The A.B.C.Ds of Schubert calculus

Abstract

We collect Atiyah-Bott Combinatorial Dreams (A•B•C•Ds) in Schubert calculus. One result relates equivariant structure coefficients for two isotropic flag manifolds, with consequences to the thesis of C. Monical. We contextualize using work of N. Bergeron-F. Sottile, S. Billey-M. Haiman, P. Pragacz, and T. Ikeda-L. Mihalcea-I. Naruse. The relation complements a theorem of A. Kresch-H. Tamvakis in quantum cohomology. Results of A. Buch-V. Ravikumar rule out a similar correspondence in K-theory.

Equivariant cohomology, Schubert calculus, and edge labeled tableaux

“…The non-equivariant story is explained in Section 7. We explain the problem in Section 8 together with some recent developments of C. Monical [44] and of the authors [55]. The latter work shows that the combinatorial problems concerning the two spaces are essentially equivalent.…”

“…For the remainder of this section, assume C ν λ,µ is expressed (uniquely) as a polynomial in the variables β i := t i − t i+1 . We now state a few open problems/conjectures introduced in [55], for the special case of Grassmannians.…”