Algebra Seminar

Quotients of symmetric polynomial rings deforming the cohomology of the Grassmannian

Darij Grinberg (University of Minnesota)

Wednesday, December 5, 2018 11:15 am
MONT 313 (Storrs)

The cohomology ring of a Grassmannian $Gr(k, n)$ is long known to be a quotient of the ring $S$ of symmetric polynomials in k variables. More precisely, it is the quotient of S by the ideal generated by the $k$ consecutive complete homogeneous symmetric polynomials $h_{n-k,1},\ h_{n-k,2},\ldots,\ h_n$. We propose and begin to study a deformation of this quotient, in which the ideal is replaced by the ideal generated by $h_{n-k,1} - a_1, h_{n-k,2} - a_2, ..., h_n - a_k$ for some $k$ fixed elements $a_1, a_2, ..., a_k$ of the base ring. This generalizes both the classical and the quantum cohomology rings of $Gr(k, n)$. We find two bases for the new quotient, as well as an $S_3$-symmetry of its structure constants and a fairly complicated Pieri rule. We conjecture that the structure constants are nonnegative in an appropriate sense (treating the a_i as signed variables), which suggests a geometric or combinatorial meaning for the quotient.