Appendix: The bijection between W and alcoves in type SL3

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 SL3

The following pictures illustrate the bijection of (2.19) for type SL3. In this case, Ω∨= { 1,g∨, (g∨)2 ≅ℤ/3ℤ, } and Ω∨×𝔥ℝ* has 3 sheets. The alcoves are the triangles and the (centres of) hexagons are the elements of 𝔥ℤ*.

𝔥α2∨+d 𝔥α2∨ 𝔥-α2∨+2d 𝔥-α2∨+4d 𝔥-φ+4d 𝔥-φ∨+3d 𝔥-φ∨+2d 𝔥α0∨ 𝔥φ∨ 𝔥φ∨+d 𝔥φ∨+2d 𝔥φ∨+3d 𝔥φ∨+4d 𝔥-α1∨+d 𝔥α1∨ 𝔥α1∨+2d 𝔥α1∨+4d - + - + - + - + - + - + - + + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - Xp 1 s0∨ w0 Xw0p s1∨ s2∨ s0∨s1∨ s0∨s2∨ Xs2p Xs1p s1∨s0∨ s2∨s0∨ Xs1p Xs1s2p
Sheet 1


𝔥α2∨+d 𝔥α2∨ 𝔥-α2∨+2d 𝔥-α2∨+4d 𝔥-φ+4d 𝔥-φ∨+3d 𝔥-φ∨+2d 𝔥α0∨ 𝔥φ∨ 𝔥φ∨+d 𝔥φ∨+2d 𝔥φ∨+3d 𝔥φ∨+4d 𝔥-α1∨+d 𝔥α1∨ 𝔥α1∨+2d 𝔥α1∨+4d - + - + - + - + - + - + - + + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - κ κs2∨ Xw0ω2 κs1∨ κs0∨ Xω2 Xs2ω2
Sheet κ=(g∨)2


𝔥α2∨+d 𝔥α2∨ 𝔥-α2∨+2d 𝔥-α2∨+4d 𝔥-φ+4d 𝔥-φ∨+3d 𝔥-φ∨+2d 𝔥α0∨ 𝔥φ∨ 𝔥φ∨+d 𝔥φ∨+2d 𝔥φ∨+3d 𝔥φ∨+4d 𝔥-α1∨+d 𝔥α1∨ 𝔥α1∨+2d 𝔥α1∨+4d - + - + - + - + - + - + - + + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - g∨ g∨s2∨ Xw0ω1 g∨s0∨ g∨s1∨ Xω1 Xs1ω1
Sheet g∨

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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