scholarly journals Equivariant Schubert calculus and jeu de taquin

2018 ◽  
Vol 68 (1) ◽  
pp. 275-318 ◽  
Author(s):  
Hugh Thomas ◽  
Alexander Yong
Keyword(s):  
Schubert Calculus ◽  
Jeu De Taquin

scholarly journals EQUIVARIANT -THEORY OF GRASSMANNIANS

2017 ◽  
Vol 5 ◽  
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.

scholarly journals A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus

2009 ◽  
Vol 3 (2) ◽  
pp. 121-148 ◽  
Author(s):  
Hugh Thomas ◽  
Alexander Yong
Keyword(s):  
Schubert Calculus ◽  
Jeu De Taquin

scholarly journals Promotion of Increasing Tableaux: Frames and Homomesies

10.37236/6836 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Oliver Pechenik
Keyword(s):  
Schubert Calculus ◽  
Young Tableaux ◽  
Jeu De Taquin ◽  
Average Value ◽  
K Promotion ◽  
Standard Young Tableaux ◽  
Promotion Operator

A key fact about M.-P. Schützenberger's (1972) promotion operator on rectangular standard Young tableaux is that iterating promotion once per entry recovers the original tableau. For tableaux with strictly increasing rows and columns, H. Thomas and A. Yong (2009) introduced a theory of $K$-jeu de taquin with applications to $K$-theoretic Schubert calculus. The author (2014) studied a $K$-promotion operator $\mathcal{P}$ derived from this theory, but observed that this key fact does not generally extend to $K$-promotion of such increasing tableaux. Here, we show that the key fact holds for labels on the boundary of the rectangle. That is, for $T$ a rectangular increasing tableau with entries bounded by $q$, we have $\mathsf{Frame}(\mathcal{P}^q(T)) = \mathsf{Frame}(T)$, where $\mathsf{Frame}(U)$ denotes the restriction of $U$ to its first and last row and column. Using this fact, we obtain a family of homomesy results on the average value of certain statistics over $K$-promotion orbits, extending a $2$-row theorem of J. Bloom, D. Saracino, and the author (2016) to arbitrary rectangular shapes.

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

2017 ◽  
Vol 153 (4) ◽  
pp. 667-677 ◽  
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.

scholarly journals Schubert calculus and threshold polynomials of affine fusion

2000 ◽  
Vol 584 (3) ◽  
pp. 795-809 ◽  
Author(s):  
S.E. Irvine ◽  
M.A. Walton
Keyword(s):  
Schubert Calculus

scholarly journals Dynamic Pole Assignment and Schubert Calculus

1996 ◽  
Vol 34 (3) ◽  
pp. 813-832 ◽  
Author(s):  
M. S. Ravi ◽  
Joachim Rosenthal ◽  
Xiaochang Wang
Keyword(s):  
Pole Assignment ◽  
Schubert Calculus

scholarly journals The elliptic Weyl character formula

2014 ◽  
Vol 150 (7) ◽  
pp. 1196-1234 ◽  
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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