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K-State Today

February 12, 2015

Mathematics Colloquium lecture Feb. 12

By Reta McDermott

Mikhail Mazin, Kansas State University, will present "Shuffle Conjectures and Schur Positivity" as part of the Mathematics Department Colloquium lecture series at 2:30 p.m. Thursday, Feb. 12, in 122 Cardwell Hall.

The abstract for the lecture is: The ring of diagonal coinvariants R_n is defined as the quotient of the polynomial ring C[X,Y]$ in two sets of variables X={x_1,...,x_n} and Y={y_1,...,y_n} by the ideal generated by the positive degree invariants of the diagonal S_n action. It was intensively studied by Garsia and Haiman in relation to the Macdonald positivity conjecture. R_n is naturally bigraded and carries an S_n action. Haiman showed that it is isomophic to the space of global sections of the so-called Procesi bundle over the Hilbert scheme of points on the complex plane and related it to the Macdonald polynomials. Shuffle conjecture provides a combinatorial formula for the Frobenius characteristic of R_n in terms of parking functions.

In this talk I will explain the classical relation between S_n-modules and symmetric functions, sketch Haiman's construction of the Procesi bundle, and formulate the Shuffle conjecture and its generalizations. I will then discuss how the combinatorial side of the Shuffle conjecture is related to the affine Symmetric group and the cohomology of certain affine Springer fibers. The talk is partially based on a joint work with Monica Vazirani and Eugene Gorsky (to appear in TAMS).