Last update: 17 September 2013

Let $G=GL(n,\u2102)$ and let
$B={B}_{n}$ be the subgroup of upper triangular matrices in $GL(n,\u2102)\text{.}$
A *line bundle* on $G/B$ is a pair $(\mathcal{L},p)$
where $\mathcal{L}$ is an algebraic variety and $p$ is a map (morphism of algebraic varieties)
$$p:\mathcal{L}\u27f6G/B,$$
such that the fibers of $p$ are lines and such that $\mathcal{L}$ is a locally trivial family of lines. In this definition, *fibers*
means the sets ${p}^{-1}\left(x\right)$ for
$x\in G/B$ 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
$(\mathcal{L},p)$ is simply denoted by $\mathcal{L}\text{.}$
Conceptually, a line bundle on $G/B$ means that we are putting a one-dimensional vector space over each point in
$G/B\text{.}$

A *global section* of the line bundle $\mathcal{L}$ is a map (morphism of algebraic varieties)
$$s:G/B\to \mathcal{L}$$
such that $p\circ s$ is the identity map on $G/B\text{.}$
In other words a global section is any possible “right inverse map” to the line bundle.

Each partition $\lambda =({\lambda}_{1},\dots ,{\lambda}_{n})$ determines a character (i.e. 1-dimensional representation) of the group ${T}_{n}$ of diagonal matrices in $GL(n,\u2102)$ via $$\lambda \left(\left(\begin{array}{cccc}{t}_{1}& 0& \cdots & 0\\ 0& {t}_{2}& & \vdots \\ \vdots & & \ddots & 0\\ 0& \dots & 0& {t}_{n}\end{array}\right)\right)={t}_{1}^{{\lambda}_{1}}{t}_{2}^{{\lambda}_{2}}\cdots {t}_{n}^{{\lambda}_{n}}\text{.}$$ Extend this character to be a character of $B={B}_{n}$ by letting $\lambda $ ignore the strictly upper triangular part of the matrix, that is $\lambda \left(u\right)=1,$ for all $u\in {U}_{n}\text{.}$ Let ${\mathcal{L}}_{\lambda}$ be the fiber product $G{\times}_{B}\lambda ,$ i.e. the set of equivalence classes of pairs $(g,c),$ $g\in G,$ $c\in {\u2102}^{*},$ under the equivalence relation $$(gb,c)\sim (g,\lambda \left({b}^{-1}\right)c),\phantom{\rule{2em}{0ex}}\text{for all}\hspace{0.17em}b\in B\text{.}$$ Then ${\mathcal{L}}_{\lambda}=G{\times}_{B}\lambda $ with the map $$\begin{array}{cccc}p:& G{\times}_{B}\lambda & \u27f6& G/B\\ & (g,c)& \u27fc& gB\end{array}$$ is a line bundle on $G/B\text{.}$

The Borel-Weil-Bott theorem says that the irreducible representation ${V}^{\lambda}$ of ${GL}_{n}\left(\u2102\right)$ is $${V}^{\lambda}\cong {H}^{0}(G/B,{\mathcal{L}}_{\lambda}),$$ where ${H}^{0}(G/B,{\mathcal{L}}_{\lambda})$ is the space of global sections of the line bundle ${\mathcal{L}}_{\lambda}\text{.}$

**References**

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..