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تسجيل مستخدم جديدA Chevalley formula for semi-infinite flag manifolds and quantum K-theory (Extended abstract)
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We give a combinatorial Chevalley formula for an arbitrary weight, in the torus-equivariant K-theory of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for anti-dominant fundamental weights in the (small) torus-equivariant quantum K-theory of the flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of the mentioned quantum K-theory. Moreover, in type A, we prove that the so-called quantum Grothendieck polynomials indeed represent Schubert classes in the (non-equivariant) quantum K-theory of the corresponding flag manifold.
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We give a combinatorial Chevalley formula for an arbitrary weight, in 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 anti
-dominant fundamental weights in the (small) torus-equivariant quantum $K$-theory $QK_T(G/B)$ of the flag manifold $G/B$; this has been a longstanding conjecture about the multiplicative structure of $QK_T(G/B)$. Moreover, in type $A_{n-1}$, we prove that the so-called quantum Grothendieck polynomials indeed represent 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.
We propose a definition of equivariant (with respect to an Iwahori subgroup) $K$-theory of the formal power series model $mathbf{Q}_{G}$ of semi-infinite flag manifold and prove the Pieri-Chevalley formula, which describes the product, in the $K$-the
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We prove a Pieri-Chevalley formula for anti-dominant weights and also a Monk formula in the torus-equivariant $K$-group of the formal power series model of semi-infinite flag manifolds, both of which are described explicitly in terms of semi-infinite
Lakshmibai-Seshadri paths (or, equivalently, quantum Lakshmibai-Seshadri paths). In view of recent results of Kato, these formulas give an explicit description of the structure constants for the Pontryagin product in the torus-equivariant $K$-group of affine Grassmannians and that for the quantum multiplication of the torus-equivariant (small) quantum $K$-group of finite-dimensional flag manifolds. Our proof of these formulas is based on standard monomial theory for semi-infinite Lakshmibai-Seshadri paths, which is established in our previous work, and also uses a string property of Demazure-like subsets of the crystal basis of a level-zero extremal weight module over a quantum affine algebra.
We prove an explicit inverse Chevalley formula in the equivariant $K$-theory of semi-infinite flag manifolds of simply-laced type. By an inverse Chevalley formula, we mean a formula for the product of an equivariant scalar with a Schubert class, expr
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In this paper, we give an explicit formula of Chevalley type, in terms of the Bruhat graph, for the quantum multiplication with the class of the line bundle associated to the anti-dominant minuscule fundamental weight $- varpi_{k}$ in the torus-equiv
ariant quantum $K$-group of the partial flag manifold $G/P_{J}$ (where $J = I setminus {k}$) corresponding to the maximal (standard) parabolic subgroup $P_{J}$ of minuscule type in type $A$, $D$, $E$, or $B$. This result is obtained by proving a similar formula in a torus-equivariant $K$-group of the semi-infinite partial flag manifold $mathbf{Q}_{J}$ of minuscule type, and then by making use of the isomorphism between the torus-equivariant quantum $K$-group of $G/P_{J}$ and the torus-equivariant $K$-group of $mathbf{Q}_{J}$, recently established by Kato.
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