The dual PBW and dual canonical bases of <math> <msub><mi>U</mi><mi>q</mi></msub><msup><mi>𝔫</mi><mi>-</mi></msup> </math>: Page History

## The dual PBW and dual canonical bases of ${U}_{q}{𝔫}^{-}$

Let $ℬ + = good words .$ The dual PBW basis of ${U}_{q}{𝔫}^{-}$ is $\left\{{E}_{g}^{*}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}g\in {ℬ}^{+}\right\}$ where $` E l β * = constant l β E l β 2 * ⊔⊔ E l β 1 * - q β 1 β 2 E l β 1 * ⊔⊔ E l β 2 * ,ifl β =l β 1 l β 2 ∈ ℛ + ,$ with $l\left({\beta }_{1}\right) (so that $l\left({\beta }_{1}\right)$ is the left Lyndon factor of $l\left(\beta \right)$ of maximal length), and $E g * = constant g E l k * ⊔⊔ n k ⊔⊔…⊔⊔ E l 1 * ⊔⊔ n 1 ,ifg= l 1 n 1 … l k n k ,$ with ${l}_{1}>\dots >{l}_{k}$ good Lyndon. The constants are important, but we will attend to these later.

The dual canonical basis is the basis $\left\{{b}_{g}^{*}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}g\in {ℬ}^{+}\right\}$ of ${U}_{q}{𝔫}^{-}$ given by $b g * = E g * + ∑ h∈ ℬ + ,h and $b g * = κ g g+ ∑ h∈ ℬ + ,h (See [Le, Prop 39]).

[Le, Cor 41] If $l\in {ℛ}^{+}$ then ${b}_{l}^{*}={E}_{l}^{*}.$

This paragraph is about the constants: Leclerc sets (see [Le, 28], first displayed equation in proof of [Le, Prop. 30] and definition right before [Le, Prop 22]) $κ β E l β *= C β r l β = C β r l β 2 * ⊔⊔ r l β 1 * - q β 1 β 2 r l β 1 * ⊔⊔ r l β 2 * ,$ if $l\left(\beta \right)=l\left({\beta }_{1}\right)l\left({\beta }_{2}\right)\in {ℛ}^{+},$ with $l\left({\beta }_{1}\right) (so that $l\left({\beta }_{1}\right)$ is the left Lyndon factor of $l\left(\beta \right)$ of maximal length.) In general, set $d i = α i α i 2 and d β = ββ 2$ for a positive root $\beta \in {R}^{+}.$ Here (see last line of the proof of [Le, Prop 31]) $C β = -1 𝓁 l -1 1- q deg l deg l q Ndeg l ∏ i=1 r 1- q α i α i c i = q d β - q - d β q d 1 - q - d 1 c 1 … q d r - q - d r c r ,$ where $\beta =\mathrm{deg}\left(l\right)={c}_{1}{\alpha }_{1}+\dots +{c}_{r}{\alpha }_{r},$$N deg l = 1 2 deg l deg l - c 1 α 1 α1 -…- c r α r α r = d β - c 1 d 1 +…+ c r d r ,$ and (see [Le, 5.5.2]).

Next $constant g = q c g ,$ where $c g = a 1 2 d l 1 +…+ a k 2 d l k ,with d β = ββ 2 .$ Also, $κ g = κ l 1 a 1 a 1 l 1 ! κ l 2 a 2 a 2 l 2 !… κ l k a k a k l k !.$