Differential operators and reection group of type Dn

In this note, we study the actions of rational quantum Olshanetsky-Perelomov systems for finite reflections groups of type D n . we endowed the polynomial ring C[x 1,..., xn ] with a differential structure by using directly the action of the Weyl algebra associated with the ring C[x 1,..., xn ] W of invariant polynomials under the reflections groups W after a localization. Then we study the polynomials representation of the ring of invariant differential operators under the reflections groups. We use the higher Specht polynomials associated with the representation of the reflections group W to exhibit the generators of its simple components.

Differential operators and reection group of type Bn

In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group W of type Bn. We endowed the polynomial ring C[x 1,..., xn ] with a structure of module over the Weyl algebra associated with the ring C[x 1,..., xn]W of invariant polynomials under a reflections group W of type Bn . Then we study the polynomials representation of the ring of invariant differential operators under the reflections group W. We make use of the theory of representation of groups namely the higher Specht polynomials associated with the reflection group W to yield a decomposition of that structure by providing explicitly the generators of its simple components.

Triality for Homogeneous Polynomials

Through the triality of SO(8,C), we study three interrelated homogeneous basis of the ring of invariant polynomials of Lie algebras, which give the basis of three Hitchin fibrations, and identify the explicit automorphisms that relate them.

Multicritical hypercubic models

We study renormalization group multicritical fixed points in the ϵ-expansion of scalar field theories characterized by the symmetry of the (hyper)cubic point group HN. After reviewing the algebra of HN-invariant polynomials and arguing that there can be an entire family of multicritical (hyper)cubic solutions with ϕ2n interactions in $$ d=\frac{2n}{n-1}-\epsilon $$ d = 2 n n − 1 − ϵ dimensions, we use the general multicomponent beta functionals formalism to study the special cases d = 3 − ϵ and $$ d=\frac{8}{3}-\epsilon $$ d = 8 3 − ϵ , deriving explicitly the beta functions describing the flow of three- and four-critical (hyper)cubic models. We perform a study of their fixed points, critical exponents and quadratic deformations for various values of N, including the limit N = 0, that was reported in another paper in relation to the randomly diluted single-spin models, and an analysis of the large N limit, which turns out to be particularly interesting since it depends on the specific multicriticality. We see that, in general, the continuation in N of the random solutions is different from the continuation coming from large-N, and only the latter interpolates with the physically interesting cases of low-N such as N = 3. Finally, we also include an analysis of a theory with quintic interactions in $$ d=\frac{10}{3}-\epsilon $$ d = 10 3 − ϵ and, for completeness, the NNLO computations in d = 4 − ϵ.

JORDAN–KRONECKER INVARIANTS OF LIE ALGEBRA REPRESENTATIONS AND DEGREES OF INVARIANT POLYNOMIALS

For an arbitrary representation ρ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan–Kronecker invariants of ρ. Among other interesting properties, these numbers provide lower bounds for degrees of polynomial invariants of ρ. Furthermore, we prove that these lower bounds are exact if and only if the invariants are independent outside of a set of large codimension. Finally, we show that under certain additional assumptions our bounds are exact if and only if the algebra of invariants is freely generated.

On Modular Vector Invariant Fields

Let [Formula: see text] be a finite field of any characteristic and [Formula: see text] be the general linear group over [Formula: see text]. Suppose W denotes the standard representation of [Formula: see text], and [Formula: see text] acts diagonally on the direct sum of W and its dual space W∗. Let G be any subgroup of [Formula: see text]. Suppose the invariant field [Formula: see text], where [Formula: see text] in [Formula: see text] are homogeneous invariant polynomials. We prove that there exist homogeneous polynomials [Formula: see text] in the invariant ring [Formula: see text] such that the invariant field [Formula: see text] is generated by [Formula: see text] over [Formula: see text].

Quasi-Invariants in Characteristic p and Twisted Quasi-Invariants

The spaces of quasi-invariant polynomials were introduced by Chalykh and Veselov [Comm. Math. Phys. 126 (1990), 597-611]. Their Hilbert series over fields of characteristic 0 were computed by Feigin and Veselov [Int. Math. Res. Not. 2002 (2002), 521-545]. In this paper, we show some partial results and make two conjectures on the Hilbert series of these spaces over fields of positive characteristic. On the other hand, Braverman, Etingof and Finkelberg [arXiv:1611.10216] introduced the spaces of quasi-invariant polynomials twisted by a monomial. We extend some of their results to the spaces twisted by a smooth function.

A Many-Body, Fully Polarizable Approach to QM/MM Simulations

<div> <div> <div> <p>We present a new development in quantum mechanics/molecular mechanics (QM/MM) methods by replacing conventional MM models with data-driven many-body (MB) representations rigorously derived from high-level QM calculations. The new QM/MM approach builds on top of mutually polarizable QM/MM schemes developed for polarizable force fields with inducible dipoles and uses permutationally invariant polynomials to effectively account for quantum- mechanical contributions (e.g., exchange-repulsion, and charge transfer and penetration) that are difficult to describe by classical expressions adopted by conventional MM models. Us- ing the many-body MB-pol and MB-DFT potential energy functions for water, which include explicit 2-body and 3-body terms fitted to reproduce the corresponding CCSD(T) and PBE0 2- body and 3-body energies for water, we demonstrate a smooth energetic transition as molecules are transferred between QM and MM regions, without the need of a transition layer. By effectively elevating the accuracy of both the MM region and the QM/MM interface to that of the QM region, the new QM/MB-MM approach achieves an accuracy comparable to that obtained with a fully QM treatment of the entire system. </p> </div> </div> </div>