Special Unitary Group

The special unitary group ${\it SU}_n(q)$ is the set of $n\times n$ Unitary Matrices with Determinant $+1$ (having $n^2-1$ independent parameters). ${\it SU}(2)$ is Homeomorphic with the Orthogonal Group $O_3^+(2)$. It is also called the Unitary Unimodular Group and is a Lie Group. The special unitary group can be represented by the Matrix

$$U(a,b) = \left[{\matrix{a & b \cr -b^* & a^*}}\right],$$ (1)

where $a^*a+b^*b = 1$ and $a, b$ are the Cayley-Klein Parameters. The special unitary group may also be represented by the Matrix

$$U(\xi,\eta,\zeta) = \left[{\matrix{ e^{i\xi}\cos\eta & e^{i... ...eta \cr -e^{-i\zeta}\sin\eta & e^{-i\xi}\cos\eta\cr}}\right],$$ (2)

or the matrices

$$U_x({\textstyle{1\over 2}}\phi) = \left[\begin{array}{cc}\cos({\textstyle{1\over 2}}\phi) & i\sin({... ...\textstyle{1\over 2}}\phi) & \cos({\textstyle{1\over 2}}\phi)\end{array}\right]$$ (3)

$$U_y({\textstyle{1\over 2}}\beta) = \left[\begin{array}{cc}\cos({\textstyle{1\over 2}}\beta) & \sin({... ...extstyle{1\over 2}}\beta) & \cos({\textstyle{1\over 2}}\beta)\end{array}\right]$$ (4)

$$U_z(\xi) = \left[\begin{array}{cc}e^{i\xi} & 0 \\ 0 & e^{-i\xi} \end{array}\right].$$ (5)

The order $2j+1$ representation is

${U_{p,q}}^{(j)} (\alpha,\beta,\gamma)= \sum_m {{{(-1)}^{m-q-p}\sqrt{(j+p)!(j-p)!(j+q)!(j-q)!}} \over {(j-p-m)!(j+q-m)!(m+p-q)!m!}}$
${}\times e^{iq\alpha} \cos^{2j+q-p-2m}({\textstyle{1\over 2}}\beta)\sin^{p+2m-q}({\textstyle{1\over 2}}\beta) e^{ip\gamma}.\quad$	(6)

The summation is terminated by putting ${1/(-N)!} = 0$. The Character is given by

$$\chi^{(j)}(\alpha) = \left\{\begin{array}{ll} 1+2\cos\alpha + \ldots + 2\cos(j\alpha) ... ...{\textstyle{3\over 2}}\alpha)+\ldots+\cos(j\alpha)] & \mbox{}\end{array}\right.$$

$$= \left\{\begin{array}{ll} {\sin[(j+{\textstyle{1\over 2}})\alpha]\... ...= {\textstyle{1\over 2}}, {\textstyle{3\over 2}}, \ldots.$\ }\end{array}\right.$$ (7)

See also Orthogonal Group, Special Linear Group, Special Orthogonal Group

References

Arfken, G. ``Special Unitary Group, ${\it SU}(2)$ and ${\it SU}(2)$-$O_3^+$ Homomorphism.'' Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 253-259, 1985.

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. ``The Groups ${\it GU}_n(q)$, ${\it SU}_n(q)$, ${\it PGU}_n(q)$, and ${\it PSU}_n(q)={\it U}_n(q)$.'' §2.2 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, p. x, 1985.