The dihedral group $G\left(r,r,2\right)$ of order $2r$

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 28 May 2010

The dihedral group $G\left(r,r,2\right)$ of order $2r$

The dihedral group $G\left(r,r,2\right)$ is the group of $2\times 2$ matrices given by $$G\left(r,r,2\right)=\left\{\left(\begin{array}{cc}{\xi }^k& 0\\ 0& {\xi }^{-k}\end{array}\right),\left(\begin{array}{cc}0& {\xi }^k\\ {\xi }^{-k}& 0\end{array}\right)|k=0,1,\dots ,r-1\right\},\text{where }\xi =e^{2\pi i/r}.$$ Conjugation by the matrix $P=$????? shows that $G\left(r,r,2\right)$ is isomorphic to the group of matrices $2\times 2$ matrices given by $$G\left(r,r,2\right)=\left\{\left(\begin{array}{cc}{\xi }^k& 0\\ 0& {\xi }^{-k}\end{array}\right),\left(\begin{array}{cc}0& {\xi }^k\\ {\xi }^{-k}& 0\end{array}\right)|k=0,1,\dots ,r-1\right\},\text{where }\xi =e^{2\pi i/r}.$$

In this form, $G\left(r,r,2\right)$ is the group of symmetries of a regular $r$-gon (embedded in ${ℝ}^2$ with its center at the origin),

with $s_1$ being the reflection in $H_{{\alpha }_2}$ and $s_2$ the reflection in $H_{{\alpha }_2}.$

The generators $t=\left(\begin{array}{cc}\xi & 0\\ 0& {\xi }^{-1}\end{array}\right)$ and $s=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ and the relations $$t^r=1,s^2=1,st=t^{-1}s,$$ form a presentation of $G\left(r,r,2\right).$

The generators $s_1=\left(\begin{array}{cc}0& \xi \\ {\xi }^{-1}& 0\end{array}\right)$ and $s_2=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ and the relations $$\left(s_1{s}_2{s}_1{s}_2\dots \left(m\text{ factors}\right)\right)=\left(s_2{s}_1{s}_2{s}_1\dots \left(m\text{ factors}\right)\right),s_1^2=1,s_2^2=1,$$ form a presentation of $G\left(r,r,2\right).$

The conjugacy classes of $G\left(r,r,2\right)$ are $${𝒞}_1=\left\{1\right\},{𝒞}_{w_0}=\left\{\left(s_1{s}_2{s}_1{s}_2\dots \left(r\text{ factors}\right)\right)\right\}$$ $${𝒞}_{s_1}=\left\{\left(\dots s_1{s}_2\left(k\text{ factors}\right)s_1\left(s_2{s}_1\dots \left(k\text{ factors}\right)\right)\right)|0\le k<r/2\right\},$$ $${𝒞}_{s_2}=\left\{\left(\dots s_1{s}_2\left(k\text{ factors}\right)s_2\left(s_2{s}_1\dots \left(k\text{ factors}\right)\right)\right)|0\le k<r/2\right\},$$and $${𝒞}_k=\left\{{\left(s_1{s}_2\right)}^k,{\left(s_2{s}_1\right)}^k|1\le k<r/2\right\}.$$

The irreducible representations of the dihedral group $G\left(r,r,2\right)$ are given as follows. The one dimensional representations ${\rho }^{++}$ and ${\rho }^{--}$ are given by $${\rho }^{++}\left(s_1\right)=1,{\rho }^{++}\left(s_2\right)=1\text{ and }{\rho }^{--}\left(s_1\right)=-1,{\rho }^{--}\left(s_2\right)=-1,$$ and, if $r$ is even, there are additional one dimensional representations ${\rho }^{+-}$ and ${\rho }^{+-}$ given by The two dimensional representations $S^{\lambda },0<\lambda <r/2,$ are given by $$S^{\lambda }\left(s_2\right)=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\text{ and }{S}^{\lambda }\left(s_1\right)=\left(\begin{array}{cc}0& {\xi }^{\lambda }\\ {\xi }^{-\lambda }& 0\end{array}\right),\text{where }{e}^{2\pi i/r}.$$

		Proof.
	There are four things to show: The ${\rho }^{\lambda }$ are irreducible; The ${\rho }^{\lambda }$ are representations of $G\left(r,r,2\right),$ The ${\rho }^{\lambda }$ are all non-isomorphic, The ${\rho }^{\lambda }$ are a complete set of irreducible representations. It is straightforward to check that ${\rho }^{\lambda }{\left(s_1\right)}^2={\rho }_{\lambda }{\left(s_2\right)}^2=\mathrm{Id}$ for all the given ${\rho }^{\lambda }.$ It remains to check that ${\rho }^{\lambda }{\left(s_1{s}_2\right)}^r=\mathrm{Id}.$ This follows since ${\rho }^{\lambda }{\left(s_1{s}_2\right)}^r=1,$ if $r$ is odd and ${\rho }^{\lambda }$ is one dimensional, ${\rho }^{\lambda }{\left(s_1{s}_2\right)}^r=\pm 1,$ if $r$ is even and ${\rho }^{\lambda }$ is one dimensional, ${\rho }^{\lambda }\left(s_1{s}_2\right)=\left(\begin{array}{cc}{\xi }^{\lambda }& 0\\ 0& {\xi }^{-\lambda }\end{array}\right),$ if ${\rho }^{\lambda }$ is two dimensional. All one dimensional representations are irreducible. Let ${\rho }^{\lambda }$ be a given type of two dimensional representations and let $M_1$ with bsis $\left\{m_1,m_2\right\}$ be the corresponding $G\left(r,r,2\right)$ module. Let $N$ be a nonzero submodule of $M$ and let $n=c_1{m}_1+c_2{m}_2$ be a nonzero vector in $N.$ Suppose $c_1\ne 0.$ Then $$\left(\frac{s_1{s}_2-{\xi }^{-\lambda }}{{\xi }^{\lambda }-{\xi }^{-\lambda }}\right)n=c_1{m}_1\in N,$$ and so $m_1\in N$ and $m_2=s_2{m}_1\in N.$ So $N=M.$ So $M$ is irreducible. Since the values $${\chi }^{\lambda }\left(s_1{s}_2\right)={\xi }^{\lambda }+{\xi }^{-\lambda }=2\cos \left(2\pi \lambda /r\right),1\le \lambda <r/2,$$ are all distinct, the characters of all the two dimensional representations $\rho \lambda ,1\le \lambda <r/2,$ are distinct. Thus these representations are non-isomorphic. If $m$ is odd, $$\sum _{\lambda }{d}_{\lambda }^2=1^2+1^2+\sum _{\lambda =1}^{r/2-1/2}{2}^2=2+4\left(r/2-1/2\right)=2r,$$ and, if $m$ is even, $$\sum _{\lambda }{d}_{\lambda }^2=1^2+1^2+1^2+1^2+\sum _{\lambda =1}^{r/2-1}{2}^2=2+4\left(r/2-1\right)=2r.$$ In either case, ${\sum }_{\lambda }{d}_{\lambda }^2=\left|G\left(r,r,2\right)\right|,$ and so the ${\rho }^{\lambda }$ are a complete set of inequivalent representations of $G\left(r,r,2\right)$ and re irreducible. $\square$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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