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We study the back stable $K$-theory Schubert calculus of the infinite flag variety. We define back stable (double) Grothendieck polynomials and double $K$-Stanley functions and establish coproduct expansion formulae. Applying work of Weigandt, we extend our previous results on bumpless pipedreams from cohomology to $K$-theory. We study finiteness and positivity properties of the ring of back stable Grothendieck polynomials, and divided difference operators in $K$-homology.
We study the back stable Schubert calculus of the infinite flag variety. Our main results are: 1) a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part; 2) a novel definition of do
We first define an action of the double coinvariant algebra $DR_n$ on the homology of the affine flag variety $widetilde{Fl}_n$ in type $A$, and use affine Schubert calculus to prove that it preserves the image of the homology of the rational $(n,m)$
The cohomology of the affine flag variety of a complex reductive group is a comodule over the cohomology of the affine Grassmannian. We give positive formulae for the coproduct of an affine Schubert class in terms of affine Stanley classes and finite
The Murnaghan-Nakayama rule expresses the product of a Schur function with a Newton power sum in the basis of Schur functions. We establish a version of the Murnaghan-Nakayama rule for Schubert polynomials and a version for the quantum cohomology rin
The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant weight, it is pr