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CAGE: Philadelphia Area Combinatorics and Alg. Geometry Seminar

Thursday, October 17, 2019 - 3:00pm

Darij Grinberg



University of Pennsylvania


One of the many connections between Grassmannians and combinatorics is cohomological: The cohomology ring of a Grassmannian Gr(k, n) is 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}, ..., h_n. We propose and begin to study a deformation of this quotient, in which the ideal is instead 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, a "rim hook rule" for straightening arbitrary Schur polynomials, and a fairly complicated Pieri rule. We conjecture that the structure constants are nonnegative in an appropriate sense (treating the a_i as signed indeterminate), which suggests a geometric or combinatorial meaning for the quotient.

Proof methods and open questions will be briefly discussed.