Tesler Matrices and Lusztig Data

The type A Kostant partition function (KPF) enumerates several families of objects that arise in representation theory and combinatorics, including Tesler matrices, Kostant pictures, Lusztig data, and integral flows. In this paper, we establish logarithmic asymptotics for several classes of KPF by linking them to integer partitions. To this end, we introduce height diagrams, integral and row Tesler matrices. As an application, we obtain the logarithmic asymptotics of regular Tesler matrices. We also compare the known poset structures on the aforementioned KPF objects. We prove that the Lusztig data partial order induced on Kostant pictures refines the natural partial order on those, which we also show to be equivalent to the partial order on Tesler matrices.