Writing Milestone Seminar: Flags, spanning lines, and Grothendieck shenanigans

The flag variety is a well-studied object in combinatorial algebraic geometry. It is a celebrated theorem by Borel that the cohomology ring of the flag variety is isomorphic to the coinvariant algebra. Pawlowski and Rhoades (2017) constructed the space of spanning line configurations X_{n,k}, a quasiprojective variety whose cohomology ring is isomorphic to a generalized coinvariant algebra R_{n,k}. The space X_{n,k} provides a geometric foundation for the combinatorics of Fubini words.

The Grothendieck group K_0 classifies vector bundles over an algebraic variety and has close connections to cohomology. In this talk, we show that the Grothendieck group K_0(X_{n,k}) is isomorphic to the generalized coinvariant algebra R_{n,k}. Along the way, we discuss Schubert and Grothendieck polynomials, Fubini words, and the relevant combinatorics.