves the total weight $μ + η$ .

3.1F2step 2.1

Vectorwise finiteness and the uniform tail bound. The action on a vector $m_{μ} \otimes n_{η}$ is the sum of the finitely many terms with $U_{q}^{+} [ν] m_{μ} \neq 0$ and $U_{q}^{-} [- ν] n_{η} \neq 0$ , and [F2] provides exactly the hypotheses under which this set of $ν$ is finite and uniformly bounded on each fixed total-weight space of finite-weight-space $O_{i n t}$ pairs. Hence the family defines a vectorwise finite action, and no integrality of coefficients or diagonal same-weight term is produced.

4.1step 1.1step 2.1step 3.1∎

Conclusion. Steps 1.1 and 2.1 give the stated basis independence, $Θ_{0} = 1$ , finite-dimensional components and degree shift, and step 3.1 gives the exact finiteness scope; no integral algebra coefficient property is asserted.

Depends on

Used by

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Sources

Jonathan Brundan, Dual Canonical Bases and Kazhdan–Lusztig Polynomials, arXiv:math/0509700v2 (Transform. Groups 14 (2009), 535–558) (standard reference, not scraped)
Antoine Lacabanne, Drinfeld Double of Quantum Groups, Tilting Modules and Z-Modular Data Associated to Complex Reflection Groups, arXiv:1807.00770v2 (standard reference, not scraped)
George Lusztig, Introduction to Quantum Groups, author-hosted complete scan (standard reference, not scraped)