December  2019, 11(4): 575-599. doi: 10.3934/jgm.2019029

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

Dept of Mathematics, University of Manchester, Manchester M13 9PL, UK

* Corresponding author

Dedicated to Darryl Holm on the occasion of his 70th birthday

Received  May 2018 Revised  June 2019 Published  November 2019

This is the first of two companion papers. The joint aim is to study a generalization to higher dimension of the familiar point vortex systems in 2 dimensions. In this paper we classify the momentum polytopes for the action of the Lie group SU(3) on products of copies of complex projective 2-space (a real 4-dimensional manifold). For 2 copies, the momentum polytope is simply a line segment, which can sit in the positive Weyl chamber in a small number of ways. For a product of 3 copies there are 8 different types of generic momentum polytope, and numerous transition polytopes, all of which are classified here. The type of polytope depends on the weights of the symplectic form on each copy of projective space. In the second paper we use techniques of symplectic reduction to study the possible dynamics of interacting generalized point vortices.

The results of this paper can be applied to determine the inequalities satisfied by the eigenvalues of the sum of up to three 3x3 Hermitian matrices where each has a double eigenvalue.

Citation: James Montaldi, Amna Shaddad. Non-Abelian momentum polytopes for products of $ \mathbb{CP}^2 $. Journal of Geometric Mechanics, 2019, 11 (4) : 575-599. doi: 10.3934/jgm.2019029
References:
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M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14 (1982), 1-15.  doi: 10.1112/blms/14.1.1.  Google Scholar

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L. Bedulli and A. Gori, On deformations of Hamiltonian actions, Arch. Math., 88 (2007), 468-480.  doi: 10.1007/s00013-006-1944-y.  Google Scholar

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V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math., 67 (1982), 491-513.  doi: 10.1007/BF01398933.  Google Scholar

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A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math., 76 (1954), 620-630.  doi: 10.2307/2372705.  Google Scholar

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F. C. Kirwan, Convexity properties of the moment mapping. Ⅲ, Invent. Math., 77 (1984), 547-552.  doi: 10.1007/BF01388838.  Google Scholar

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A. Knutson, The symplectic and algebraic geometry of Horn's problem, Linear Alg. Appl., 319 (2000), 61-81.  doi: 10.1016/S0024-3795(00)00220-2.  Google Scholar

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B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ecole Norm. Sup., S'erie 4, 6 (1974), 413-455.   Google Scholar

[9]

J. Montaldi and M. Roberts, Stratification of the momentum map, in preparation. Google Scholar

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J. Montaldi and A. Shaddad, Generalized point vortex dynamics on $\mathbb{CP}^2$, J. Geom. Mechanics, (2019) (this volume). Google Scholar

[11]

J. Montaldi and T. Tokieda, Openness of momentum maps and persistence of extremal relative equilibria, Topology, 42 (2003), 833-844.  doi: 10.1016/S0040-9383(02)00047-2.  Google Scholar

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J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, 222. Birkhäuser Boston, Inc., Boston, MA, 2004. doi: 10.1007/978-1-4757-3811-7.  Google Scholar

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I. Schur, Über eine klasse von mittelbildungen mit anwendungen auf der determinantentheorie (On a class of averaging with application to the theory of determinants), Sitzunsberichte der Berliner Mathematischen Gesellschaft, 22 (1923), 9-20.   Google Scholar

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A. Shaddad, The Classification and Dynamics of the Momentum Polytopes of the SU(3) Action on Points in the Complex Projective Plane with an Application to Point Vortices, Ph.D. thesis, University of Manchester, 2018. Google Scholar

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R. Sjamaar, Convexity properties of the moment mapping re-examined, Advances in Math., 138 (1998), 46-91.  doi: 10.1006/aima.1998.1739.  Google Scholar

show all references

References:
[1]

M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14 (1982), 1-15.  doi: 10.1112/blms/14.1.1.  Google Scholar

[2]

L. Bedulli and A. Gori, On deformations of Hamiltonian actions, Arch. Math., 88 (2007), 468-480.  doi: 10.1007/s00013-006-1944-y.  Google Scholar

[3]

V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math., 67 (1982), 491-513.  doi: 10.1007/BF01398933.  Google Scholar

[4] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1984.   Google Scholar
[5]

A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math., 76 (1954), 620-630.  doi: 10.2307/2372705.  Google Scholar

[6]

F. C. Kirwan, Convexity properties of the moment mapping. Ⅲ, Invent. Math., 77 (1984), 547-552.  doi: 10.1007/BF01388838.  Google Scholar

[7]

A. Knutson, The symplectic and algebraic geometry of Horn's problem, Linear Alg. Appl., 319 (2000), 61-81.  doi: 10.1016/S0024-3795(00)00220-2.  Google Scholar

[8]

B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ecole Norm. Sup., S'erie 4, 6 (1974), 413-455.   Google Scholar

[9]

J. Montaldi and M. Roberts, Stratification of the momentum map, in preparation. Google Scholar

[10]

J. Montaldi and A. Shaddad, Generalized point vortex dynamics on $\mathbb{CP}^2$, J. Geom. Mechanics, (2019) (this volume). Google Scholar

[11]

J. Montaldi and T. Tokieda, Openness of momentum maps and persistence of extremal relative equilibria, Topology, 42 (2003), 833-844.  doi: 10.1016/S0040-9383(02)00047-2.  Google Scholar

[12]

J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, 222. Birkhäuser Boston, Inc., Boston, MA, 2004. doi: 10.1007/978-1-4757-3811-7.  Google Scholar

[13]

I. Schur, Über eine klasse von mittelbildungen mit anwendungen auf der determinantentheorie (On a class of averaging with application to the theory of determinants), Sitzunsberichte der Berliner Mathematischen Gesellschaft, 22 (1923), 9-20.   Google Scholar

[14]

A. Shaddad, The Classification and Dynamics of the Momentum Polytopes of the SU(3) Action on Points in the Complex Projective Plane with an Application to Point Vortices, Ph.D. thesis, University of Manchester, 2018. Google Scholar

[15]

R. Sjamaar, Convexity properties of the moment mapping re-examined, Advances in Math., 138 (1998), 46-91.  doi: 10.1006/aima.1998.1739.  Google Scholar

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 $
Table 1.  Transition values of $ \Gamma_j $ ; similar transitions occur permuting the indices. '$ x\in\text{Wall} $' means that the point $ x $ belongs to a wall of the Weyl chamber. See Figure 5; further details are shown in Section 4.2 and Figures 9-13
condition degeneracy
$\Gamma_1=0$ $a=c_1, \; b=c_2=c_3$
$\Gamma_1=\Gamma_2$ $b\in \text{Wall}, \;c_2=c_3$
$\Gamma_1+\Gamma_2=0$ $a=c_3\, (\in \text{Wall})$
$\Gamma_1=\Gamma_2+\Gamma_3$ $c_1\in \text{Wall}$
$\Gamma_1+\Gamma_2+\Gamma_3=0$ $a=0$
condition degeneracy
$\Gamma_1=0$ $a=c_1, \; b=c_2=c_3$
$\Gamma_1=\Gamma_2$ $b\in \text{Wall}, \;c_2=c_3$
$\Gamma_1+\Gamma_2=0$ $a=c_3\, (\in \text{Wall})$
$\Gamma_1=\Gamma_2+\Gamma_3$ $c_1\in \text{Wall}$
$\Gamma_1+\Gamma_2+\Gamma_3=0$ $a=0$
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