arXiv subjects

Teresa Lüdenbach

Publications and source records attributed to Teresa Lüdenbach.

2 recordsLinked to original sources

arXiv2024

Mirror symmetry, tropical geometry and representation theory

An ideal filling is a combinatorial object introduced by Judd that amounts to expressing a dominant weight $λ$ of $SL_n$ 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 $SL_n/B$ 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 $GL_n$, we also define a new family of polytopes in $\mathbb{R}^{R_+}$, where $R_+$ denotes the positive roots of $GL_n$, 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 $GL_n/B$ case. Finally, we demonstrate a close relation between our new interpretation of ideal fillings and factorisations of Toeplitz matrices into simple root subgroups.

math.RT↗

arXiv2022

Ideal Polytopes for Representations of $GL_n(\mathbb{C})$

In this paper we use the superpotential for the flag variety $GL_n/B$ and particular coordinate systems that we call ideal coordinates for $\mathbf{i}$, to construct polytopes $\mathcal{P}^{\mathbf{i}}_λ$ inside $\mathbb{R}^{R_+}$, associated to highest weight representations $V_λ$ of $GL_n(\mathbb{C})$. Here $\mathbf{i}$ is a reduced expression of the longest element of the Weyl group and $R_+$ is the set of positive roots of $GL_n$. The lattice points of $\mathcal{P}^{\mathbf{i}}_λ$ can be used to encode a basis of the representation $V_λ$. In particular, for a specific choice of $\mathbf{i}$, the polytope $\mathcal{P}^{\mathbf{i}}_λ$ is unimodularly equivalent to a Gelfand-Tsetlin polytope. The construction of the polytopes involves tropicalisation of the superpotential. Using work of Judd (arXiv:1606.06883) we have that there is a unique positive critical point of the superpotential $\mathcal{W}_{t^λ}$ over the field of Puiseux series. Its coordinates, in terms of the ideal coordinates for $\mathbf{i}$, 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 $\mathbf{i}$, gives a point in $\mathbb{R}^{R_+}$ that lies in the interior of the polytope $\mathcal{P}^{\mathbf{i}}_λ$. We prove that this tropical critical point is independent of the reduced expression $\mathbf{i}$, 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 (arXiv:math/0511124) 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 factorized into simple root subgroups, the valuations of the factors give an ideal filling.

math.RT↗