Schur Functions Research Articles

Tableaux and plane partitions of truncated shapes (extended abstract)

We consider a new kind of straight and shifted plane partitions/Young tableaux — ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function. Nous considérons un nouveau type de partitions planes, ou de tableaux de Young, droits ou décalés, obtenus en privant leurs diagrammes de certaines cellules en haut à droite, et dans certains cas nous trouvons des formules d'énumération pour les tableaux standard. Les preuves impliquent le calcul de la fonction génératrice pour les partitions planes correspondantes, en utilisant des interprétations et des formules pour les sommes de fonctions de Schur restreintes et leurs spécialisations. Le nombre de tableaux standard est alors obtenu comme une certaine limite de cette fonction.

Skew quantum Murnaghan-Nakayama rule

In this extended abstract, we extend recent results of Assaf and McNamara, the skew Pieri rule and the skew Murnaghan-Nakayama rule, to a more general identity, which gives an elegant expansion of the product of a skew Schur function with a quantum power sum function in terms of skew Schur functions. We give two proofs, one completely bijective in the spirit of Assaf-McNamara's original proof, and one via Lam-Lauve-Sotille's skew Littlewood-Richardson rule. Dans cet article nous élargissons le cadre de résultats récents de Assaf et McNamara, la règle dissymétrique de Pieri et la règle dissymétrique de Murnaghan-Nakayama, pour obtenir une identité plus générale donnant un développement élégant du produit de la fonction de Schur dissymétrique par une somme de puissances quantiques, en termes de fonctions de Schur dissymétriques. Nous donnons deux démonstrations, la première suivant l'approche de Assaf-McNamara et la deuxième par le biais de la règle dissymétrique de Littlewood-Richardson obtenue par Lam-Lauve-Sotille.

Euler characters and super Jacobi polynomials

We prove that Euler supercharacters for orthosymplectic Lie superalgebras can be obtained as a certain specialization of super Jacobi polynomials. A new version of Weyl type formula for super Schur functions and specialized super Jacobi polynomials play a key role in the proof.

Sylvester double sums and subresultants

Sylvester double sums, introduced first by Sylvester (see Sylvester (1840, 1853)), are symmetric expressions of the roots of two polynomials, while subresultants are defined through the coefficients of these polynomials (see Apery and Jouanolou (2006) and Basu et al. (2003) for references on subresultants). As pointed out by Sylvester, the two notions are very closely related: Sylvester double sums and subresultants are equal up to a multiplicative non-zero constant in the ground field. Two proofs are already known: that of Lascoux and Pragacz (2003), using Schur functions, and that of d’Andrea et al. (2007), using manipulations of matrices. The purpose of this paper is to give a new simple proof using similar inductive properties of double sums and subresultants.

Refinements of the Littlewood-Richardson rule

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.

On Ranges and Moore–Penrose Inverses Related to Matrix Carathéodory and Schur Functions

By Cayley transformation, there is an interplay between matrix-valued Caratheodory and Schur functions in the unit disk. One of the main aims of the paper is to study this interplay with a view to the fact that the matrix function given by the values of a Caratheodory function via Moore–Penrose inverses is also a Caratheodory function. In doing so, we also analyze ranges and null spaces of the values of the matrix function in question and get some results of independence concerning the concretely chosen point of the domain.

Plethysms, replicated Schur functions and series, with applications to vertex operators

Specializations of Schur functions are exploited to define and evaluate the Schur functions sλ[αX] and plethysms sλ[αsν(X))] for any α—integer, real or complex. Plethysms are then used to define pairs of mutually inverse infinite series of Schur functions, Mπ and Lπ, specified by arbitrary partitions π. These are used in turn to define and provide generating functions for formal characters, s(π)λ, of certain groups Hπ, thereby extending known results for orthogonal and symplectic group characters. Each of these formal characters is then given a vertex operator realization, first in terms of the series M = M(0) and various L⊥σ dual to Lσ, and then more explicitly in the exponential form. Finally the replicated form of such vertex operators are written down.The characters of the orthogonal and symplectic groups have been found by Schur [34] and Weyl [35] respectively. The method used is transcendental, and depends on integration over the group manifold. These characters, however, may be obtained by purely algebraic methods,…. This algebraic method would seem to offer a better prospect of successful application to other restricted groups than the method of group integration.Littlewood D E 1944 Phil. Trans. R. Soc. London, Ser. A 239 (809) 392

Log-convexity properties of Schur functions and generalized hypergeometric functions of matrix argument

We establish a positivity property for the difference of products of certain Schur functions, sλ(x), where λ varies over a fundamental Weyl chamber in ℝn and x belongs to the positive orthant in ℝn. Further, we generalize that result to the difference of certain products of arbitrary numbers of Schur functions. We also derive a log-convexity property of the generalized hypergeometric functions of two Hermitian matrix arguments, and we show how that result may be extended to derive higher-order log-convexity properties.

Enumeration formulas for young tableaux in a diagonal strip

We derive combinatorial identities, involving the Bernoulli and Euler numbers, for the numbers of standard Young tableaux of certain skew shapes. This generalizes the classical formulas of D. André on the number of up-down permutations. The analysis uses a transfer operator approach extending the method of Elkies, combined with an identity expressing the volume of a certain polytope in terms of a Schur function.

