at the

American Institute of Mathematics, Palo Alto, California

organized by

Michael Kapovich, Arun Ram, and Monica Vazirani

It has emerged from recent works of Littelmann-Gaussent, Kapovich-Leeb-Millson, Haines, and others, that Bruhat--Tits buildings play an essential, not yet well-understood role in combinatorial representation theory by providing a geometric realization to existing combinatorial models and linking them to the algebro-geometric tools of representation theory.

In particular the workshop goals include examining and comparing the different approaches to the saturation theorem, with an emphasis on the role of buildings, to get more precise answers (in all types) and improve the proofs, and possibly also make a sensible Horn conjecture in other types.

We further aim to understand the different combinatorial models involved (such as Knutson-Tao honeycombs, MV polytopes, Littelmann path models, canonical bases), provide a dictionary between them, and lay the groundwork to enable researchers to apply these tools toward a host of related problems.

The workshop schedule.

A report on the workshop activities.

- Arun Ram, Introduction to Buildings and Combinatorial Representation Theory: pdf ps
- Michael Kapovich, Overview of connections between buildings and representation theory, and open problems: pdf ps dvi
- Arkady Berenstein, Polytopal models and tropical geometry: pdf ps dvi
- Jenia Tevelev, Tropical Geometry and Affine Buildings: pdf ps dvi
- Stephane Gaussent, LS Galleries and MV Cycles: pdf ps dvi
- Joel Kamnitzer, Mirkovic-Vilonen cycles and polytopes: pdf ps dvi
- David Nadler, Langlands: pdf ps dvi
- Allen Knutson, Honeycombs: pdf ps dvi
- Michael Kapovich, Saturation: pdf ps dvi
- Cristian Lenart, Models for crystals: pdf ps

- Daniele Alessandrini. Tropicalization of group representations.
- Arkady Berenstein
and Andrei
Zelevinsky. Tensor product multiplicities,
canonical bases and totally positive varieties.
*Inventiones Mathematicae*, 143 (1): 77 – 128, 2001. - Anders S. Buch. The saturation conjecture (after A. Knutson and T. Tao).
- Joel Kamnitzer. Mirkovic Vilonen cycles and polytopes.
- M. Kapovich. Generalized triangle inequalities and their applications.
- Allen Knutson and
Terence Tao.
The honeycomb model of GL(
*n*) tensor products I: proof of the saturation conjecture.*Journal of the American Mathematical Society*, 12 (4): 1055 – 1090, 1999. - Sophie
Morier-Genoud.
*Relèvement Géométrique de l'involution de Schützenberger et Applications*. PhD thesis, l'Université Claude Bernard - Lyon 1, 2006. - Arun Ram.
Alcove walks, Hecke algebras, spherical functions,
crystals and column strict tableaux.
*Pure and Applied Mathematics Quarterly*, 2: 135 – 183, 2006. - Guy Rousseau. Euclidean buildings. Lecture notes from Summer School 2004: Non-positively curved geometries, discrete groups and rigidities.

- Notes on
buildings, Chevalley groups, the flag variety, the affine flag variety,
affine Hecke algebras, loop groups, central extensions, and GL(
*n*) (Arun Ram) - Software package for the LS-gallery/alcove path model (Cristian Lenart)

For your entertainment (or challenge).