Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 17 September 2013

Appendix A

A6. The Borel-Weil-Bott construction

Let $G=GL(n,ℂ)$ and let $B=B_n$ be the subgroup of upper triangular matrices in $GL(n,ℂ)\text{.}$ A line bundle on $G/B$ is a pair $(ℒ,p)$ where $ℒ$ is an algebraic variety and $p$ is a map (morphism of algebraic varieties) $$p:ℒ⟶G/B,$$ such that the fibers of $p$ are lines and such that $ℒ$ 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 $(ℒ,p)$ is simply denoted by $ℒ\text{.}$ 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 $ℒ$ is a map (morphism of algebraic varieties) $$s:G/B\to ℒ$$ 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 $GL(n,ℂ)$ via $$\lambda \left(\left(\begin{array}{cccc}{t}_1& 0& \cdots & 0\\ 0& t_2& & ⋮\\ ⋮& & \ddots & 0\\ 0& \dots & 0& t_n\end{array}\right)\right)=t_1^{{\lambda }_1}{t}_2^{{\lambda }_2}\cdots t_n^{{\lambda }_n}\text{.}$$ 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 ${ℒ}_{\lambda }$ be the fiber product $G{\times }_B\lambda ,$ i.e. the set of equivalence classes of pairs $(g,c),$ $g\in G,$ $c\in {ℂ}^{*},$ under the equivalence relation $$(gb,c)\sim (g,\lambda \left(b^{-1}\right)c),\text{for all}\hspace{0.17em}b\in B\text{.}$$ Then ${ℒ}_{\lambda }=G{\times }_B\lambda$ with the map $$\begin{array}{cccc}p:& G{\times }_B\lambda & ⟶& G/B\\ & (g,c)& ⟼& gB\end{array}$$ is a line bundle on $G/B\text{.}$

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

References

See [FHa1991] and G. Segal’s article in [CMS1995] for further information and references on this very important construction.

Notes and references

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..

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