# Affine Schubert calculus and diagonal coinvariants

Erik Carlsson, UC Davis
Wednesday, May 22, 2019 - 3:30pm
PDL C-401

The diagonal coinvariant algebra \$DR_n\$ is the ring of polynomials in 2n variables \$\mathbb{Q}[x_1,\dots,x_n,y_1,\dots,y_n]\$ modulo the ideal generated by the nonconstant "diagonally symmetric functions," i.e. functions that are symmetric under simultaneously permuting the x and y variables. It is a well-studied vector space whose dimension was shown by Haiman to be \$(n+1)^{n-1}\$. Whereas the single variable version is well-known to be isomorphic to the homology of the complex flag variety, it was observed by Hikita, Gorsky-Mazin, and others that the diagonal coinvariant ring has the same Hilbert series as the homology of a much more intricate space called the "(n,n+1) affine springer fiber," and it was hoped that this fact would lead to a proof of the "shuffle conjecture," which is a combinatorial formula for the decomposition of each bigraded component of \$DR_n\$ into irreducible \$S_n\$-representations, later proved by Mellit and myself. I'll present a more recent result with A. Oblomkov that constructs a particular isomorphism of \$DR_n\$ with the aforementioned homology group, and explain how this leads to an explicit vector basis of \$DR_n\$ by monomials.

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