The center of the symmetric group ring plays an important role for mathematics. Less known is that it can be interpreted as a commutative Hecke ring.

Motivated by this observation, in this talk, in analogy with the set of Jucys- Murphy elements we present a construction of a set of ring generators for the Hecke algebra of the Gel'fand pair $(S_{2n},B_n)$, where $B_n$ is the hyperoctahedral subgroup of the symmetric group $S_{2n}$. Along the way we present various applications of our work, ranging from the cohomology ring of the Hilbert scheme of points in the plane to the Weingarten matrices.

This is a joint work with Kursat Aker.