Laws of trigonometry on $\textrm {SU}(3)$

Abstract:

The orbit space of congruence classes of triangles in $SU(3)$ has dimension $8$. Each corner is given by a pair of tangent vectors $(X,Y)$, and we consider the $8$ functions ${\text {tr}}{X^2},i{\text {tr}}{X^3},{\text {tr}}{Y^2},i{\text {tr}}{Y^3},{\text {tr}}XY,i{\text {tr}}{X^2}Y,i{\text {tr}}X{Y^2}$ and ${\text {tr}}{X^2}{Y^2}$ which are invariant under the full isometry group of $SU(3)$. We show that these $8$ corner invariants determine the isometry class of the triangle. We give relations (laws of trigonometry) between the invariants at the different corners, enabling us to determine the invariants at the remaining corners, including the values of the remaining side and angles, if we know one set of corner invariants. The invariants that only depend on one tangent vector we will call side invariants, while those that depend on two tangent vectors will be called angular invariants. For each triangle we then have $6$ side invariants and $12$ angular invariants. Hence we need $18 - 8 = 10$ laws of trigonometry. If we restrict to $SU(2)$, we get the cosine laws of spherical trigonometry. The basic tool for deriving these laws is a formula expressing ${\text {tr}}({\operatorname {exp}}X{\operatorname {exp}}Y)$ in terms of the corner invariants.