2024/11/07 by Lüdenbach, Teresa
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2411.04636
An ideal filling is a combinatorial object introduced by Judd that amounts to expressing a dominant weight λ of SLn as a rational sum of the positive roots in a canonical way, such that the coefficients satisfy a max relation. He proved that whenever an ideal filling has integral coefficients it corresponds to a lattice point in the interior of the string polytope which parametrises the canonical basis of the representation with highest weight λ. The work of Judd makes use of a construction of string polytopes via the theory of geometric crystals, and involves tropicalising the superpotential of the flag variety SLn/B in certain `string' coordinates. He shows that each ideal filling relates to a positive critical point of the superpotential over the field of Puiseux series, through a careful analysis of the critical point conditions. In this thesis we give a new interpretation of ideal fillings, together with a parabolic generalisation. For every dominant weight λ of GLn, we also define a new family of polytopes in ℝR+, where R+ denotes the positive roots of GLn, with one polytope for each reduced expression of the longest element of the Weyl group. These polytopes are related by piecewise-linear transformations which fix the ideal filling associated to λ as a point in the interior of each of these polytopes. Our main technical tool is a new coordinate system in which to express the superpotential, which we call the `ideal' coordinates. We describe explicit transformations between these coordinates and string coordinates in the GLn/B case. Finally, we demonstrate a close relation between our new interpretation of ideal fillings and factorisations of Toeplitz matrices into simple root subgroups.