Compactifications of algebraic groups and Schubert calculus

SIN RESUMEN

To further investigate the equivariant cohomology of projective group embeddings, beyond the toric case. Moreover, to describe the algebra H∗T×T (Pϵ(M)) explicitly in terms of generators and relations, using the combinatorial data of the monoid M.

OE1: To construct generalized Schubert polynomials {Ψv,w}: these polynomials will be indexed by a finite set WJ×WJ , a quotient of W×W, which depends on J. When J = ∅, then WJ = W. If J is not empty, then WJ is no longer a group. Using the results of [R3], one can define a descent set DJ (v), for each v ∈ WJ . Our polynomials will include a correction factor coming from the various DJ (v), which takes into account that X is not a projective homogeneous space. OE2: To propose a definition of generalized divided differences operators {∂s}, for each s ∈ WJ . This is based on ideas of Lascoux [L]. Using such operators we construct our polynomials {Ψv,w} by looking at the product of two finite graphs, a procedure that mimics Schubert calculus on flag varieties. OE3: To provide a neat geometric interpretation of our generalized Schubert polynomials as equivariant pushforwards of characteristic classes associated to the boundary divisors of X. This is possible due to the fact that equivariant (co)homology and equivariant Chow (co)homology agree on X [G5]. This should yield the desired algebro-geometric picture of the algebra H∗G×G(X).

Investigacion basica

Disciplinario

CONCURSO ANUAL DE INVESTIGACIÓN

• 75 — Matemáticas puras

Ciencias naturales - Matemáticas - Matemáticas puras

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