Algebras Generated by Elements with Given Spectrum and Scalar Sum and Kleinian Singularities

For a certain class of (nonunital) subalgebras of deformed preprojective algebra of affine type we describe their centers as certain deformation of Kleinian singularity and find their PI-degree. These results can be applied to algebras generated by elements with given spectrum and scalar sum.Comment: 18 page

Plethystic identities and mixed Hodge structures of character varieties

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 varieties

Integrality of HLV kernels

We prove that the coefficients of the generating function of Hausel, Letellier, Villegas, and its recent generalization by Carlsson and Villegas, which according to various conjectures should compute mixed Hodge numbers of character varieties and moduli spaces of Higgs bundles of curves of genus $g$ with $n$ punctures, are polynomials in $q$ and $t$ with integer coefficients for any $g,n\geq 0$.Comment: Revised and expanded. Accepted in Duk

Homology of torus knots

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, Gorsky-Negut and Oblomkov-Rasmussen-Shende

Elliptic dilogarithms and parallel lines

We prove Boyd's conjectures relating Mahler's measures and values of L-functions of elliptic curves in the cases when the corresponding elliptic curve has conductor 14

Certain Examples of Deformed Preprojective Algebras and Geometry of Their *-Representations

We consider algebras $e_i \Pi^\lambda(Q) e_i$ obtained from deformed preprojective algebra of affine type $\Pi^\lambda(Q)$ and an idempotent $e_i$ for certain concrete value of the vector $\lambda$ which corresponds to the traces of $-1\in SU(2, C)$ in irreducible representations of finite subgroups of $SU(2, C)$. We give a certain realization of these algebras which allows us to construct the $C^*$-enveloping algebras for them. Some well-known results, including description of four projections with sum 2 happen to be a particular case of this picture.Comment: 17 page

Poincar\'e polynomials of moduli spaces of Higgs bundles and character varieties (no punctures)

Using our earlier results on polynomiality properties of plethystic logarithms of generating series of certain type we show that Schiffmann's formulas for various counts of Higgs bundles over finite fields can be reduced to much simpler formulas conjectured by Mozgovoy. In particular, our result implies the conjecture of Hausel and Rodriguez-Villegas on the Poincar\'e polynomials of twisted character varieties and the conjecture of Hausel and Thaddeus on independence of $E$-polynomials on the degree

Dwork's congruences for the constant terms of powers of a Laurent polynomial

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 satisfies a non-trivial analytic continuation property, similar to what B. Dwork showed for a class of hypergeometric series

Five-term relation and Macdonald polynomials

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.Comment: Some mistakes are corrected and the proof is substantially expanded. Accepted in JCT

A proof of the shuffle conjecture

We present a proof of the compositional shuffle conjecture, which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector space $V_*$ whose degree zero part is the ring of symmetric functions $Sym[X]$ over $\mathbb{Q}(q,t)$. We then extend these operators to an action of an algebra $\tilde{\AA}$ acting on this space, and interpret the right generalization of the $\nabla$ using an involution of the algebra which is antilinear with respect to the conjugation $(q,t)\mapsto (q^{-1},t^{-1})$.Comment: some proofs are expanded. Accepted in JAM