Ben Brubaker's 2021 Project
A group representation is just a vector space equipped with a nice group action. In combinatorial representation theory, one tries to find "good" bases for these vector spaces, where "good" depends on what one wants to study. If our group is the set of n x n invertible matrices with complex entries, often written GL C, there's a good basis for any finite dimensional representation made from triangular arrays of integers called "Gelfand-Tsetlin" patterns. This project will study functions on these triangular arrays and identities between them. You don't need to have studied representation theory before in order to investigate these questions; rather, we hope that the project motivates you to want to study more representation theory (especially of matrix groups like GL C known as Lie groups) later.
To be more precise, here's an example of a Gelfand-Tsetlin pattern T :
12 10 4 0
T = 10 5 3
It represents one basis vector in a representation of GL C. It turns out that (irreducible, polynomial) representations of this group can be indexed by partitions - sequences of integers m ≥ m ≥ m ≥ m . In this example, the top row tells us the partition: (m , m , m , m ) = (12, 10, 4, 0). And a basis for the representation is given by all triangular arrays of integers with this top row. The numbers below the top row are subject to one condition - each such integer must lie between the two integers above it. So the 10 in the second row is between the 10 and the 12 in the top row, etc.
If you plotted all such Gelfand-Tsetlin patterns as integer points in Euclidean space, they form a polytope. We'll be studying functions on these polytopes and equalities between them. Some further details are presented in the early chapters of this book, but we'll start from the very beginning in the Polymath Jr.