2023/04/05 by Can Yaylali, Yaylali, Can
Mathematics · #14C15 (Primary) 14L30 #14M15 (Secondary) #Advanced Combinatorial Mathematics #Affine transformation #Algebra over a field #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Cohomology #Computer science #Equivariant cohomology #Equivariant map #FOS: Mathematics #Flag (linear algebra) #Generalized flag variety #Homotopy #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #K-theory (physics) #Lie group #Mathematics #Physics #Pure mathematics #Ring (chemistry) #Spectrum (functional analysis) #Stack (abstract data type) #Variety (cybernetics)
paper · pdf · doi:10.48550/arxiv.2304.02288
openalex publication_date 2023/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We use the construction of the stable homotopy category by Khan-Ravi to calculate the integral T-equivariant K-theory spectrum of a flag variety over an affine scheme, where T is a split torus associated to the flag variety. More precisely, we show that the T-equivariant K-theory ring spectrum of a flag variety is decomposed into a direct sum of K-theory spectra of the classifying stack BT indexed by the associated Weyl group. We also explain how to relate these results to the motivic world and deduce classical results for T-equivariant intersection theory and K-theory of flag varieties.\par For this purpose, we analyze the motive of schemes stratified by affine spaces with group action, that preserves these stratifications. We work with cohomology theories, that satisfy certain vanishing conditions, which are satisfied for example by motivic cohomology and K-Theory.