Root system type $C_2$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 26 June 2012

Root system type $C_2$

An example is when the lattice $P=\mathbb{Z}{\epsilon }_1+\mathbb{Z}{\epsilon }_2$ with $\{{\epsilon }_1,{\mathrm{\&epsilon}}_2\}$ an orthonormal basis of ${𝔥}_{ℝ}^{*}\cong {ℝ}^2$ and $W=\{1,s_1,s_2,s_1{s}_2,s_2{s}_1,s_1{s}_2{s}_1,s_2{s}_1{s}_2,s_1{s}_2{s}_1{s}_2\}$ is the dihedral group of order 8 generated by the reflections $s_1$ and $s_2$ in the hyperplanes $H_{{\alpha }_1}$ and $H_{{\alpha }_2},$ respectively, where $$H_{{\alpha }_1}=\{x\in {𝔥}_{ℝ}^{*}|⟨x,{\epsilon }_1⟩=0\},and{H}_{{\alpha }_2}=\{x\in {𝔥}_{ℝ}^{*}|⟨x,{\epsilon }_2-{\epsilon }_1⟩=0\}.$$

This is type $C_2.$

Define $$P^{+}=P\cap \stackrel{\_}{C}and{P}^{++}=P\cap C$$ so that $P^{+}$ is a set of representatives of the orbits of the action of $W$ on $P.$ The fundamental weights are the generators ${\omega }_1,...,{\omega }_n$ of the ${\mathbb{Z}}_{\ge 0}-$module $P^{+}$ so that $$\begin{array}{ll}C=\sum _{i=1}^n{ℝ}_{\ge 0}{\omega }_i,P^{+}=\sum _{i=1}^n{\mathbb{Z}}_{\ge 0}{\omega }_i,and{P}^{++}=\sum _{i=1}^n{\mathbb{Z}}_{>0}{\omega }_i.& (TC2\; 1)\end{array}$$ The lattice $P$ has $\mathbb{Z}-$basis ${\omega }_1,...,{\omega }_n$ and the map $$\begin{array}{ll}\begin{array}{rcl}{P}^{+}& \to & P^{++}\\ \lambda & \mapsto & \rho +\lambda \end{array}where\rho ={\omega }_1+\cdots +{\omega }_n& (TC2\; 2)\end{array}$$ is a bijection.

In the case of type $C_2$ the picture is $$\begin{array}{cc}\begin{array}{c} \[ H_{{\alpha }_1} \] \[ H_{{\alpha }_2} \] \[ C \] \[ s_{1}C \] \[ s_{2}C \] \[ s_1{s}_{2}C \] \[ s_2{s}_{1}C \] \[ s_1{s}_2{s}_{1}C \] \[ s_2{s}_1{s}_{2}C \] \[ s_1{s}_2{s}_1{s}_{2}C \] \[ {\epsilon }_2={\omega }_2 \] \[ {\omega }_1 \] \[ {\epsilon }_1 \] \[ 0 \]\end{array}& \begin{array}{c} \[ H_{{\alpha }_1} \] \[ H_{{\alpha }_2} \] \[ C \] \[ s_{1}C \] \[ s_{2}C \] \[ s_1{s}_{2}C \] \[ s_2{s}_{1}C \] \[ s_1{s}_2{s}_{1}C \] \[ s_2{s}_1{s}_{2}C \] \[ s_1{s}_2{s}_1{s}_{2}C \] \[ {\epsilon }_2 \] \[ {\epsilon }_1 \] \[ \rho \]\end{array}\\ The\; set{P}^{+}& The\; set{P}^{++}\end{array}$$ with $$\begin{array}{rclrclrcl}{\omega }_1& =& {\epsilon }_1+{\epsilon }_2,& {\alpha }_1& =& 2{\epsilon }_1,& {\alpha }_1^{\vee }& =& {\epsilon }_1,\\ {\omega }_2& =& {\epsilon }_2,& {\alpha }_2& =& {\epsilon }_2-{\epsilon }_1,& {\alpha }_2^{\vee }& =& {\epsilon }_2,\end{array}$$ and $$R=\{\pm {\alpha }_1,\pm {\alpha }_2,\pm ({\alpha }_1+{\alpha }_2),\pm ({\alpha }_1+2{\alpha }_2)\}.$$

Let $⟨\phantom{A},\phantom{A}⟩:{𝔥}_{ℝ}^{*}\times {𝔥}_{ℝ}^{*}\to ℝ$ be a nondegenerate $W-$invariant symmetric bilinear form on ${𝔥}_{ℝ}^{*}.$ Any symmetric bilinear form $(\phantom{A},\phantom{A}):{𝔥}_{ℝ}^{*}\times {𝔥}_{ℝ}^{*}\to ℝ$ can be made into a $W-$invariant form $⟨\phantom{A},\phantom{A}⟩$ by defining $$⟨x,y⟩=\sum _{w\in W}(wx,wy),for\; x,y\in {𝔥}_{ℝ}^{*}.$$ The simple coroots are ${\alpha }_1^{\vee },...,{\alpha }_n^{\vee }$ the dual basis to the fundamental weights, $$\begin{array}{ll}⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}.& (TC2\; 3)\end{array}$$ Define $$\begin{array}{ll}\stackrel{\_}{C^{\vee }}=\sum _{i=1}^n{ℝ}_{\le 0}{\alpha }_i^{\vee }and{C}^{\vee }=\sum _{i=1}^n{ℝ}_{<0}{\alpha }_i^{\vee }.& (TC2\; 4)\end{array}$$ The dominance order is the partial order on ${𝔥}_{ℝ}^{*}$ given by $$\begin{array}{ll}\mu \le \lambda if\mu \in \lambda +\stackrel{\_}{C^{\vee }}.& (TC2\; 5)\end{array}$$

Notes and References

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

References

[RR] A. Ram and J. Ramagge, Affine Hecke Algebras, http://researchers.ms.unimelb.edu.au/~aram@unimelb/publications.html.

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