2025/05/05 by Ikeda, Takeshi, Iwao, Shinsuke, Naito, Satoshi +1 · 1 citation
#05E05 #14N15 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #K-Theory and Homology (math.KT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2505.02941
We investigate the phenomenon known as ``quantum equals affine'' in the setting of T-equivariant quantum K-theory of the flag variety G/B, as established by Kato for any semisimple algebraic group G. In particular, we focus on the K-Peterson isomorphism between the T-equivariant quantum K-ring QKT(SLn(ℂ)/B) and the T-equivariant K-homology ring K_*T(GrSLn) of the affine Grassmannian, after suitable localizations on both sides. Building on an earlier work by Ikeda, Iwao, and Maeno, we present an explicit algebraic realization of the K-Peterson map via a rational substitution that sends the generators of the quantum K-theory ring to explicit rational expressions in the fundamental generators of K_*T(GrSLn), thereby matching the Schubert bases on both sides. Our approach builds on recent developments in the theory of QKT(SLn(ℂ)/B) by Maeno, Naito, and Sagaki, as well as the theory of K-theoretic double k-Schur functions introduced by Ikeda, Shimozono, and Yamaguchi. This concrete formulation provides new insight into the combinatorial structure of the K-Peterson isomorphism in the equivariant setting. As an application, we establish a factorization formula for the K-theoretic double k-Schur function associated with the maximal k-irreducible k-bounded partition.