Cohomology and $K$-theory rings of the space of commuting elements in $SU(2)$

In this paper, we compute explicitly both the $K$-theory and integral cohomology rings of the space of commuting elements in $SU(2)$ via the $K$-theory of its desingularization. We also briefly discuss the different behavior of its cohomology with complex and ${\mathbb{Z}}_2$ coefficients in the context of representation stability and FI-modules.