Terms Of Schur Functions Research Articles

Schur functions in noncommuting variables

In 2004 Rosas and Sagan asked whether there was a way to define a basis in the algebra of symmetric functions in noncommuting variables, NCSym, having properties analogous to the classical Schur functions. This was because they had constructed a partial such set that was not a basis. We answer their question by defining Schur functions in noncommuting variables using a noncommutative analogue of the Jacobi-Trudi determinant. Our Schur functions in NCSym map to classical Schur functions under commutation, and a subset of them indexed by set partitions forms a basis for NCSym. Amongst other properties, Schur functions in NCSym also satisfy a noncommutative analogue of the product rule for classical Schur functions in terms of skew Schur functions.We also show how Schur functions in NCSym are related to Specht modules, and naturally refine the Rosas-Sagan Schur functions. Moreover, by generalizing Rosas-Sagan Schur functions to skew Schur functions in the natural way, we prove noncommutative analogues of the Littlewood-Richardson rule and coproduct rule for them. Finally, we relate our functions to noncommutative symmetric functions by proving a subset of our functions are natural extensions of noncommutative ribbon Schur functions, and immaculate functions indexed by integer partitions.

Combinatorial relations on skew Schur and skew stable Grothendieck polynomials

We give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard tableaux of skew shape. This expansion unifies some previous results: it generalizes a combinatorial formula obtained in earlier joint work with López Martín and Teixidor i Bigas concerning Brill–Noether curves, and it generalizes a 2000 formula of Lenart and a recent result of Reiner–Tenner–Yong to skew shapes. We also give an expansion in the other direction: expressing skew Schur functions in terms of skew Grothendieck polynomials.

Positivity of cylindric skew Schur functions

Cylindric skew Schur functions, a generalization of skew Schur functions, are closely related to the well-known problem of finding a combinatorial formula for the 3-point Gromov-Witten invariants of Grassmannians. In this paper, we prove cylindric Schur positivity of cylindric skew Schur functions, as conjectured by McNamara. We also show that all the coefficients appearing in the expansion are the same as the 3-point Gromov-Witten invariants. We start by discussing the properties of affine Stanley symmetric functions for general affine permutations and 321-avoiding affine permutations, and we explain how these functions are related to cylindric skew Schur functions. In addition, we provide an effective algorithm to compute the expansion of cylindric skew Schur functions in terms of cylindric Schur functions, as well as the expansion of affine Stanley symmetric functions in terms of affine Schur functions.

Multigraded Hilbert series of noncommutative modules

In this paper, we propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory of regular languages, we provide conditions when the methods are effective and hence the sum of the Hilbert series is a rational function. Moreover, a characterization of finite-dimensional algebras is obtained in terms of the nilpotency of a key matrix involved in the computations. Using this result, efficient variants of the methods are also developed for the computation of Hilbert series of truncated infinite-dimensional algebras whose (non-truncated) Hilbert series may not be rational functions. We consider some applications of the computation of multigraded Hilbert series to algebras that are invariant under the action of the general linear group. In fact, in this case such series are symmetric functions which can be decomposed in terms of Schur functions. Finally, we present an efficient and complete implementation of (standard) graded and multigraded Hilbert series that has been developed in the kernel of the computer algebra system Singular. A large set of tests provides a comprehensive experimentation for the proposed algorithms and their implementations.

Correlations of occupation numbers in the canonical ensemble and application to a Bose-Einstein condensate in a one-dimensional harmonic trap

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.

