A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory

Cristian Lenart ^[1] ; Satoshi Naito ^[2] ; Daisuke Sagaki ^[3]
1. [1] State University of New York
  
  State University of New York
  
  City of Albany, Estados Unidos
2. [2] Tokyo Institute of Technology
  
  Tokyo Institute of Technology
  
  Japón
3. [3] University of Tsukuba
  
  University of Tsukuba
  
  Japón
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Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 3, 2024, págs. 1-44
Idioma: inglés
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Resumen
- We give a Chevalley formula for an arbitrary weight for the torus-equivariant Kgroup of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum K-theory QKT (G/B) of a (finite-dimensional) flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of QKT (G/B). In type An−1, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum K-theory QK(SLn/B); we also obtain very explicit information about the coefficients in the respective Chevalley formula.