We study the isospectral Hilbert scheme $ X_ {n} $, defined as the reduced fiber product of
$(\mathbb {C}^{2})^{n} $ with the Hilbert scheme $ H_ {n} $ of points in the plane $\mathbb …

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 …

Let R n be the ring of coinvariants for the diagonal action of the symmetric group S n. It is
known that the character of R n as a doubly graded S n-module can be expressed using the …

In an earlier paper [14], we showed that the Hilbert scheme of points in the plane H n= Hilb n
(ℂ 2) can be identified with the Hilbert scheme of regular orbits ℂ 2 n//S n. Using this result …

This work contains detailed descriptions of developments in the combinatorics of the space
of diagonal harmonics, a topic at the forefront of current research in algebraic combinatorics …

We survey the proof of a series of conjectures in combinatorics using new results on the
geometry of Hilbert schemes. The combinatorial results include the positivity conjecture for …

We present here a proof that a certain rational function Cn (q, t) which has come to be known
as the “q, t-Catalan” is in fact a polynomial with positive integer coefficients. This has been …

After recalling the basic notions of combinatorics, representation theory, and commutative
algebra, the main material in this book provides links between the study of …

We exhibit a strong link between the Hall algebra HX of an elliptic curve X defined over a
finite field 𝔽l (or, more precisely, its spherical subalgebra U+ X) and Cherednik's double …

In 2008, Haglund et al. formulated a Compositional form of the Shuffle Conjecture of
Haglund et al.. In very recent work, Gorsky and Negut, by combining their discoveries [,,] with …