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Chow rings of flag varieties have bases of Schubert cycles \sigma_u, indexed by permutations. A major problem of algebraic combinatorics is to give a positive combinatorial formula for the structure constants of this basis. The celebrated Littlewood-Richardson rules solve this problem for special products \sigma_u \cdot \sigma_v where u and v are p-Grassmannian permutations. Building on work of Wyser, we introduce backstable clans to prove such a rule for the problem of computing the product \sigma_u \cdot \sigma_v when u is p-inverse Grassmannian and v is q-inverse Grassmannian. By establishing several new families of linear relations among structure constants, we further extend this result to obtain a positive combinatorial rule for \sigma_u \cdot \sigma_v in the case that u is covered in weak Bruhat order by a p-inverse Grassmannian permutation and v is a q-inverse Grassmannian permutation.
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Positive Configuration Space
Abstract We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply thenonnegative configuration space. This space has a natural stratification bypositive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang–Mills scattering amplitudes, which will be discussed in a sequel.
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- PAR ID:
- 10221246
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Communications in Mathematical Physics
- Volume:
- 384
- Issue:
- 2
- ISSN:
- 0010-3616
- Page Range / eLocation ID:
- p. 909-954
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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