A Fast Algorithm for the Construction of Integrity Bases Associated to Symmetry-Adapted Polynomial Representations. Application to Tetrahedral XY4 Molecules

Invariant theory provides more efficient tools, such as Molien generating functions and integrity bases, than basic group theory, that relies on projector techniques for the construction of symmetry--adapted polynomials in the symmetry coordinates of a molecular system, because it is based on a finer description of the mathematical structure of the latter. The present article extends its use to the construction of polynomial bases which span possibly, non--totally symmetric irreducible representations of a molecular symmetry group. Electric or magnetic observables can carry such irreducible representations, a common example is given by the electric dipole moment surface. The elementary generating functions and their corresponding integrity bases, where both the initial and the final representations are irreducible, are the building blocks of the algorithm presented in this article, which is faster than algorithms based on projection operators only. The generating functions for the full initial representation of interest are built recursively from the elementary generating functions. Integrity bases which can be used to generate in the most economical way symmetry--adapted polynomial bases are constructed alongside in the same fashion. The method is illustrated in detail on XY4 type of molecules. Explicit integrity bases for all five possible final irreducible representations of the tetrahedral group have been calculated and are given in the supplemental material associated with this paper

Integrable Lattice Realizations of N=1 Superconformal Boundary Conditions

We construct integrable boundary conditions for sl(2) coset models with central charges c=3/2-12/(m(m+2)) and m=3,4,... The associated cylinder partition functions are generating functions for the branching functions but these boundary conditions manifestly break the superconformal symmetry. We show that there are additional integrable boundary conditions, satisfying the boundary Yang-Baxter equation, which respect the superconformal symmetry and lead to generating functions for the superconformal characters in both Ramond and Neveu-Schwarz sectors. We also present general formulas for the cylinder partition functions. This involves an alternative derivation of the superconformal Verlinde formula recently proposed by Nepomechie.Comment: 22 pages, 12 figures; section 2 rewritten; journal-ref. adde

The Weak Bruhat Order and Separable Permutations

In this paper we consider the rank generating function of a separable permutation $\pi$ in the weak Bruhat order on the two intervals $[\text{id}, \pi]$ and $[\pi, w_0]$, where $w_0 = n,(n-1),..., 1$. We show a surprising result that the product of these two generating functions is the generating function for the symmetric group with the weak order. We then obtain explicit formulas for the rank generating functions on $[\text{id}, \pi]$ and $[\pi, w_0]$, which leads to the rank-symmetry and unimodality of the two graded posets

Gauge symmetry and Slavnov-Taylor identities for randomly stirred fluids

The path integral for randomly forced incompressible fluids is shown to have an underlying Becchi-Rouet-Stora (BRS) symmetry as a consequence of Galilean invariance. This symmetry must be respected to have a consistent generating functional, free from both an overall infinite factor and spurious relations amongst correlation functions. We present a procedure for respecting this BRS symmetry, akin to gauge fixing in quantum field theory. Relations are derived between correlation functions of this gauge fixed, BRS symmetric theory, analogous to the Slavnov-Taylor identities of quantum field theory.Comment: 5 pages, no figures, In Press Physical Review Letters, 200