Mavlyutov's mirror construction and string polytopes of homogeneous spaces

Brion and Alexeev have found an example of a homogeneous space of type $E$ such that the string polytope corresponding to the representation in the space of global sections of the anticanonical bundle is not reflexive, but a rational polytope. Recently Mavlyutov has discovered a new combinatorial duality for rational polytopes which generalizes the polar duality for reflexive polytopes. He has introduced the notion of $Q$-reflexive polytopes and shown that this notion coincides with the notion of usual refelexive polytopes if the dimension is less or equal to 4. Our aim is to explain the Mavlyutov's mirror construction and its relation to toric degenerations of homogeneous spaces via string polytopes. This is a joint work with Makoto Miura.