Puzzles and (equivariant) cohomology of Grassmannians

By Allen Knutson, Terence Tao. 2001

252 citations recorded by the source corpus.

We generalize our puzzle formula for ordinary Schubert calculus on Grassmannians, to a formula for the T-equivariant Schubert calculus. The structure constants to be calculated are polynomials in {y_{i+1} - y_i}; they were shown (abstractly) to have positive coefficients in [Graham] math.AG/9908172. Our formula is the first to be manifestly positive in this sense. In particular this gives a new and self-contained proof of the ordinary puzzle formula, by an induction backwards from the "most equivariant" case. The proof of the formula is mostly combinatorial, but requires no prior combinatorics, and only a modicum of equivariant cohomology (which we include). This formula is closely related to the one in [Molev-Sagan] q-alg/9707028 for multiplying factorial Schur functions in three sets of variables, although their rule does not give a positive formula in the sense of [Graham]. We include a cohomological interpretation of this problem, and a puzzle formulation for it.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0846da36b5b8c328d9c9e75d719f33d7862c181a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

A Littlewood–Richardson rule for two-step flag varieties (2009, background): “For instance, in recent years, new Littlewood–Richardson rules for the Grassmannians have enabled Klyachko, Knutson, Tao, Woodward and their collaborators to resolve long standing problems such as Horn’s Conjecture and the Saturation Conjecture (see [15] and [16]).”
k-Schur Functions and Affine Schubert Calculus (2013, methodology): “Robinson–Schensted algorithm, jeu-de-taquin, the plactic monoid (see for example [140]), crystal bases [128], and puzzles [74], now underlies Schubert calculus and in particular produces a direct formula for the Littlewood–Richardson coefficients.”
Toric topology (2012, background): “We refer the reader to [30], [31], [48], [53], [85] on this topic.”
The honeycomb model of _{}(ℂ) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone (2001, background): “(A direct proof will appear in [ KT1 ], in turn giving an independent proof of the honeycomb rule.) First we need a lemma on honeycombs.”
A positive Monk formula in the S1‐equivariant cohomology of type A Peterson varieties (2009, methodology): “In the case of Gr(k, C ) or F ags(C), the equivariant Schubert classes give precisely such a basis, thus allowing for effective use of the GKM theory in both classical and modern Schubert calculus [23].”
Double Schubert polynomials for the classical groups (2008, background): “Let denote the Bruhat–Chevalley ordering on W ; then ev ∈ Xw if and only if w v. Proposition 2.1.”
Flag arrangements and triangulations of products of simplices (2006, methodology): “Can we use the complete triangulations and fine mixed subdivisions to understand more subtle geometric questions about flags? Does the geometry of flags tell us something new about triangulations of products of simplices, and their multiple appearances in tropical geometry, optimization, and other s”
K-classes for matroids and equivariant localization (2010, background): “The reader may want to compare the description of equivariant cohomology in [13].”
Poset pinball, GKM-compatible subspaces, and Hessenberg varieties (2010, background): “the product structure of equivariant cohomology rings (see for instance [18,20,22,24,35]).”
Littlewood-Richardson polynomials (2007, unknown): “We also use the Littlewood–Richardson polynomials to describe the multiplication rule in the algebra of the Casimir elements for the general linear Lie algebra in the basis of the quantum immanants constructed by A. Okounkov and G. Olshanski.”