Degeneracy Loci Classes in $K$-theory - Determinantal and Pfaffian Formula -

We prove a determinantal formula and Pfaffian formulas that respectively describe the $K$-theoretic degeneracy loci classes for Grassmann bundles and for symplectic Grassmann and odd orthogonal bundles. The former generalizes Damon–Kempf–Laksov’s determinantal formula and the latter generalize Pragacz–Kazarian’s formula for the Chow ring. As an application, we introduce the factorial $G\mathrm{\Theta }\mathrm{/}G{\mathrm{\Theta }}^{\mathrm{\prime }}$-functions representing the torus equivariant $K$-theoretic Schubert classes of the symplectic and the odd orthogonal Grassmannians, which generalize the (double) theta polynomials of Buch–Kresch–Tamvakis and Tamvakis–Wilson.