On Dunkl angular momenta algebra

We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincare-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl(N) version of the subalgebra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.Comment: 27 pages; small changes, concluding remarks expande

On the asymptotic S_n-structure of invariant differential operators on symplectic manifolds

We consider the space of polydifferential operators on n functions on symplectic manifolds invariant under symplectic automorphisms, whose study was initiated by Mathieu in 1995. Permutations of inputs yield an action of S_n, which extends to an action of S_{n+1}. We study this structure viewing n as a parameter, in the sense of Deligne's category. For manifolds of dimension 2d, we show that the isotypic part of this space of <= 2d+1-th tensor powers of the reflection representation h=C^n of S_{n+1} is spanned by Poisson polynomials. We also prove a partial converse, and compute explicitly the isotypic part of <= 4-th tensor powers of the reflection representation. We give generating functions for the isotypic parts corresponding to Young diagrams which only differ in the length of the top row, and prove that they are rational functions whose denominators are related to hook lengths of the diagrams obtained by removing the top row. This also gives such a formula for the same isotypic parts of induced representations from Z/(n+1) to S_{n+1} where n is viewed as a parameter. We apply this to the Poisson and Hochschild homology associated to the singularity C^{2dn}/S_{n+1}. Namely, the Brylinski spectral sequence from the zeroth Poisson homology of the S_{n+1}-invariants of the n-th Weyl algebra of C^{2d} with coefficients in the whole Weyl algebra degenerates in the 2d+1-th tensor power of h, as well as its fourth tensor power. Furthermore, the kernel of this spectral sequence has dimension on the order of 1/n^3 times the dimension of the homology group.Comment: v2: 47 pages; removed what was part (ii) of Theorem 1.3.45 since its proof was invalid. Nothing else was affected. v3: Several corrections; final version to be published in J. Algebr

Zeroth Hochschild homology of preprojective algebras over the integers

We determine the Z-module structure of the preprojective algebra and its zeroth Hochschild homology, for any non-Dynkin quiver (and hence the structure working over any base commutative ring, of any characteristic). This answers (and generalizes) a conjecture of Hesselholt and Rains, producing new $p$-torsion classes in degrees 2p^l, l >= 1, We relate these classes by p-th power maps and interpret them in terms of the kernel of Verschiebung maps from noncommutative Witt theory. An important tool is a generalization of the Diamond Lemma to modules over commutative rings, which we give in the appendix. In the previous version, additional results are included, such as: the Poisson center of $\text{Sym } HH_0(\Pi)$ for all quivers, the BV algebra structure on Hochschild cohomology, including how the Lie algebra structure $HH_0(\Pi_Q)$ naturally arises from it, and the cyclic homology groups of $\Pi_Q$.Comment: 69 pages, 2 figures; final pre-publication version; many corrections and improvements throughout. Note though the first version has additional results (for instance, it computes the higher Hochschild (co)homology and its structures

Recognizing Graph Theoretic Properties with Polynomial Ideals

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Groebner bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure

Complete N-Point Superstring Disk Amplitude II. Amplitude and Hypergeometric Function Structure

Using the pure spinor formalism in part I [1] we compute the complete tree-level amplitude of N massless open strings and find a striking simple and compact form in terms of minimal building blocks: the full N-point amplitude is expressed by a sum over (N-3)! Yang-Mills partial subamplitudes each multiplying a multiple Gaussian hypergeometric function. While the former capture the space-time kinematics of the amplitude the latter encode the string effects. This result disguises a lot of structure linking aspects of gauge amplitudes as color and kinematics with properties of generalized Euler integrals. In this part II the structure of the multiple hypergeometric functions is analyzed in detail: their relations to monodromy equations, their minimal basis structure, and methods to determine their poles and transcendentality properties are proposed. Finally, a Groebner basis analysis provides independent sets of rational functions in the Euler integrals.Comment: 68 pages, harvmac Te