The $B_{\text{dR}}^{+}$-affine Grassmannian

2020 ◽  
pp. 169-181
Keyword(s):  
Affine Grassmannian

scholarly journals Schubert Polynomials and $k$-Schur Functions

10.37236/4139 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Carolina Benedetti ◽  
Nantel Bergeron
Keyword(s):  
Finite Type ◽  
Schur Functions ◽  
Type A ◽  
Bruhat Order ◽  
Schur Function ◽  
Quasisymmetric Functions ◽  
Affine Grassmannian ◽  
Schubert Polynomial ◽  
Schubert Polynomials ◽  
General Program

The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood combinatorially from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's $r$-Bruhat order, along with certain operators associated to this order. Then, we connect this poset with a graph on dual $k$-Schur functions given by studying the affine grassmannian order of  Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual $k$-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem. This is the first step of our more general program of showing combinatorially  the positivity of the multiplication of a dual $k$-Schur function by a Schur function.

scholarly journals Peterson Isomorphism in K-theory and Relativistic Toda Lattice

2018 ◽  
Vol 2020 (19) ◽  
pp. 6421-6462 ◽  
Author(s):  
Takeshi Ikeda ◽  
Shinsuke Iwao ◽  
Toshiaki Maeno
Keyword(s):  
Symmetric Functions ◽  
Toda Lattice ◽  
Flag Variety ◽  
The Other ◽  
Initial Condition ◽  
Birational Morphism ◽  
Affine Grassmannian ◽  
Other Hand ◽  
K Theory ◽  
Relativistic Toda Lattice

Abstract The K-homology ring of the affine Grassmannian of $SL_{n}(\mathbb{C})$ was studied by Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf subring of the ring of symmetric functions. On the other hand, for the quantum K-theory of the flag variety $F\,\! l_{n}$, Kirillov and Maeno provided a conjectural presentation based on the results obtained by Givental and Lee. We construct an explicit birational morphism between the spectrums of these two rings. Our method relies on Ruijsenaars’s relativistic Toda lattice with unipotent initial condition. From this result, we obtain a K-theory analogue of the so-called Peterson isomorphism for (co)homology. We provide a conjecture on the detailed relationship between the Schubert bases, and, in particular, we determine the image of Lenart–Maeno’s quantum Grothendieck polynomial associated with a Grassmannian permutation.

scholarly journals ON A REDUCEDNESS CONJECTURE FOR SPHERICAL SCHUBERT VARIETIES AND SLICES IN THE AFFINE GRASSMANNIAN

2017 ◽  
Vol 23 (3) ◽  
pp. 707-722 ◽  
Author(s):  
JOEL KAMNITZER ◽  
DINAKAR MUTHIAH ◽  
ALEX WEEKES
Keyword(s):  
Schubert Varieties ◽  
Affine Grassmannian

Steenrod operators, the Coulomb branch and the Frobenius twist

2021 ◽  
Vol 157 (11) ◽  
pp. 2494-2552
Author(s):  
Gus Lonergan
Keyword(s):  
Dual Group ◽  
Coulomb Branch ◽  
Affine Grassmannian ◽  
Steenrod Operations ◽  
Fundamental Relationship ◽  
Theoretic Version

Abstract We observe a fundamental relationship between Steenrod operations and the Artin–Schreier morphism. We use Steenrod's construction, together with some new geometry related to the affine Grassmannian, to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We also demonstrate the corresponding result for the $K$ -theoretic version of the quantum Coulomb branch. At the end of the paper, we investigate what our ideas produce on the categorical level. We find that they yield, after a little fiddling, a construction which corresponds, under the geometric Satake equivalence, to the Frobenius twist functor for representations of the Langlands dual group. We also describe the unfiddled answer, conditional on a conjectural ‘modular derived Satake’, and, though it is more complicated to state, it is in our opinion just as neat and even more compelling.

scholarly journals Pursuing the double affine Grassmannian II: Convolution

2012 ◽  
Vol 230 (1) ◽  
pp. 414-432 ◽  
Author(s):  
Alexander Braverman ◽  
Michael Finkelberg
Keyword(s):  
Affine Grassmannian

scholarly journals Projectivity of the Witt vector affine Grassmannian

2016 ◽  
Vol 209 (2) ◽  
pp. 329-423 ◽  
Author(s):  
Bhargav Bhatt ◽  
Peter Scholze
Keyword(s):  
Witt Vector ◽  
Affine Grassmannian

DIFFERENTIAL OPERATORS ON AND THE AFFINE GRASSMANNIAN

2014 ◽  
Vol 14 (3) ◽  
pp. 493-575 ◽  
Author(s):  
Victor Ginzburg ◽  
Simon Riche
Keyword(s):  
Algebraic Group ◽  
Affine Space ◽  
Equivariant Cohomology ◽  
Differential Operators ◽  
Dual Group ◽  
Intertwining Operators ◽  
Verma Modules ◽  
Reductive Algebraic Group ◽  
Affine Grassmannian ◽  
Basic Affine

We describe the equivariant cohomology of cofibers of spherical perverse sheaves on the affine Grassmannian of a reductive algebraic group in terms of the geometry of the Langlands dual group. In fact we give two equivalent descriptions: one in terms of $\mathscr{D}$-modules of the basic affine space, and one in terms of intertwining operators for universal Verma modules. We also construct natural collections of isomorphisms parameterized by the Weyl group in these three contexts, and prove that they are compatible with our isomorphisms. As applications we reprove some results of the first author and of Braverman and Finkelberg.

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