Equivariant -theory of Grassmannians II: the Knutson–Vakil conjecture

Oliver Pechenik; Alexander Yong

doi:10.1112/s0010437x16008186

Equivariant -theory of Grassmannians II: the Knutson–Vakil conjecture

Compositio Mathematica ◽

10.1112/s0010437x16008186 ◽

2017 ◽

Vol 153 (4) ◽

pp. 667-677 ◽

Cited By ~ 4

Author(s):

Oliver Pechenik ◽

Alexander Yong

Keyword(s):

Schubert Calculus ◽

Equivariant Theory

In 2005, Knutson–Vakil conjectured apuzzlerule for equivariant$K$-theory of Grassmannians. We resolve this conjecture. After giving a correction, we establish a modified rule by combinatorially connecting it to the authors’ recently proved tableau rule for the same Schubert calculus problem.

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EQUIVARIANT -THEORY OF GRASSMANNIANS

Forum of Mathematics Pi ◽

10.1017/fmp.2017.4 ◽

2017 ◽

Vol 5 ◽

Cited By ~ 4

Author(s):

OLIVER PECHENIK ◽

ALEXANDER YONG

Keyword(s):

Equivariant Cohomology ◽

Homogeneous Spaces ◽

Acta Math ◽

Schubert Calculus ◽

Duke Math ◽

Jeu De Taquin ◽

Lecture Notes ◽

Equivariant Theory

We address a unification of the Schubert calculus problems solved by Buch [A Littlewood–Richardson rule for the $K$-theory of Grassmannians, Acta Math. 189 (2002), 37–78] and Knutson and Tao [Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J.119(2) (2003), 221–260]. That is, we prove a combinatorial rule for the structure coefficients in the torus-equivariant $K$-theory of Grassmannians with respect to the basis of Schubert structure sheaves. This rule is positive in the sense of Anderson et al. [Positivity and Kleiman transversality in equivariant $K$-theory of homogeneous spaces, J. Eur. Math. Soc.13 (2011), 57–84] and in a stronger form. Our work is based on the combinatorics of genomic tableaux and a generalization of Schützenberger’s [Combinatoire et représentation du groupe symétrique, in Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976, Lecture Notes in Mathematics, 579 (Springer, Berlin, 1977), 59–113] jeu de taquin. Using our rule, we deduce the two other combinatorial rules for these coefficients. The first is a conjecture of Thomas and Yong [Equivariant Schubert calculus and jeu de taquin, Ann. Inst. Fourier (Grenoble) (2013), to appear]. The second (found in a sequel to this paper) is a puzzle rule, resolving a conjecture of Knutson and Vakil from 2005.

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Involution words: Counting problems and connections to Schubert calculus for symmetric orbit closures

Journal of Combinatorial Theory Series A ◽

10.1016/j.jcta.2018.06.012 ◽

2018 ◽

Vol 160 ◽

pp. 217-260 ◽

Cited By ~ 6

Author(s):

Zachary Hamaker ◽

Eric Marberg ◽

Brendan Pawlowski

Keyword(s):

Schubert Calculus ◽

Counting Problems ◽

Orbit Closures ◽

Symmetric Orbit

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Schubert calculus and threshold polynomials of affine fusion

Nuclear Physics B ◽

10.1016/s0550-3213(00)00404-1 ◽

2000 ◽

Vol 584 (3) ◽

pp. 795-809 ◽

Cited By ~ 1

Author(s):

S.E. Irvine ◽

M.A. Walton

Keyword(s):

Schubert Calculus

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Dynamic Pole Assignment and Schubert Calculus

SIAM Journal on Control and Optimization ◽

10.1137/s036301299325270x ◽

1996 ◽

Vol 34 (3) ◽

pp. 813-832 ◽

Cited By ~ 24

Author(s):

M. S. Ravi ◽

Joachim Rosenthal ◽

Xiaochang Wang

Keyword(s):

Pole Assignment ◽

Schubert Calculus

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Schubert Calculus and Schur Functions

Contemporary Geometry ◽

10.1007/978-1-4684-7950-8_18 ◽

1991 ◽

pp. 449-462

Author(s):

Hung-Hsi Wu

Keyword(s):

Schur Functions ◽

Schubert Calculus

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The elliptic Weyl character formula

Compositio Mathematica ◽

10.1112/s0010437x1300777x ◽

2014 ◽

Vol 150 (7) ◽

pp. 1196-1234 ◽

Cited By ~ 9

Author(s):

Nora Ganter

Keyword(s):

Line Bundle ◽

Lie Groups ◽

Theoretical Framework ◽

Flag Variety ◽

Schubert Calculus ◽

Character Formula ◽

Elliptic Cohomology ◽

Image Position ◽

Weyl Character Formula

AbstractWe calculate equivariant elliptic cohomology of the partial flag variety$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G/H$, where$H\subseteq G$are compact connected Lie groups of equal rank. We identify the${\rm RO}(G)$-graded coefficients${\mathcal{E}} ll_G^*$as powers of Looijenga’s line bundle and prove that transfer along the map$$\begin{equation*} \pi \,{:}\,G/H\longrightarrow {\rm pt} \end{equation*}$$is calculated by the Weyl–Kac character formula. Treating ordinary cohomology,$K$-theory and elliptic cohomology in parallel, this paper organizes the theoretical framework for the elliptic Schubert calculus of [N. Ganter and A. Ram,Elliptic Schubert calculus, in preparation].

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Multiplicity-Free Schubert Calculus

Canadian Mathematical Bulletin ◽

10.4153/cmb-2010-032-x ◽

2010 ◽

Vol 53 (1) ◽

pp. 171-186 ◽

Cited By ~ 5

Author(s):

Hugh Thomas ◽

Alexander Yong

Keyword(s):

Algebraic Geometry ◽

Schubert Calculus ◽

Chow Ring ◽

Free Products ◽

Linear Basis ◽

Combinatorial Classification ◽

Schubert Classes ◽

Multiplicity Free

AbstractMultiplicity-free algebraic geometry is the study of subvarieties Y ⊆ X with the “smallest invariants” as witnessed by a multiplicity-free Chow ring decomposition of [Y] ∈ A*(X) into a predetermined linear basis.This paper concerns the case of Richardson subvarieties of the Grassmannian in terms of the Schubert basis. We give a nonrecursive combinatorial classification of multiplicity-free Richardson varieties, i.e., we classify multiplicity-free products of Schubert classes. This answers a question of W. Fulton.

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