Appendix: The bijection between $W$ and alcoves in type $S{L}_{3}$
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 18 February 2013

Appendix: The bijection between $W$ and alcoves in type $S{L}_{3}$
The following pictures illustrate the bijection of (2.19) for type $S{L}_{3}\text{.}$ In this case,
${\Omega}^{\vee}=\{1,{g}^{\vee},{\left({g}^{\vee}\right)}^{2}\cong \mathbb{Z}/3\mathbb{Z},\}$
and ${\Omega}^{\vee}\times {\U0001d525}_{\mathbb{R}}^{*}$
has 3 sheets. The alcoves are the triangles and the (centres of) hexagons are the elements of ${\U0001d525}_{\mathbb{Z}}^{*}\text{.}$

$${\U0001d525}^{{\alpha}_{2}^{\vee}+d}$$
$${\U0001d525}^{{\alpha}_{2}^{\vee}}$$
$${\U0001d525}^{-{\alpha}_{2}^{\vee}+2d}$$
$${\U0001d525}^{-{\alpha}_{2}^{\vee}+4d}$$
$${\U0001d525}^{-\phi +4d}$$
$${\U0001d525}^{-{\phi}^{\vee}+3d}$$
$${\U0001d525}^{-{\phi}^{\vee}+2d}$$
$${\U0001d525}^{{\alpha}_{0}^{\vee}}$$
$${\U0001d525}^{{\phi}^{\vee}}$$
$${\U0001d525}^{{\phi}^{\vee}+d}$$
$${\U0001d525}^{{\phi}^{\vee}+2d}$$
$${\U0001d525}^{{\phi}^{\vee}+3d}$$
$${\U0001d525}^{{\phi}^{\vee}+4d}$$
$${\U0001d525}^{-{\alpha}_{1}^{\vee}+d}$$
$${\U0001d525}^{{\alpha}_{1}^{\vee}}$$
$${\U0001d525}^{{\alpha}_{1}^{\vee}+2d}$$
$${\U0001d525}^{{\alpha}_{1}^{\vee}+4d}$$
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$${X}^{p}$$
$$1$$
$${s}_{0}^{\vee}$$
$${w}_{0}$$
$${X}^{{w}_{0}p}$$
$${s}_{1}^{\vee}$$
$${s}_{2}^{\vee}$$
$${s}_{0}^{\vee}{s}_{1}^{\vee}$$
$${s}_{0}^{\vee}{s}_{2}^{\vee}$$
$${X}^{{s}_{2}p}$$
$${X}^{{s}_{1}p}$$
$${s}_{1}^{\vee}{s}_{0}^{\vee}$$
$${s}_{2}^{\vee}{s}_{0}^{\vee}$$
$${X}^{{s}_{1}p}$$
$${X}^{{s}_{1}{s}_{2}p}$$
Sheet 1

$${\U0001d525}^{{\alpha}_{2}^{\vee}+d}$$
$${\U0001d525}^{{\alpha}_{2}^{\vee}}$$
$${\U0001d525}^{-{\alpha}_{2}^{\vee}+2d}$$
$${\U0001d525}^{-{\alpha}_{2}^{\vee}+4d}$$
$${\U0001d525}^{-\phi +4d}$$
$${\U0001d525}^{-{\phi}^{\vee}+3d}$$
$${\U0001d525}^{-{\phi}^{\vee}+2d}$$
$${\U0001d525}^{{\alpha}_{0}^{\vee}}$$
$${\U0001d525}^{{\phi}^{\vee}}$$
$${\U0001d525}^{{\phi}^{\vee}+d}$$
$${\U0001d525}^{{\phi}^{\vee}+2d}$$
$${\U0001d525}^{{\phi}^{\vee}+3d}$$
$${\U0001d525}^{{\phi}^{\vee}+4d}$$
$${\U0001d525}^{-{\alpha}_{1}^{\vee}+d}$$
$${\U0001d525}^{{\alpha}_{1}^{\vee}}$$
$${\U0001d525}^{{\alpha}_{1}^{\vee}+2d}$$
$${\U0001d525}^{{\alpha}_{1}^{\vee}+4d}$$
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$$\kappa $$
$$\kappa {s}_{2}^{\vee}$$
$${X}^{{w}_{0}{\omega}_{2}}$$
$$\kappa {s}_{1}^{\vee}$$
$$\kappa {s}_{0}^{\vee}$$
$${X}^{{\omega}_{2}}$$
$${X}^{{s}_{2}{\omega}_{2}}$$
Sheet $\kappa ={\left({g}^{\vee}\right)}^{2}$

$${\U0001d525}^{{\alpha}_{2}^{\vee}+d}$$
$${\U0001d525}^{{\alpha}_{2}^{\vee}}$$
$${\U0001d525}^{-{\alpha}_{2}^{\vee}+2d}$$
$${\U0001d525}^{-{\alpha}_{2}^{\vee}+4d}$$
$${\U0001d525}^{-\phi +4d}$$
$${\U0001d525}^{-{\phi}^{\vee}+3d}$$
$${\U0001d525}^{-{\phi}^{\vee}+2d}$$
$${\U0001d525}^{{\alpha}_{0}^{\vee}}$$
$${\U0001d525}^{{\phi}^{\vee}}$$
$${\U0001d525}^{{\phi}^{\vee}+d}$$
$${\U0001d525}^{{\phi}^{\vee}+2d}$$
$${\U0001d525}^{{\phi}^{\vee}+3d}$$
$${\U0001d525}^{{\phi}^{\vee}+4d}$$
$${\U0001d525}^{-{\alpha}_{1}^{\vee}+d}$$
$${\U0001d525}^{{\alpha}_{1}^{\vee}}$$
$${\U0001d525}^{{\alpha}_{1}^{\vee}+2d}$$
$${\U0001d525}^{{\alpha}_{1}^{\vee}+4d}$$
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$${g}^{\vee}$$
$${g}^{\vee}{s}_{2}^{\vee}$$
$${X}^{{w}_{0}{\omega}_{1}}$$
$${g}^{\vee}{s}_{0}^{\vee}$$
$${g}^{\vee}{s}_{1}^{\vee}$$
$${X}^{{\omega}_{1}}$$
$${X}^{{s}_{1}{\omega}_{1}}$$
Sheet ${g}^{\vee}$

Notes and References
This is an excerpt from a paper entitled A combinatorial formula for Macdonald polynomials authored by Arun Ram and Martha Yip. It was dedicated to Adriano Garsia.

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