Ideal polytopes for representations of GLn(C)

In this paper we use the superpotential Wq for the flag variety GLn/B and particular coordinate systems that we call 'ideal coordinates for i', to construct polytopes Pλi inside RR+, associated to highest weight representations Vλ of GLn(C). Here i is a reduced expression of the longest element of the Weyl group and R+ is the set of positive roots of GLn. The lattice points of Pλi can be used to encode a basis of the representation Vλ, with the weights encoded in a projection onto the weight polytope. For a specific choice of i, the polytope Pλi is unimodularly equivalent to a Gelfand-Tsetlin polytope. The construction of the polytopes involves tropicalisation of the superpotential, namely Pλi is {Trop(Wtλi)≥0} written in a tropical version of the ideal coordinates for i. Using work of Judd ([13]) we have that there is a unique 'positive' critical point of Wtλ over the field of Puiseux series, which projects onto the centre of mass of the weight polytope. Its coordinates, in terms of the ideal coordinates for i, are positive in the sense of having positive leading term. The remarkable property of our new polytopes relates to the tropical version of this critical point, which, for every choice of i, gives a point in RR+ that lies in the interior of the polytope Pλi. We prove that this tropical critical point is independent of the reduced expression i, and that it is given by a pattern called the 'ideal filling' for λ that was introduced by Judd. Finally, combining these results with work of Rietsch ([22]) relating critical points of the superpotential with Toeplitz matrices, we show that for a totally positive lower-triangular Toeplitz matrix over the field of Puiseux series factorised into simple root subgroups, the valuations of the factors give an ideal filling.