A Schur type analysis of the minimal weak unitary Hilbert space dilations of a Kreĭn space bicontraction and the Relaxed Commutant Lifting Theorem in a Kreĭn space setting

A parameterization of the minimal weak unitary Hilbert space dilations of a given continuous bicontractive operator on a Kreĭn space by means of operator-valued Schur functions is obtained from the Arov–Grossman functional model. The result is combined with the coupling method to give a parametric description of the interpolants in a Kreĭn space version of the Relaxed Commutant Lifting Theorem.

Wallis-Ramanujan-Schur-Feynman

One of the earliest examples of analytic representations for π is given by an infinite product provided by Wallis in 1655. The modern literature often presents this evaluation based on the integral formula In trying to understand the behavior of this integral when the integrand is replaced by the inverse of a product of distinct quadratic factors, the authors encounter relations to some formulas of Ramanujan, expressions involving Schur functions, and Matsubara sums that have appeared in the context of Feynman diagrams.

Multi-Point Degenerate Interpolation Problem for Generalized Schur Functions: Description of All Solutions

We consider the Nevanlinna-Pick-Caratheodory-Fejer interpolation problem with finitely many interpolation conditions in the class Sgk of meromorphic functions f with ∥f∥l∞(t) ≤ 1 and with κ poles inside the unit disk D. The problem has infinitely many solutions if and only if κ is greater than or equal to the number of non-positive eigenvalues (counted with multiplicities) of the Pick matrix P constructed from interpolation data. For each such κ, we describe the solution set of the problem in terms of a family of linearfractional transformations with disjoint ranges. The parameters defining this family are free and independent.

Nested Quantum Dyck Paths and (s )

We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron-Garsia nabla operator. The formula involves nested labelled Dyck paths weighted by area and a suitable “diagonal inversion” statistic. Our model includes as special cases many previous conjectures connecting the nabla operator to quantum lattice paths. The combinatorics of the inverse Kostka matrix leads to an elementary proof of our proposed formula when q = 1. We also outline a possible approach for proving all the extant nabla conjectures that reduces everything to the construction of sign-reversing involutions on explicit collections of signed, weighted objects.

From quasisymmetric expansions to Schur expansions via a modified inverse Kostka matrix

Every symmetric function f can be written uniquely as a linear combination of Schur functions, say f = ∑ λ x λ s λ , and also as a linear combination of fundamental quasisymmetric functions, say f = ∑ α y α Q α . For many choices of f arising in the theory of Macdonald polynomials and related areas, one knows the quasisymmetric coefficients y α and wishes to compute the Schur coefficients x λ . This paper gives a general combinatorial formula expressing each x λ as a linear combination of the y α ’s, where each coefficient in this linear combination is + 1 , − 1 , or 0. This formula arises by suitably modifying Eğecioğlu and Remmel’s combinatorial interpretation of the inverse Kostka matrix involving special rim-hook tableaux.

Skew Littlewood-Richardson Rules from Hopf Algebras

We use Hopf algebras to prove a version of the Littlewood–Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood–Richardson rules for Schur P- and Q-functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group.

On the calculation of inner products of Schur functions

Two methods for calculating inner products of Schur functions in terms of outer products and plethysms are given and they are easy to implement on a machine. One of these is derived from a recent analysis of the SO(8) proton-neutron pairing model of atomic nuclei. The two methods allow for generation of inner products for the Schur functions of degree up to 20 and even beyond.

Extended Bressoud–Wei and Koike skew Schur function identities

The Jacobi–Trudi identity expresses a skew Schur function as a determinant of complete symmetric functions. Bressoud and Wei extend this idea, introducing an integer parameter t ⩾ − 1 and showing that signed sums of skew Schur functions of a certain shape are expressible once again as a determinant of complete symmetric functions. Koike provides a Jacobi–Trudi-style definition of universal rational characters of the general linear group and gives their expansion as a signed sum of products of Schur functions in two distinct sets of variables. Here we extend Bressoud and Wei's formula by including an additional parameter and extending the result to the case of all integer t. Then we introduce this parameter idea to the Koike formula, extending it in the same way. We prove our results algebraically using Laplace determinantal expansions.

A Pieri rule for skew shapes

The Pieri rule expresses the product of a Schur function and a single row Schur function in terms of Schur functions. We extend the classical Pieri rule by expressing the product of a skew Schur function and a single row Schur function in terms of skew Schur functions. Like the classical rule, our rule involves simple additions of boxes to the original skew shape. Our proof is purely combinatorial and extends the combinatorial proof of the classical case.

Shift operators contained in contractions, pseudocontinuable Schur functions and orthogonal systems on the unit circle

The main aim of this paper is to establish the connection between well-known criteria for the pseudocontinuability of a non-inner Schur function Θ in the unit disk (see Theorems 3.9, 4.2). In a canonical way we associate a probability measure μ on the unit circle with Θ. One of the two criteria will be reformulated in the face of μ, whereas the other one is drafted in view of a completely nonunitary contraction T having Θ as corresponding characteristic function. Our main result clarifies an immediate connection between the above-mentioned two criteria. In this way, we rewrite these criteria in terms of orthogonal systems with respect to the measure μ (see Theorem 7.2).

Matrix Schur functions, permutation matrices, and young operators as inner product spaces

An inner product is defined on the space of permutation matrices and the space of matrices dual to the permutation matrices is given. The relationship of permutation matrices to the expansion of matrices of fixed line-sum is discussed. This inner product carries over to the space of linear Young operators, as does the notion of the dual space. Motivation from physics for considering such algebraic structures is also given; in particular, the real, orthogonal, irreducible representations of the symmetric group originating from the matrix Schur functions are given.