A generalization of Schur functions: Applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks

Recent work on return properties of quantum walks (the quantum analogue of random walks) has identified their generating functions of first returns as Schur functions. This is connected with a representation of Schur functions in terms of the operators governing the evolution of quantum walks, i.e. the unitary operators on Hilbert spaces.In this paper we propose a generalization of Schur functions by extending the above operator representation to arbitrary closed operators on Banach spaces. Such generalized ‘Schur functions’ meet the formal structure of first return generating functions, thus we call them FR-functions. We derive some general properties of FR-functions, among them a simple relation with an operator version of Stieltjes functions which generalizes the renewal equation already known for random and quantum walks. We also prove that FR-functions satisfy splitting properties which extend useful factorizations of Schur functions.When specialized to self-adjoint operators on Hilbert spaces, we show that FR-functions become Nevanlinna functions. This allows us to obtain properties of Nevanlinna functions which, as far as we know, seem to be new. The FR-function structure leads to a new operator representation of Nevanlinna functions in terms of self-adjoint operators, whose spectral measures provide also new integral representations of such functions. This allows us to characterize each Nevanlinna function by a measure on the real line, which we refer to as ‘the measure of the Nevanlinna function’. In contrast to standard operator and integral representations of Nevanlinna functions, these new ones are exact analogues of those already known for Schur functions. The above results are also the source of a very simple ‘Schur algorithm’ for Nevanlinna functions based on interpolations at points on the real line, which we refer to as the ‘Schur algorithm on the real line’.The paper is completed with several applications of FR-functions to orthogonal polynomials and random and quantum walks which illustrate their wide interest: an analogue for orthogonal polynomials on the real line of the Khrushchev formula for orthogonal polynomials on the unit circle, and the use of FR-functions to study recurrence in random walks, quantum walks and open quantum walks. These applications provide numerous explicit examples of FR-functions, clarifying the meaning of these functions –as first return generating functions– and their splittings –which become recurrence splitting rules. They also show that these new tools, despite being extensions of very classical ones, play an important role in the study of physical problems of a highly topical nature.

Crystal Analysis of type C Stanley Symmetric Functions

Combining results of T.K. Lam and J. Stembridge, the type $C$ Stanley symmetric function $F_w^C(\mathbf{x})$, indexed by an element $w$ in the type $C$ Coxeter group, has a nonnegative integer expansion in terms of Schur functions. We provide a crystal theoretic explanation of this fact and give an explicit combinatorial description of the coefficients in the Schur expansion in terms of highest weight crystal elements.

A Murnaghan–Nakayama rule for noncommutative Schur functions

We prove a Murnaghan–Nakayama rule for noncommutative Schur functions introduced by Bessenrodt, Luoto and van Willigenburg. In other words, we give an explicit combinatorial formula for expanding the product of a noncommutative power sum symmetric function and a noncommutative Schur function in terms of noncommutative Schur functions. In direct analogy to the classical Murnaghan–Nakayama rule, the summands are computed using a noncommutative analogue of border strips, and have coefficients ±1 determined by the height of these border strips. The rule is proved by interpreting the noncommutative Pieri rules for noncommutative Schur functions in terms of box-adding operators on compositions.

On the expansion coefficients of Tau-function of the BKP hierarchy

We study the series expansion of the tau function of the BKP hierarchy applying the addition formulae of the BKP hierarchy. Any formal power series can be expanded in terms of Schur functions. It is known that, under the condition , a formal power series is a solution of the KP hierarchy if and only if its coefficients of Schur function expansion are given by the so called Giambelli type formula. A similar result is known for the BKP hierarchy with respect to Schur’s Q-function expansion under a similar condition. In this paper we generalize this result to the case of .

Superlattices with coupled degenerated spectrum

Two new analytical results are given which are of great help to understand superlattices with coupled modes. The first is an explicit relation for the transfer matrices in terms of Schur functions and Chebyshev polynomials. The second is a condition which generalizes the old well-known Floquet-Bloch trace condition to determine the spectrum. These improvements allow a fast computation of scattering amplitudes without obscuring the calculation with complicated numerical methods. As the energy grows, the eigenvalues degeneracy determines two types of transmission gaps. It is shown that these results could make it possible to design in greater detail the energy spectrum, for the very interesting case including modes coupling and degeneracy. They keep the understanding on the same footing as that of the traditional basic uncoupled problem considered at the beginning of the study of superlattices, like the Kroning-Penning model, or by Floquet-Bloch's theorem, or later by Esaki for heterostructures.

