Last update: 17 September 2013
Let and let be the subgroup of upper triangular matrices in A line bundle on is a pair where is an algebraic variety and is a map (morphism of algebraic varieties) such that the fibers of are lines and such that is a locally trivial family of lines. In this definition, fibers means the sets for and lines means one-dimensional vector spaces. For the definition of locally trivial family of lines see [Sha1996] Chapt. VI §1.2. By abuse of language, a line bundle is simply denoted by Conceptually, a line bundle on means that we are putting a one-dimensional vector space over each point in
A global section of the line bundle is a map (morphism of algebraic varieties) such that is the identity map on In other words a global section is any possible “right inverse map” to the line bundle.
Each partition determines a character (i.e. 1-dimensional representation) of the group of diagonal matrices in via Extend this character to be a character of by letting ignore the strictly upper triangular part of the matrix, that is for all Let be the fiber product i.e. the set of equivalence classes of pairs under the equivalence relation Then with the map is a line bundle on
The Borel-Weil-Bott theorem says that the irreducible representation of is where is the space of global sections of the line bundle
See [FHa1991] and G. Segal’s article in [CMS1995] for further information and references on this very important construction.
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..