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Stable Grothendieck polynomials and $$K$$-theoretic factor sequences. (English) Zbl 1157.14036
Schubert polynomials are the building blocks of several cohomological degeneracy locus formulas. Their role in $$K$$-theory is played by so-called Grothendieck polynomials. In particular, “stable” versions of Schubert and Grothendieck polynomials turn up naturally, for example in degeneracy locus problems associated with certain quiver representations.
The paper under review studies the expansion of stable Grothendieck polynomials in the basis of stable Grothendieck polynomials for partitions. This generalizes the result of Fomin-Green in the cohomological setting, where the analogues of stable Grothendick polynomials for partitions are the Schur polynomials. The existence of such a finite, integer linear combination expansion was proved by Buch, and the sign of the coefficients were determined by Lascoux. In this paper the authors give a new, non-recursive combinatorial rule for the coefficients. Namely, they prove that the coefficients, up to explicit sign, count the number of increasing tableau of a given shape, with an associated word having explicit combinatorial properties stemming from the combinatorics of the 0-Hecke monoid. The main ingredient of the proof is a generalized, so-called Hecke insertion algorithm.
The main application showed in the paper is a $$K$$-theoretic analogue of the factor sequence formula of Buch-Fulton for the cohomological quiver polynomials (of equioriented type A).

MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 05E15 Combinatorial aspects of groups and algebras (MSC2010) 19E08 $$K$$-theory of schemes 05E05 Symmetric functions and generalizations
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