The double affine braid group

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

Last update: 24 December 2012

The double affine braid group

The double affine braid group $\stackrel{\sim }{ℬ}$ is the group generated by $T_0,\dots ,T_n,\Omega$ and $X$ with relations $$\begin{array}{cc}\underset{\underset{m_{ij}}{⏟}}{T_i{T}_j\dots }=\underset{\underset{m_{ij}}{⏟}}{T_j{T}_i\dots },g{T}_i{g}^{-1}=T_{\sigma \left(i\right)},g{X}^{\mu }=X^{g\mu }g,& \text{(2.22)}\end{array}$$ for $g\in \Omega$, and $$\begin{array}{cc}\begin{array}{ccc}{T}_i{X}^{\mu }=X^{s_i\mu }{T}_i,& & \text{if}⟨\mu ,{\alpha }_i^{\vee }⟩=0,\\ T_i{X}^{\mu }{T}_i=X^{s_i\mu },& & \text{if}⟨\mu ,{\alpha }_i^{\vee }⟩=1,\end{array}\text{for}i=0,1,\dots ,n,& \text{(2.23)}\end{array}$$ where the action of $W^{\vee }$ on ${𝔥}_{\mathbb{Z}}^{*}\oplus \mathbb{Z}\delta$ is as in (2.10). The element $$\begin{array}{cc}q=X^{\delta }\text{is in the center of}\stackrel{\sim }{ℬ}.& \text{(2.24)}\end{array}$$ For $w\in W^{\vee }$, view a reduced word $w=g{s}_{i_1}\dots s_{i_{\ell }}$ as a minimal length path $p$ from the fundamental alcove to $w$ in ${𝔥}_{ℝ}$ and define with respect to the periodic orientation (see (2.20) and the pictures in Appendix A). For $v\in W$, view a reduced word $v=g^{\vee }{s}_{i_{\ell }}^{\vee }\dots s_{i_{\ell }}^{\vee }$ as a minimal length path $p^{\vee }$ from the fundamental alcove to $v$ in ${𝔥}_{ℝ}^{*}$ and define

Let $T_i^{\vee }=T_i,$ for $i=1,2,\dots ,n,$ $$\begin{array}{cc}{g}^{\vee }=X^{{\omega }_g}{T}_{w_g{w}_0}^{\vee },{\left(T_0^{\vee }\right)}^{-1}=X^{\phi }{T}_{s_{\phi }}^{\vee },g=Y^{{\omega }_g^{\vee }}{T}_{w_0{w}_g}^{-1},T_0=Y^{{\phi }^{\vee }}{T}_{s_{\phi }}^{-1}\text{.}& \text{(2.27)}\end{array}$$

where $\phi$ and ${\phi }^{\vee }$ are as in (2.11) and, using the action in (2.15), ${\omega }_g=g^{\vee }·0$ and $w_g$ is the longest element of the stabilizer of ${\omega }_g$ in $W_0\text{.}$

The following theorem, discovered by Cherednik [Ch, Thm. 2.2], is proved in [Mac4, 3.5-3.7], in [Io], and in [Hai, 4.13-4.18].

(Duality) Let $Y^d=q^{-1}\text{.}$ The double affine braid group $\stackrel{\sim }{ℬ}$ is generated by $T_0^{\vee },T_1^{\vee },\dots ,T_n^{\vee },{\Omega }^{\vee }$ and $Y$ with relations $$\begin{array}{cc}\underset{\underset{m_{ij}^{\vee }}{⏟}}{T_i^{\vee }{T}_j^{\vee }\dots }=\underset{\underset{m_{ij}^{\vee }}{⏟}}{T_j^{\vee }{T}_i^{\vee }\dots },g^{\vee }{T}_i^{\vee }{\left(g^{\vee }\right)}^{-1}=T_{{\sigma }^{\vee }\left(i\right)}^{\vee },g^{\vee }{Y}^{{\lambda }^{\vee }}=Y^{g^{\vee }{\lambda }^{\vee }}{g}^{\vee },& \text{(2.28)}\end{array}$$

for $g^{\vee }\in {\Omega }^{\vee },$ and

$$\begin{array}{cc}\begin{array}{ccc}{T}_i^{\vee }=Y^{s_i^{\vee }{\lambda }^{\vee }}{T}_i^{\vee },& & \text{if}⟨{\lambda }^{\vee },{\alpha }_i⟩=0,\\ {\left(T_i^{\vee }\right)}^{-1}{Y}^{{\lambda }^{\vee }}{\left(T_i^{\vee }\right)}^{-1}=Y^{s_i^{\vee }{\lambda }^{\vee }},& & \text{if}⟨{\lambda }^{\vee },{\alpha }_i⟩=1,\end{array}\text{for}i=0,1,\dots ,n,& \text{(2.29)}\end{array}$$

where the action of $W$ on ${𝔥}_{\mathbb{Z}}\oplus \mathbb{Z}d$ is as in (2.10).

Notes and References

This page is taken from a paper entitled A combinatorial formula for Macdonald polynomials by Arun Ram and Martha Yip. (2.7) is a reference to the section entitled The double affine Weyl group.

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