## Appendix: The bijection between $W$ and alcoves in type $S{L}_{3}$

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, ${\Omega }^{\vee }=\left\{1,{g}^{\vee },{\left({g}^{\vee }\right)}^{2}\cong ℤ/3ℤ,\right\}$ and ${\Omega }^{\vee }×{𝔥}_{ℝ}^{*}$ has 3 sheets. The alcoves are the triangles and the (centres of) hexagons are the elements of ${𝔥}_{ℤ}^{*}\text{.}$

Sheet 1

Sheet $\kappa ={\left({g}^{\vee }\right)}^{2}$

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.