The non-commutative five-term relation $T\_{1,0} T\_{0,1}=T\_{0,1} T\_{1,1} T\_{1,0}$ is shown to hold
for certain operators acting on symmetric functions. The "generalized recursion"
conjecture of Bergeron and Haiman is a corollary of this result. …
We prove that the coefficients of the generating function of Hausel, Letellier, and
Rodriguez-Villegas and its recent generalization by Carlsson and Rodriguez-Villegas,
which according to various conjectures should compute mixed Hodge numbers of …
We present a proof of the compositional shuffle conjecture by Haglund, Morse and Zabrocki
[Canad. J. Math., 64 (2012), 822-844], which generalizes the famous shuffle conjecture
for the character of the diagonal coinvariant algebra by Haglund, …
Using the method of Elias-Hogancamp and combinatorics of toric braids we give an explicit formula for the triply graded Khovanov-Rozansky homology of an arbitrary torus knot, thereby proving some of the conjectures of Aganagic-Shakirov, Cherednik, …
I demonstrate how certain identities for Macdonald's polynomials established by Garsia, Haiman and Tesler, together with the conjecture of Hausel, Letellier and Villegas imply explicit relations between mixed Hodge polynomials of different character …
The Dyck path algebra construction of Carlsson and Mellit from arXiv:1508.06239 is interpreted as a representation of "the positive part" of the group of toric braids. Then certain sums over $(m,n)$-parking functions are related to evaluations of …
We show that values of finite hypergeometric functions defined over Q correspond to point counting results on explicit varieties defined over finite fields.
In this paper, we prove that the constant terms of powers of a Laurent polynomial satisfy certain congruences modulo prime powers. As a corollary, the generating series of these numbers considered as a function of a p-adic variable admits an analytic …
We conjecture that derived categories of coherent sheaves on fake projective $n$-spaces have a semi-orthogonal decomposition into a collection of exceptional objects and a category with vanishing Hochschild homology. We prove this for fake projective …