The Schur function realization of vertex operators

In the prequel to this paper, we showed how results of Mason involving a new combinatorial formula for polynomials that are now known as Demazure atoms (characters of quotients of Demazure modules, called standard bases by Lascoux and Schützenberger) could be used to define a new basis for the ring of quasisymmetric functions we call “Quasisymmetric Schur functions” (QS functions for short). In this paper we develop the combinatorics of these polynomials further, by showing that the product of a Schur function and a Demazure atom has a positive expansion in terms of Demazure atoms. We use these techniques, together with the fact that both a QS function and a Demazure character have explicit expressions as a positive sum of atoms, to obtain the expansion of a product of a Schur function with a QS function (Demazure character) as a positive sum of QS functions (Demazure characters). Our formula for the coefficients in the expansion of a product of a Demazure character and a Schur function into Demazure characters is similar to known results and includes in particular the famous Littlewood-Richardson rule for the expansion of a product of Schur functions in terms of the Schur basis.

We consider a flagged form of the Cauchy determinant, for which we provide a combinatorial interpretation in terms of nonintersecting lattice paths. In combination with the standard determinant for the enumeration of nonintersecting lattice paths, we are able to give a new proof of the Cauchy identity for Schur functions. Moreover, by choosing different starting and end points for the lattice paths, we are led to a lattice path proof of an identity of Gessel which expresses a Cauchy-like sum of Schur functions in terms of the complete symmetric functions.

We study statistical properties of $N$ non-interacting identical bosons or fermions in the canonical ensemble. We derive several general representations for the $p$-point correlation function of occupation numbers $\overline{n_1\cdots n_p}$. We demonstrate that it can be expressed as a ratio of two $p\times p$ determinants involving the (canonical) mean occupations $\overline{n_1}$, ..., $\overline{n_p}$, which can themselves be conveniently expressed in terms of the $k$-body partition functions (with $k\leq N$). We draw some connection with the theory of symmetric functions, and obtain an expression of the correlation function in terms of Schur functions. Our findings are illustrated by revisiting the problem of Bose-Einstein condensation in a 1D harmonic trap, for which we get analytical results. We get the moments of the occupation numbers and the correlation between ground state and excited state occupancies. In the temperature regime dominated by quantum correlations, the distribution of the ground state occupancy is shown to be a truncated Gumbel law. The Gumbel law, describing extreme value statistics, is obtained when the temperature is much smaller than the Bose-Einstein temperature.

Using the vertex operator representations for symplectic and orthogonal Schur functions, we define two families of symmetric functions and show thatthey are the skew symplectic and skew orthogonal Schur polynomials defined implicitly by Koike and Terada and satisfy the general branching rules. Furthermore, we derive the Jacobi-Trudi identities and Gelfand-Tsetlin patterns for these symmetric functions. Additionally, the vertex operator method yields their Cauchy-type identities. This demonstrates that vertex operator representations serve not only as a tool for studying symmetric functions but also offers unified realizations for skew Schur functions of types A, C, and D.

We present an algorithm for DFT calculations employing Gaussian basis sets for the wave function and a Fourier basis for the potential representation. In particular, a numerically very efficient calculation of the local potential matrix elements and the charge density is described. Special emphasis is placed on the consequences of periodicity and explicit $\mathbit{k}$-vector dependence. The algorithm is tested by comparison with more straightforward ones for the case of adsorption of ethylene on the silicon-rich $\mathrm{Si}\mathrm{C}(001)\text{\ensuremath{-}}(3\ifmmode\times\else\texttimes\fi{}2)$ surface clearly revealing its substantial advantages. A complete self-consistency cycle is speeded up by roughly one order of magnitude since the calculation of matrix elements and of the charge density are accelerated by factors of 10 and 80, respectively, as compared to their straightforward calculation. Our results for ${\mathrm{C}}_{2}{\mathrm{H}}_{4}:\mathrm{Si}\mathrm{C}(001)\text{\ensuremath{-}}(3\ifmmode\times\else\texttimes\fi{}2)$ show that ethylene molecules preferentially adsorb in on-top positions above Si dimers on the substrate surface saturating both dimer dangling bonds per unit cell. In addition, a twist of the molecules around a surface-perpendicular axis is slightly favored energetically similar to the case of a complete monolayer of ethylene adsorbed on the $\mathrm{Si}(001)\text{\ensuremath{-}}(2\ifmmode\times\else\texttimes\fi{}1)$ surface.

The dual immaculate functions are a basis of the ring QSym of quasisymmetric functions and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an immaculate tableau is defined similarly to a semistandard Young tableau, but the shape is a composition rather than a partition, and only the first column is required to strictly increase (whereas the other columns can be arbitrary, but each row has to weakly increase). Dual immaculate functions were introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki in arXiv:1208.5191, and have since been found to possess numerous nontrivial properties.In this note, we prove a conjecture of M. Zabrocki that provides an alternative construction for the dual immaculate functions in terms of certain “vertex operators”. The proof uses a dendriform structure on the ring QSym; we discuss the relation of this structure to known dendriformstructures on the combinatorial Hopf algebras FQSym andWQSym.

