The dual PBW and dual canonical bases of $U_q{𝔫}^{-}$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 2 March 2010

The dual PBW and dual canonical bases of $U_q{𝔫}^{-}$

Let $${ℬ}^{+}=\left\{\text{good words}\right\}.$$ The dual PBW basis of $U_q{𝔫}^{-}$ is $\left\{E_g^{*}|g\in {ℬ}^{+}\right\}$ where $$`E_{l\left(\beta \right)}^{*}={\left(\text{constant}\right)}_{l\left(\beta \right)}\left(E_{l\left({\beta }_2\right)}^{*}\bigsqcup \bigsqcup E_{l\left({\beta }_1\right)}^{*}-q^{⟨{\beta }_1,{\beta }_2⟩}{E}_{l\left({\beta }_1\right)}^{*}\bigsqcup \bigsqcup E_{l\left({\beta }_2\right)}^{*}\right),\text{if}l\left(\beta \right)=l\left({\beta }_1\right)l\left({\beta }_2\right)\in {ℛ}^{+},$$ with $l\left({\beta }_1\right)<l\left({\beta }_2\right)$ (so that $l\left({\beta }_1\right)$ is the left Lyndon factor of $l\left(\beta \right)$ of maximal length), and $$E_g^{*}={\left(\text{constant}\right)}_g\left({\left(E_{l_k}^{*}\right)}^{\bigsqcup \bigsqcup n_k}\bigsqcup \bigsqcup \dots \bigsqcup \bigsqcup {\left(E_{l_1}^{*}\right)}^{\bigsqcup \bigsqcup n_1}\right),\text{if}g=l_1^{n_1}\dots l_k^{n_k},$$ 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 $\left\{b_g^{*}|g\in {ℬ}^{+}\right\}$ of $U_q{𝔫}^{-}$ given by $$b_g^{*}=E_g^{*}+\sum _{h\in {ℬ}^{+},h<g}{p}_{gh}{E}_h^{*},\text{with}{p}_{gh}\in q\mathbb{Z}\left[q\right],$$ and $$b_g^{*}={\kappa }_{g}g+\sum _{h\in {ℬ}^{+},h<g}{\kappa }_{gh}h,\text{with}{\kappa }_{gh}\in \mathbb{Z}\left[q+q^{-1}\right].$$ (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]) $${\kappa }_{\beta }{E}_{l\left(\beta \right)}^{*}=C_{\beta }{r}_{l\left(\beta \right)}=C_{\beta }\left(r_{l\left({\beta }_2\right)}^{*}\bigsqcup \bigsqcup r_{l\left({\beta }_1\right)}^{*}-q^{⟨{\beta }_1,{\beta }_2⟩}{r}_{l\left({\beta }_1\right)}^{*}\bigsqcup \bigsqcup r_{l\left({\beta }_2\right)}^{*}\right),$$ if $l\left(\beta \right)=l\left({\beta }_1\right)l\left({\beta }_2\right)\in {ℛ}^{+},$ with $l\left({\beta }_1\right)<l\left({\beta }_2\right)$ (so that $l\left({\beta }_1\right)$ is the left Lyndon factor of $l\left(\beta \right)$ of maximal length.) In general, set $$d_i=\frac{⟨{\alpha }_i,{\alpha }_i⟩}{2}\text{and}{d}_{\beta }=\frac{⟨\beta ,\beta ⟩}{2}$$ for a positive root $\beta \in R^{+}.$ Here (see last line of the proof of [Le, Prop 31]) $$C_{\beta }=\frac{{\left(-1\right)}^{𝓁\left(l\right)-1}\left(1-q^{⟨\mathrm{deg}\left(l\right),\mathrm{deg}\left(l\right)⟩}\right)}{q^{N\mathrm{deg}\left(l\right)}\prod _{i=1}^r{\left(1-q^{⟨{\alpha }_i,{\alpha }_i⟩}\right)}^{c_i}}=\frac{\left(q^{d_{\beta }}-q^{-d_{\beta }}\right)}{{\left(q^{d_1}-q^{-d_1}\right)}^{c_1}\dots {\left(q^{d_r}-q^{-d_r}\right)}^{c_r}},$$ where $\beta =\mathrm{deg}\left(l\right)=c_1{\alpha }_1+\dots +c_r{\alpha }_r,$$$N\left(\mathrm{deg}\left(l\right)\right)=\frac{1}{2}\left(⟨\mathrm{deg}\left(l\right)⟩,\mathrm{deg}\left(l\right)\right)-c_1⟨{\alpha }_1,{\alpha }_1⟩-\dots -c_r⟨{\alpha }_r,{\alpha }_r⟩=d_{\beta }-\left(c_1{d}_1+\dots +c_r{d}_r\right),$$ and $${\kappa }_l=\left(\text{coefficient of }l\text{ in }{E}_l^{*}\right)\text{and}{\kappa }_l^2=\left(\text{coefficient of }l\text{ in }{C}_{\beta }{r}_l\right),$$ (see [Le, 5.5.2]).

Next $${\left(\text{constant}\right)}_g=q^{c_g},$$ where $$c_g=\left(\genfrac{}{}{0ex}{}{a_1}{2}\right)d_{l_1}+\dots +\left(\genfrac{}{}{0ex}{}{a_k}{2}\right)d_{l_k},\text{with}{d}_{\beta }=\frac{⟨\beta ,\beta ⟩}{2}.$$ Also, $${\kappa }_g={\kappa }_{l_1}^{a_1}{\left[a_1\right]}_{l_1}!{\kappa }_{l_2}^{a_2}{\left[a_2\right]}_{l_2}!\dots {\kappa }_{l_k}^{a_k}{\left[a_k\right]}_{l_k}!.$$

References

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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