2015/12/19 by Maria Monks Gillespie, Gillespie, Maria Monks, Jake Levinson +1
Mathematics · #05E99 (Primary) 14N15 #14H30 #14P25 #19M05 (Secondary) #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #K-Theory and Homology (math.KT) #math.AG #math.CO #math.KT #msc:05E99 #msc:14H30 #msc:14N15 #msc:14P25 #msc:19M05
paper · pdf · doi:10.48550/arxiv.1512.06259
arxiv created 2015/12/19 · arxiv updated 2015/12/22
We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves S(λ_\bullet), which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map ω on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of \mathbbRP1, with ω as the monodromy operator. We provide a fast, local algorithm for computing ω without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux, which enumerate the K-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. Using this bijection, we give purely combinatorial proofs of several numerical results involving the K-theory and real geometry of S(λ_\bullet).