## The double affine braid group

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 $TiTj … ⏟mij = TjTi… ⏟mij , gTi g-1 = Tσ(i) , gXμ= Xgμg, (2.22)$ for $g\in \Omega$, and $TiXμ= Xsiμ Ti , if ⟨μ, αi∨⟩ =0, TiXμ Ti =Xsiμ , if ⟨μ, αi∨⟩ =1 , for i=0,1, …,n, (2.23)$ where the action of ${W}^{\vee }$ on ${𝔥}_{ℤ}^{*}\oplus ℤ\delta$ is as in (2.10). The element $q=Xδ is in the center of ℬ∼. (2.24)$ 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 $i=1,2,\dots ,n,$ $g∨= Xωg Twg w0∨, (T0∨)-1= XφTsφ∨, g= Yωg∨ Tw0 wg-1, T0= Yφ∨ Tsφ-1. (2.27)$

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 $Ti∨Tj∨… ⏟mij∨ = Tj∨Ti∨… ⏟mij∨ , g∨Ti∨ (g∨)-1= Tσ∨(i)∨, g∨Yλ∨= Yg∨λ∨ g∨, (2.28)$

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

$Ti∨= Ysi∨λ∨ Ti∨ , if ⟨λ∨,αi⟩ =0 , (Ti∨) -1 Yλ∨ (Ti∨) -1 =Ysi∨λ∨ , if ⟨λ∨,αi⟩ =1 , fori=0,1, …,n, (2.29)$

where the action of $W$ on ${𝔥}_{ℤ}\oplus ℤd$ 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.