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Abstract Grothendieck polynomials were introduced by Lascoux and Schützenberger and play an important role in K-theoretic Schubert calculus. In this paper, we give a new definition of double stable Grothendieck polynomials based on an iterated residue operation. We illustrate the power of our definition by calculating the Grothendieck expansion of K-theoretic Thom polynomials of $${\mathcal {A}}_{2}$$ singularities. We present this expansion in two versions: one displays its stabilization property, while the other displays its expected finiteness property.
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Abstract We consider a pair of quiver varieties $$(X;X^{\prime})$$ related by 3D mirror symmetry, where $$X =T^*{Gr}(k,n)$$ is the cotangent bundle of the Grassmannian of $$k$$-planes of $$n$$-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on $$X \times X^{\prime} $$ (the mother function) whose restrictions to $$X$$ and $$X^{\prime} $$ are the elliptic stable envelopes of those varieties. This implies that the restriction matrices of the elliptic stable envelopes for $$X$$ and $$X^{\prime}$$ are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side.more » « less
