Last update: 17 September 2013
To each of the “types”, etc., there is an associated hyperplane arrangement in The space has the usual Euclidean inner product For each hyperplane in the arrangement we choose two vectors orthogonal to the hyperplane and pointing in opposite directions. This set of chosen vectors is called the root system associated to There is a convention for choosing the lengths of these vectors but we shall not worry about that here.
Choose a chamber (connected component) of For each root we say that is positive if it points toward the same side of the hyperplane as is and negative if points toward the opposite side. It is standard notation to write The positive roots which are associated to hyperplanes which form the walls of are the simple roots The fundamental weights are the vectors in such that Then are the lattice of integral weights and the cone of dominant integral weights, respectively. There is a one-to-one correspondence between the irreducible representations of and the elements of the cone in the lattice
Let be a point in Then the straight line path from to is the map The set is given by where are the path operators introduced in [Lit1995]. These paths might look like They are always piecewise linear and end in a point in
The basics of root systems can be found in [Hum1978]. The reference for the path model of Littelmann is [Lit1995].
This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.
Key words and phrases. Algebraic combinatorics, representations.
Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..