Root system type C2

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 26 June 2012

Root system type C2

An example is when the lattice P = ε1 + ε2 with {ε1,&epsilon2} an orthonormal basis of 𝔥* 2 and W = { 1,s1, s2, s1s2, s2s1, s1s2s1, s2s1s2, s1s2s1s2 } is the dihedral group of order 8 generated by the reflections s1 and s2 in the hyperplanes Hα1 and Hα2, respectively, where Hα1 = { x𝔥* | x,ε1=0 }, and Hα2 = { x𝔥* | x,ε2-ε1 = 0 }.

Hα1+α2 Hα1 Hα2 Hα1+2α2 C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C α1+2α2 ε2 α2 α1+α2 ε1 α1

This is type C2.

Define P+ = PC_ and P++ = PC so that P+ is a set of representatives of the orbits of the action of W on P. The fundamental weights are the generators ω1,...,ωn of the 0-module P+ so that C = i=1n 0 ωi, P+ = i=1n 0 ωi, and P++ = i=1n >0 ωi. (TC2 1) The lattice P has -basis ω1,...,ωn and the map P+ P++ λ ρ+λ where ρ = ω1++ωn (TC2 2) is a bijection.

In the case of type C2 the picture is Hα1 Hα2 C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C ε2 = ω2 ω1 ε1 0 Hα1 Hα2 C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C ε2 ε1 ρ The set P+ The set P++ with ω1 = ε1+ε2, α1 = 2ε1, α1 = ε1, ω2 = ε2, α2 = ε2-ε1, α2 = ε2, and R = { ±α1, ±α2, ±(α1+α2), ±(α1+2α2) }.

Let A,A: 𝔥*×𝔥* be a nondegenerate W-invariant symmetric bilinear form on 𝔥*. Any symmetric bilinear form (A,A): 𝔥*×𝔥* can be made into a W-invariant form A,A by defining x,y = wW (wx,wy), for   x,y𝔥*. The simple coroots are α1,...,αn the dual basis to the fundamental weights, ωi, αj = δij. (TC2 3) Define C_ = i=1n 0 αi and C = i=1n <0 αi. (TC2 4) The dominance order is the partial order on 𝔥* given by μλ if μλ+C_. (TC2 5)


Notes and References

An alternate reference for this material is [Bou] (Bourbaki Chpt. IV-VI).


[RR] A. Ram and J. Ramagge, Affine Hecke Algebras,

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