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Séminaire Lotharingien de Combinatoire, 85B.22 (2021), 12 pp.

# Cristian Lenart, Satoshi Naito and Daisuke Sagaki

# A Combinatorial Chevalley Formula for Semi-Infinite Flag Manifolds and its Applications

**Abstract.**
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. We also discuss the Chevalley formula for partial flag manifolds *G*/*P*. 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*).

Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.

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