Representation theoretical methods in tropical flag varieties
TGMRC Lecture series: October, 2020 -- January, 2021
Introduction
Degeneration method plays an important role in algebraic geometry: it degenerates an algebraic variety to simpler objects (such as a toric variety). The geometric properties of the variety usually degenerate into combinatorial properties of a polyhedral complex. Intuitively this procedure resembles taking a picture of an object from different angles, at a specific angle of vision, some features of the object become visible or invisible. Therefore classifying all reasonable degenerations of a variety becomes an important question.A general procedure of producing reasonable degenerations usually comes together with a basis of the coordinate ring. Groebner theory provides a systematical way of constructing such bases, and the classification problem in this setup is solved by examining a polyhedral object called Groebner fan.
Tropical geometry is a piecewise linear version of algebraic geometry: it associates a polyhedral complex, called a tropical variety, to an embedded projective variety. They are very useful tools in the study of intersection theory, enumerative geometry and mirror symmetry. The tropical varieties turn out to be a subfan of the Groebner fan. When the dimension of the variety is high, it is very difficult to give an explicit description to the polyhedra in the tropical variety. Even for nice varieties such as Grassmannians, only the cases of two-planes are understood.
In this lecture series, I will introduce the (type A) tropical flag varieties, and use representation theory to describe explicitly a maximal dimension cone in it. The algebra, geometry and combinatorics arising from this maximal cone will also be discussed.
Contents
The lecture series consists of the following eight topics (the last one was only briefly mentioned in the lecture).