We prove that the Gromov width of a coadjoint orbit of the symplectic group through a
regular point
,
lying on some rational line, is at least equal to:
Together with the results of Zoghi and Caviedes concerning the upper
bounds, this establishes the actual Gromov width. This fits in the
general conjecture that for any compact connected simple Lie group
, the Gromov width of its
coadjoint orbit through
is given by the above formula. The proof relies on tools coming from symplectic
geometry, algebraic geometry and representation theory: we use a toric degeneration of
a coadjoint orbit to a toric variety whose polytope is the string polytope arising from a
string parametrization of elements of a crystal basis for a certain representation of the
symplectic group.