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| Bott–Samelson varieties | |
|---|---|
| Name | Bott–Samelson varieties |
| Founder | Raoul Bott, Hervé Samelson |
Bott–Samelson varieties are geometric objects arising in the intersection of Raoul Bott and Hervé Samelson's work on homotopy theory and the theory of Schubert calculus associated to Hermann Weyl groups and Élie Cartan's theory of symmetric spaces. They provide desingularizations of Schubert varietys in flag varieties related to Wilhelm Weyl's classification and connect to constructions by Jean-Pierre Serre, Armand Borel, and Koszul in representation theory. Their study involves tools from Alexander Grothendieck's scheme theory, Henri Cartan's topology of Lie groups, and techniques developed by Kazhdan and Lusztig for singularity analysis.
Bott–Samelson varieties were introduced in the context of the cohomology of compact Lie group flag manifolds by Raoul Bott and Hervé Samelson and were later developed in algebraic geometry by Jean-Pierre Serre and Armand Borel to desingularize Schubert varietys inside flag varieties associated with Kostant and Samelson's work. They play a central role in the study of the Bruhat decomposition introduced in the work of Francis Bruhat and the combinatorics of Coxeter group elements studied by H.S.M. Coxeter and Marcel Berger. Their utility spans connections to Hecke algebras, the Borel–Weil theorem of André Weil, and geometric representation theory related to George Lusztig.
A Bott–Samelson variety is built from a sequence of simple reflections in a Coxeter group associated to a reductive algebraic group such as GL_n, SL_n, Sp_{2n}, or SO_n. Given a reduced word for an element of the Weyl group studied by Hermann Weyl and Élie Cartan, one forms an iterated fibre bundle of projective lines or minimal parabolic subgroups akin to constructions used by Claude Chevalley and Armand Borel. The iterated bundle construction uses parabolic subgroup data from Chevalley's theory and maps naturally to the full flag variety studied by Élie Cartan and Hermann Weyl, providing a resolution of singularities in the sense of Alexander Grothendieck and techniques of Heisuke Hironaka.
Topologically, Bott–Samelson varieties are smooth projective varieties whose cell decompositions reflect the Bruhat decomposition of flag varieties analyzed by Frédéric Bruhat and Tits in the theory of buildings. Their geometry is governed by intersection properties akin to those in the work of René Thom and Jean Leray on fibre bundles and by characteristic classes considered by Raoul Bott and Atiyah in K-theory and Michael Atiyah's index theory. Singularities of Schubert varieties resolved by Bott–Samelson constructions relate to the stratifications studied by George Kempf and David Mumford in geometric invariant theory and to perverse sheaves developed by Alexander Beilinson and Joseph Bernstein.
Cohomology rings of Bott–Samelson varieties admit explicit presentations that mirror Schubert calculus formulas of Hermann Schubert and are computed using divided difference operators introduced by Isaac Newton's successors and formalized by Demazure. These presentations connect to Hecke algebra actions explored by Kazhdan and Lusztig and to the Borel–Moore homology frameworks of Armand Borel and Jean-Pierre Serre. Equivariant cohomology descriptions use localization techniques developed by Berline and Vergne and link to computations in equivariant K-theory by Graham and Knutson.
Bott–Samelson varieties appear in geometric representation theory related to the Beilinson–Bernstein localization of Harish-Chandra modules and to the categorification programs of Kazhdan and Lusztig. They are used in the study of quantum cohomology influenced by work of Maxim Kontsevich and in the theory of canonical bases developed by Lusztig and Kashiwara. Connections extend to mirror symmetry as investigated by Kontsevich and Strominger–Yau–Zaslow paradigms, to moduli problems in the spirit of David Mumford and Nigel Hitchin, and to combinatorial representation theory such as Young tableau theory shaped by Alfred Young.
For classical groups like SL_2, SL_3, and GL_n, Bott–Samelson varieties can be described as iterated projective bundles of P^1s and partial flag varieties studied by Élie Cartan and Hermann Weyl. Explicit coordinates follow constructions by Chevalley and matrix parametrizations familiar from Carl Friedrich Gauss's legacy in linear algebra and from work on total positivity by George Lusztig. Low-dimensional examples relate to minimal resolutions of surface singularities studied by Du Val and to combinatorial models used by Richard Stanley.
Variants include Bott–Samelson resolutions adapted to Kac–Moody groups investigated by Victor Kac and alternatives in equivariant K-theory exploited by Andrei Okounkov. Other extensions appear in affine Grassmannian contexts linked to I. G. Macdonald polynomials and to the geometric Satake correspondence developed by Lusztig and Ginzburg. Recent work connects these constructions to categorification programs of Mikhail Khovanov and to cluster algebra structures akin to those of Fomin and Zelevinsky.