Non-Abelian momentum polytopes for products of $ \mathbb{CP}^2 $

Figure 12.  The transition polytopes with repeated weights around region A

Figure 1.  On the left the roots for SU(3) and the area shaded in pink is the positive Weyl chamber $ \mathfrak{t}^*_+ $. The $ \pm\alpha_i $ are the roots. On the right are shown two orbits of the Weyl group, the black dots show a generic orbit, the blue ones a degenerate orbit

Figure 2.  This shows the plane parametrized by three real numbers $ \lambda_1, \lambda_2, \lambda_3 $ which sum to zero. The orientation is such that $ \lambda_1 $ increases to the top of the diagram. Transpositions of the three numbers correspond to reflections in the blue lines. The pink region is where $ \lambda_1\geq\lambda_2\geq\lambda_3 $. These numbers will be the eigenvalues of a trace zero Hermitian matrix. (Cf. the roots shown in Figure 1)

Figure 3.  The four generic polytopes for the action of $ SU(3) $ on $ \mathbb{CP}^2\times\mathbb{CP}^2 $. In each case $ a $ represents the image of points of the form $ (u, u) $, and $ c $ of points of the form $ (u, u^\perp) $. Notice that all these poytope-segments are parallel to one of the roots (equivalently, orthogonal to one of the walls of the Weyl chamber). Notice that figures (a) and (d) are related by the involution $ * $ of Remark 2, as are figures (b) and (c)

Figure 4.  The three transitional polytopes for the action of $ SU(3) $ on $ \mathbb{CP}^2\times\mathbb{CP}^2 $. See the caption of Figure 3 for explanations of notation, and Remark 5 for discussion. Note that the involution $ * $ exchanges figures (e) and (g) and leaves (f) unchanged

Figure 6.  The generic momentum polytopes: refer to Fig. 5 for the notation

Figure 5.  This shows the parameter plane $ \Gamma_1+\Gamma_2+\Gamma_3=\text{const} $ with const$ {}>0 $. Within the central black triangle all 3 weights are positive. The value of $ \Gamma_2 $ is constant on horizontal lines and increases vertically upwards; variations of the other variables can be deduced from this. The blue lines indicate where the polytope type changes, see Table 1. The sector between the red lines is where $ \Gamma_1\geq\Gamma_2\geq\Gamma_3 $. The generic polytope types are labelled $ A, B, \dots, H $, and illustrated in Fig. 6, and the respective transitions are labelled AB, CE etc., see Fig. 9

Figure 9.  This shows the labels of all 20 transition polytopes with $\Gamma_j\neq0$. Compare with Fig. 5. The transitions denoted D$_0$, DD$_0$, G$_0$ and GG$_0$ arise 'at infinity' in this diagram, and refer to points with $\Gamma_1+\Gamma_2+\Gamma_3=0$; the polytopes are illustrated in Figure 11. The transition between D$_0$ and G$_0$ occurs when $\Gamma_2=\Gamma_1+\Gamma_3=0$.

Figure 10.  This shows the transition B $ \to $ AB $ \to $ A, involving vertex $ c_1 $ moving to the boundary of the Weyl chamber and getting reflected back but leaving an edge 'stuck' to the boundary. See text for further explanation

Figure 11.  Polytopes arising for $ \Gamma_1+\Gamma_2+\Gamma_3=0 $, which implies $ a=0 $. Notice that D$ _0 $ and G$ _0 $ are related by a reflection in the centre line of the Weyl chamber; this is because reversing the signs of the $ \Gamma_j $ converts region G$ _0 $ to D$ _0 $, via the involution $ * $ described in Remark 2. A similar observation relates the polytopes for DD$ _0 $ and GG$ _0 $ (the latter not drawn). See Figure 9 for the regions in parameter space

Figure 13.  The remaining transition polytopes-see Fig. 9 for notation

Figure 7.  Examples showing weights at the fixed points

Figure 8.  Three possibilites for the lower part of polytope G compatible with local information at vertices $ b, c_1, c_2, c_3 $-version (a) is the correct one as shown by considering the local momentum cone at $ g $