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CAGE: Philadelphia Area Combinatorics and Alg. Geometry Seminar

Thursday, October 17, 2013 - 2:30pm

Adriano Garsia

UC San Diego

Location

University of Pennsylvania

MPA Library seminar room

Eugene Gorsky and Andrei Negut [1] have recently put the finishing touches to what may be viewed as the symmetric function side of the general $m,n$ Shuffle conjecture. Earlier, Tatsuyuki Hikita [2] gave a beautiful construction of the combinatorial side as a weighted enumeration of $m,n$-Parking Functions. All these developments are gravid with challenging Combinatorial problems. In this talk I will report on my findings in an effort to translate some of the contents of these remarkable publications in a language that is more accessible to the general combinatorial audience. In particular I have made an effort to state the resulting m,n- Shuffle Conjecture using notation that is as close as possible to the statement of the original Shuffle Conjecture. This translation would not have been possible without the invaluable and continuous help I got from Gorsky and Negut. In the first of these two talks I presented the Combinatorial side and in this second talk I will present the Symmetric Function side and occasionally the Representation Theory side.

[1] E. Gorsky and A. Negut, "Refined knot invariants and Hilbert schemes", arXiv preprint arXiv:1304.3328 (2013).

[2]} T. Hikita, "Affine Springer fibers of type A and combinatorics of diagonal coinvariants", arXiv preprint arXiv:1203.5878 (2012).