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1. A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory

   https://doi.org/10.1007/s00029-024-00924-8  

   Lenart, Cristian; Naito, Satoshi; Sagaki, Daisuke (July 2024, Selecta Mathematica)

   NA (Ed.)

   We give a Chevalley formula for an arbitrary weight for the torus-equivariant K-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum K-theory QK_{T}(G/B) of a (finite-dimensional) flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of QK_{T}(G/B). In type A_{n-1}, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum K-theory QK(SL_{n}/B); we also obtain very explicit information about the coefficients in the respective Chevalley formula. 

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2. Back Stable K -Theory Schubert Calculus

   https://doi.org/10.1093/imrn/rnac315  

   Lam, Thomas; Lee, Seung Jin; Shimozono, Mark (November 2022, International Mathematics Research Notices)

   Abstract 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. 

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3. Castelnuovo-Mumford regularity of matrix Schubert varieties

   https://doi.org/10.1007/s00029-024-00959-x  

   Pechenik, Oliver; Speyer, David E; Weigandt, Anna (July 2024, Selecta Mathematica)

   Matrix Schubert varieties are affine varieties arising in the Schubert calculus of the complete flag variety. We give a formula for the Castelnuovo–Mumford regularity of matrix Schubert varieties, answering a question of Jenna Rajchgot. We follow her proposed strategy of studying the highest-degree homogeneous parts of Grothendieck polynomials, which we call Castelnuovo–Mumford polynomials. In addition to the regularity formula, we obtain formulas for the degrees of all Castelnuovo–Mumford polynomials and for their leading terms, as well as a complete description of when two Castelnuovo–Mumford polynomials agree up to scalar multiple. The degree of the Grothendieck polynomial is a new permutation statistic which we call the Rajchgot index; we develop the properties of Rajchgot index and relate it to major index and to weak order. 

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4. Decompositions of Grothendieck Polynomials

   https://doi.org/10.1093/imrn/rnx207  

   Pechenik, Oliver; Searles, Dominic (September 2017, International Mathematics Research Notices)

   null (Ed.)

   Abstract We investigate the long-standing problem of finding a combinatorial rule for the Schubert structure constants in the $$K$$-theory of flag varieties (in type $$A$$). The Grothendieck polynomials of A. Lascoux–M.-P. Schützenberger (1982) serve as polynomial representatives for $$K$$-theoretic Schubert classes; however no positive rule for their multiplication is known in general. We contribute a new basis for polynomials (in $$n$$ variables) which we call glide polynomials, and give a positive combinatorial formula for the expansion of a Grothendieck polynomial in this basis. We then provide a positive combinatorial Littlewood–Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. Our techniques easily extend to the $$\beta$$-Grothendieck polynomials of S. Fomin–A. Kirillov (1994), representing classes in connective $$K$$-theory, and we state our results in this more general context. 

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5. On positivity for the Peterson variety

   Goldin, Rebecca (June 2024, Contemporary Mathematics, American Mathematical Society)

   Tu, Loring W (Ed.)

   We aim in this manuscript to describe a specific notion of geomet- ric positivity that manifests in cohomology rings associated to the flag variety G/B and, in some cases, to subvarieties of G/B. We offer an exposition on the the well-known geometric basis of the homology of G/B provided by Schubert varieties, whose dual basis in cohomology has nonnegative structure constants. In recent work [R. Goldin, L. Mihalcea, and R. Singh, Positivity of Peterson Schubert Calculus, arXiv2106.10372] we showed that the equivariant cohomology of Peterson varieties satisfies a positivity phenomenon similar to that for Schubert calculus for G/B. Here we explain how this positivity extends to this particular nilpotent Hessenberg variety, and offer some open questions about the ingredients for extending positivity results to other Hessenberg varieties. 

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