Quantum K-theory Chevalley formula for flag varieties

From papers

Let G/BG/B be a flag variety with Weyl group WW, let sis_i be a simple reflection, and let QK(G/B)QK(G/B) denote its quantum K-theory ring. For a root β\beta, let QβQ_\beta be the quantum Bruhat operator, and let (β1,,βl)(\beta_1,\dots,\beta_l) be a (ωi)(-\omega_i)-chain of roots. The notation [w][w] denotes the corresponding basis element of QK(G/B)QK(G/B).

Quantum K-theory Chevalley conjecture. For every wWw\in W,

[si][w]=[(1(1Qβ1)(1Qβl))(w)],[s_i]\circ [w]=\left[(1-(1-Q_{\beta_1})\cdots(1-Q_{\beta_l}))(w)\right],

where \circ denotes the product in the ring QK(G/B)QK(G/B).

This conjecture proposes a Chevalley-type formula for quantum K-theory of the flag variety, extending the corresponding quantum cohomology formula. The supplied source does not state whether it has been proved or disproved.

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Sources & referencesView supporting material

Primary source

Cristian Lenart and Alexander Postnikov, “Affine Weyl groups in K-theory and representation theory”, arXiv:math/0309207 (2005).

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