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 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 Lifting, restricting and sifting integral points on affine homogeneous varieties

2012 ◽  
Vol 148 (6) ◽  
pp. 1695-1716 ◽  
Author(s):  
Alexander Gorodnik ◽  
Amos Nevo
Keyword(s):  
Homogeneous Spaces ◽  
Acta Math ◽  
Upper Bounds ◽  
Lattice Points ◽  
List Type ◽  
Integral Points ◽  
Counting Problem ◽  
Symmetric Varieties ◽  
Almost Prime ◽  
Homogeneous Varieties

AbstractIn [Gorodnik and Nevo,Counting lattice points, J. Reine Angew. Math.663(2012), 127–176] an effective solution of the lattice point counting problem in general domains in semisimpleS-algebraic groups and affine symmetric varieties was established. The method relies on the mean ergodic theorem for the action ofGonG/Γ, and implies uniformity in counting over families of lattice subgroups admitting a uniform spectral gap. In the present paper we extend some methods developed in [Nevo and Sarnak,Prime and almost prime integral points on principal homogeneous spaces, Acta Math.205(2010), 361–402] and use them to establish several useful consequences of this property, including:(1)effective upper bounds on lifting for solutions of congruences in affine homogeneous varieties;(2)effective upper bounds on the number of integral points on general subvarieties of semisimple group varieties;(3)effective lower bounds on the number of almost prime points on symmetric varieties;(4)effective upper bounds on almost prime solutions of congruences in homogeneous varieties.

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 Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians

2013 ◽  
Vol 149 (9) ◽  
pp. 1569-1582 ◽  
Author(s):  
David Anderson ◽  
Edward Richmond ◽  
Alexander Yong
Keyword(s):  
Eigenvalue Problem ◽  
Recent Work ◽  
Equivariant Cohomology ◽  
Eigenvalue Problems ◽  
Schubert Calculus ◽  
Hermitian Matrices ◽  
Common Features ◽  
Saturation Theorem ◽  
The Common

AbstractThe saturation theorem of Knutson and Tao concerns the nonvanishing of Littlewood–Richardson coefficients. In combination with work of Klyachko, it implies Horn’s conjecture about eigenvalues of sums of Hermitian matrices. This eigenvalue problem has a generalization to majorized sums of Hermitian matrices, due to S. Friedland. We further illustrate the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians. Our main result gives a Schubert calculus interpretation of Friedland’s problem, via equivariant cohomology of Grassmannians. In particular, we prove a saturation theorem for this setting. Our arguments employ the aforementioned work together with recent work of H. Thomas and A. Yong.

scholarly journals Mod p equivariant cohomology of homogeneous spaces

2000 ◽  
Vol 147 (1) ◽  
pp. 95-105 ◽  
Author(s):  
Katsuhiko Kuribayashi
Keyword(s):  
Equivariant Cohomology ◽  
Homogeneous Spaces ◽  
Mod P

Integration over homogeneous spaces for classical Lie groups using iterated residues at infinity

2014 ◽  
Vol 12 (4) ◽  
Author(s):  
Magdalena Zielenkiewicz
Keyword(s):  
Group Action ◽  
Equivariant Cohomology ◽  
Homogeneous Spaces ◽  
Holomorphic Functions ◽  
Characteristic Classes ◽  
Integration Formula ◽  
Natural Representation ◽  
Several Variables ◽  
Special Cases ◽  
Iterated Residues

AbstractUsing the Berline-Vergne integration formula for equivariant cohomology for torus actions, we prove that integrals over Grassmannians (classical, Lagrangian or orthogonal ones) of characteristic classes of the tautological bundle can be expressed as iterated residues at infinity of some holomorphic functions of several variables. The results obtained for these cases can be expressed as special cases of one formula involving the Weyl group action on the characters of the natural representation of the torus.

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