A K_T-deformation of the ring of symmetric functions

TL;DR

This paper introduces a two-parameter deformation of the ring of symmetric functions via equivariant K-homology of Grassmannians, providing positive combinatorial formulas for structure constants using DS pipe dreams.

Contribution

It constructs a new deformation of symmetric functions using equivariant K-homology and derives positive combinatorial formulas for their multiplication coefficients.

Findings

Defines a 2-parameter deformation of symmetric functions.

Provides positive formulas for structure constants.

Connects geometric and combinatorial aspects through DS pipe dreams.

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.