Related papers: A kicking basis for the two-column Garsia-Haiman m…

In Surveys in Differential Geometry, Volume 7, published in 2002 and Philosophical Transactions of the Royal Society A, Volume 359, published in 2001, Sir Michael Atiyah introduced what is known as the Atiyah problem on configurations of…

Let $k \leq n$ be nonnegative integers and let $\lambda$ be a partition of $k$. S. Griffin recently introduced a quotient $R_{n,\lambda}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ in $n$ variables which simultaneously generalizes…

The modified Macdonald polynomials, introduced by Garsia and Haiman (1996), have many astounding combinatorial properties. One such class of properties involves applying the related $\nabla$ operator of Bergeron and Garsia (1999) to basic…

The main goal of this paper is to present an algorithm bounding the dimension of a linear system of curves of given degree (or monomial basis) with multiple points in general position. As a result we prove the Hirschowitz--Harbourne…

Gerstenhaber showed in 1961 that any commuting pair of n x n matrices over a field k generates a k-algebra A of k-dimension \leq n. A well-known example shows that the corresponding statement for 4 matrices is false. The question for 3…

A multivariable hypergeometric-type formula for raising operators of the Macdonald polynomials is conjectured. It is proved that this agrees with Jing and Jozefiak's expression for the two-row Macdonald polynomials, and also with Lassalle…

Let $R$ be the power series ring or the polynomial ring over a field $k$ and let $I $ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein $k$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the…

Earlier we showed that the Hilbert scheme of $n$ points in the plane can be identified with the Hilbert scheme of regular $S_n$ orbits on $C^{2n}$. Using this result, together with a recent theorem of Bridgeland, King and Reid on the…

We systematically extend the elementary differential and Riemannian geometry of classical $\mathrm{U}(1)$-gauge theory to the noncommutative setting by combining recent advances in noncommutative Riemannian geometry with the theory of…

This is the first in a series of papers in which we describe explicit structural properties of spaces of diagonal rectangular harmonic polynomials in $k$ sets of $n$ variables, both as $GL_k$-modules and $S_n$-modules, as well as some of…

We employ the pinching theorem, ensuring that some operators A admit any sequence of contractions as an operator diagonal of A, to deduce/improve two recent theorems of Kennedy-Skoufranis and Loreaux-Weiss for conditional expectations onto…

We propose an operator generalization of the Li-Haldane conjecture regarding the entanglement Hamiltonian of a disk in a 2+1D chiral gapped groundstate. The logic applies to regions with sharp corners, from which we derive several universal…

The aim of this work is to construct a monomial and explicit basis for the space $M_{\mu}$ relative to the $n!$ conjecture. We succeed completely for hook-shaped partitions, i.e. $\mu=(K+1,1^L)$. We are indeed able to exhibit a basis and to…

We construct a family of moduli spaces, called generalized Lagrangian flag schemes, that are reducible Gorenstein (hence singular), and that admit well-behaved pushforward and pullback operations in Hermitian $K$-theory. These schemes arise…

Dualities are hidden symmetries that map seemingly unrelated physical systems onto each other. The goal of this work is to systematically construct families of Hamiltonians endowed with a given duality and to provide a universal description…

Simion had a unimodality conjecture concerning the number of lattice paths in a rectangular grid with the Ferrers diagram of a partition removed. Hildebrand recently showed the stronger result that these numbers are log concave. Here we…

Lagrangian and Hamiltonian formulations of a free spinning particle in 2+1-dimensions or {\it anyon} are established, following closely the analysis of Hanson and Regge. Two viable (and inequivalent) Lagrangians are derived. It is also…

A bi-Hamiltonian formulation is proposed for triangular systems resulted by perturbations around solutions, from which infinitely many symmetries and conserved functionals of triangular systems can be explicitly constructed, provided that…

Let $H$ be the Iwahori--Hecke algebra associated with $S_n$, the symmetric group on $n$ symbols. This algebra has two important bases: the Kazhdan--Lusztig basis and the Murphy basis. While the former admits a deep geometric interpretation,…

Let $\Gamma$ be a discrete group of isometries acting on the complex hyperbolic $n$-space $\mathbb{H}^n_\mathbb{C}$. In this note, we prove that if $\Gamma$ is convex-cocompact, torsion-free, and the critical exponent $\delta(\Gamma)$ is…