A K_T-deformation of the ring of symmetric functions

Abstract: The ring of symmetric functions can be implemented in the homology of \union_{a,b} Gr(a,a+b), the multiplicative structure being defined from the "direct sum" map. There is a natural circle action (simultaneously on all Grassmannians) under which each direct sum map is equivariant. Upon replacing usual homology by equivariant K-homology, we obtain a 2-parameter deformation of the ring of symmetric functions. This ring has a module basis given by Schubert classes. Geometric considerations show that multiplication of Schubert classes has positive coefficients, in an appropriate sense. In this paper we give manifestly positive formulae for these coefficients: they count numbers of "DS pipe dreams'' with prescribed edge labelings.