Bott–Samelson variety

In mathematics, Bott–Samelson varieties were introduced independently by Hansen (1973) and Demazure (1974) (who named them Bott–Samelson varieties) as an algebraic group analogue of the spaces constructed for compact groups by Bott and Samelson (1958, p. 970). They are sometimes desingularizations of Schubert varieties.

A Bott–Samelson variety Z can be constructed as

$P_{1}\times _{B}P_{2}\times _{B}\times \cdots \times _{B}P_{n}/B$

where B is a Borel subgroup of a reductive algebraic group G and the Ps are minimal parabolic subgroups containing B. (An element b of B acts on P on the right as right multiplication by b and acts on P on the left as left multiplication by b⁻¹.) Taking the product of its coordinates gives a proper map from the Bott–Samelson variety Z to the flag variety G/B whose image is a Schubert variety. In some cases this map is birational and gives a desingularization of the Schubert variety.

See also Bott–Samelson resolution.

References

• Bott, Raoul; Samelson, Hans (1958), "Applications of the theory of Morse to symmetric spaces", American Journal of Mathematics, 80: 964–1029, doi:10.2307/2372843, ISSN 0002-9327, JSTOR 2372843, MR 0105694
• Demazure, Michel (1974), "Désingularisation des variétés de Schubert généralisées", Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 7: 53–88, ISSN 0012-9593, MR 0354697
• Hansen, H. C. (1973), "On cycles in flag manifolds", Mathematica Scandinavica, 33: 269–274, ISSN 0025-5521, MR 0376703