It is an important problem in algebraic combinatorics to deduce the Schur function expansion of a symmetric function whose expansion in terms of the fundamental quasisymmetric function is known. For example, formulas are known for the fundamental expansion of a Macdonald symmetric function and for the plethysm of two Schur functions, while the Schur expansions of these expressions are still elusive. Based on work of Egge, Loehr and Warrington, Garsia and Remmel provided a method to obtain the Schur expansion from the fundamental expansion by replacing each quasisymmetric function by a Schur function (not necessarily indexed by a partition) and using straightening rules to obtain the Schur expansion. Here we provide a new method that only involves the coefficients of the quasisymmetric functions indexed by partitions and the quasi-Kostka matrix. As an application, we identify the lexicographically largest term in the Schur expansion of the plethysm of two Schur functions. We provide the Schur expansion of $s_w[s_h](x,y)$ for $w=2,3,4$ using novel symmetric chain decompositions of Young's lattice for partitions in a $w\times h$ box. For $w=4$, this is the first known combinatorial expression for the coefficient of $s_{\lambda}$ in $s_{w}[s_{h}]$ for two-row partitions $\lambda$, and for $w=3$ the combinatorial expression is new.

The basic representation of $\A$ is studied. The weight vectors are represented in terms of Schur functions. A suitable base of any weight space is given. Littlewood-Richardson rule appears in the linear relations among weight vectors.

In this paper, we work toward answering the following question: given a uniformly random algebra homomorphism from the ring of symmetric functions over the integers to a finite field $\mathbb{F}_q$, what is the probability that the Schur function $s_\lambda$ maps to zero? We show that this probability is always at least $1/q$ and is asymptotically $1/q$. Moreover, we give a complete classification of all shapes that can achieve probability $1/q$. In addition, we identify certain families of shapes where the corresponding Schur functions being sent to zero are independent events, and we look into the probability that a Schur functions is mapped to nonzero values in $\mathbb{F}_q$.

Boson operator realization of Schur-functions, exploited in previous works, is developed further. The operator of the present concern is a cluster of creation (or an-nihilation) operators, and behaves itself as a generalized Young diagram. The boson operator formalism is utilized to reexpress higher order Kadomtsev-Petviashvili (K-P) equations in terms of Schur-functions. A compact presentation of the order n ≤10 equations is proposed.

Vertex Operator Algebras and the Monster

The eigenspace of the total angular and spin momentum operator Ĵ2 in the j — j coupling and of the isospin operator T2 is the most appropriate for calculation of the nuclear properties. Due to the nature of nuclear forces solutions of nuclear equations are sought in full many-particle spaces built from primitive functions, called in context of nuclear shell-model calulations’orbits’, that are almost always taken as harmonic oscillator functions (cf Wong 1981; Brussaard and Glaudemans 1977). The full spaces have very high dimensions and therefore shell-model calculations in nuclear physics are concentrated mainly in the sd shell, with only a modest studies of other shells (McGrory and Wildenthal 1980). A program determining the dimensionalities of such model spaces has recently been published (Draayer and Valdes 1985). Calculation of matrix elements in (Ĵ2, T2)-adapted space is not simple, therefore computer programs performing calculations of nuclear structure work frequently in the ‘M-scheme’ or determinantal spaces where calculation of matrix elements is simpler but the dimension of the space is much bigger (Duch 1986b).

We derive group branching laws for formal characters of subgroups of leaving invariant an arbitrary tensor T? of Young symmetry type ? where ? is an integer partition. The branchings and fixing a vector vi, a symmetric tensor gij = gji and an antisymmetric tensor fij = ?fji, respectively, are obtained as special cases. All new branchings are governed by Schur function series obtained from plethysms of the Schur function s? ? {?} by the basic M series of complete symmetric functions and the L = M?1 series of elementary symmetric functions. Our main technical tool is that of Hopf algebras and our main result is the derivation of a coproduct for any Schur function series obtained by plethysm from another such series. Therefrom one easily obtains ?-generalized Newell?Littlewood formulae and the algebra of the formal group characters of these subgroups is established. Concrete examples and extensive tabulations are displayed for and , showing their involved and nontrivial representation theory. The nature of the subgroups is shown to be in general affine and in some instances non-reductive. We discuss the complexity of the coproduct formula and give a graphical notation to cope with it. We also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2-cocycles. The algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for formal characters. Modification rules are beyond the scope of the present paper, but are briefly discussed.

ODPEVP: A program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm–Liouville problem

Starting with a family of Markov chains corresponding to a trellis (or ladder) model we show how to find symmetric functions in the eigenvalues of the transition matrix. Schur functions arise via minors of powers of the transition matrix. We give a path model for finding the Schur functions in terms of the transition probabilities of the trellis. Limit theorems for powers of the transition matrix yield limit theorems for the exterior powers of the transition matrix which in turn yield limit theorems for corresponding families of symmetric functions. We present a theorem on Ces`aro convergence of powers of a matrix that is the exterior power of a stochastic matrix.