Combinatorics Seminar
Friday, December 07, 2018 at 4:15pm to 5:15pm
Microsoft Research, MSR First Floor Research Center , One Memorial Drive, Cambridge
SPEAKER: Darij Grinberg (University of Minnesota)
TITLE: Quotients of symmettric polynomial rings deforming the cohomology of the Grassmannian
ABSTRACT:
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 hn-k+1, hn-k+2, ..., hn. We propose and begin to study a deformation of this quotient, in which the ideal is replaced by the ideal generated by hn-k+1 - a1, hn-k+2 - a2, ..., hn - ak for some k fixed elements a1, a2, ..., ak 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 S3-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 ai as signed variables), which suggests a geometric or combinatorial meaning for the quotient.
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