A new q-Selberg integral, Schur functions, and Young books

Recently, Kim and Oh expressed the Selberg integral in terms of the number of Young books which are a generalization of standard Young tableaux of shifted staircase shape. In this paper the generating function for Young books according to major index statistic is considered. It is shown that this generating function can be written as a Jackson integral which gives a new q-Selberg integral. It is also shown that the new q-Selberg integral has an expression in terms of Schur functions.

Combinatorics of a strongly coupled boson system

We introduce a quantum phase model as a limit for very strong interactions of a strongly correlated q-boson hopping model. We describe the general solution of the phase model and express scalar products of state vectors in determinant form. The representation of state vectors in terms of Schur functions allows obtaining a combinatorial interpretation of the scalar products in terms of nests of self-avoiding lattice paths. We show that under a special parameterization, the scalar products are equal to the generating functions of plane partitions confined in a finite box. We consider the two-dimensional vertex model related to the phase model and express the vertex model partition function with special boundary conditions in terms of the scalar product of the phase model state vectors.

Littlewood Complexes and Analogues of Determinantal Varieties

One interesting combinatorial feature of classical determinantal varieties is that the character of their coordinate rings give a natural truncation of the Cauchy identity in the theory of symmetric functions. Natural generalizations of these varieties exist and have been studied for the other classical groups. In this paper we develop the relevant properties from scratch. By studying the isotypic decomposition of their minimal free resolutions one can recover classical identities due to Littlewood for expressing an irreducible character of a classical group in terms of Schur functions. We propose generalizations for the exceptional groups. In type G_2, we completely analyze the variety and its minimal free resolution and get an analogue of Littlewood's identities. We have partial results for the other cases. In particular, these varieties are always normal with rational singularities.

Quantum Cohomology via Vicious and Osculating Walkers

We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder, so-called vicious and osculating walkers. These lattice paths form exactly solvable statistical mechanics models and are obtained from solutions to the Yang–Baxter equation. The eigenvectors of the transfer matrices of these models yield the idempotents of the Verlinde algebra of the gauged \({\mathfrak{\hat{u}}(n)_{k}}\)-WZNW model. The latter is known to be closely related to the small quantum cohomology ring of the Grassmannian. We establish further that the partition functions of the vicious and osculating walker model are given in terms of Postnikov’s toric Schur functions and can be interpreted as generating functions for Gromov–Witten invariants. We reveal an underlying quantum group structure in terms of Yang–Baxter algebras and use it to give a generating formula for toric Schur functions in terms of divided difference operators which appear in known representations of the nil-Hecke algebra.

Chebyshev blossoming in Müntz spaces: Toward shaping with Young diagrams

The notion of a blossom in extended Chebyshev spaces offers adequate generalizations and extra-utilities to the tools for free-form design schemes. Unfortunately, such advantages are often overshadowed by the complexity of the resulting algorithms. In this work, we show that for the case of Müntz spaces with integer exponents, the notion of a Chebyshev blossom leads to elegant algorithms whose complexities are embedded in the combinatorics of Schur functions. We express the blossom and the pseudo-affinity property in Müntz spaces in terms of Schur functions. We derive an explicit expression for the Chebyshev–Bernstein basis via an inductive argument on nested Müntz spaces. We also reveal a simple algorithm for dimension elevation. Free-form design schemes in Müntz spaces with Young diagrams as shape parameters are discussed.

The orbifold topological vertex

We develop a topological vertex formalism for computing the Donaldson–Thomas invariants of Calabi–Yau orbifolds. The basic combinatorial object is the orbifold vertex V λ μ ν G , a generating function for the number of 3D partitions asymptotic to 2D partitions λ, μ, ν and colored by representations of a finite Abelian group G acting on C 3 . In the case where G ≅ Z n acting on C 3 with transverse A n − 1 quotient singularities, we give an explicit formula for V λ μ ν G in terms of Schur functions. We discuss applications of our formalism to the Donaldson–Thomas crepant resolution conjecture and to the orbifold Donaldson–Thomas/Gromov–Witten correspondence. We also explicitly compute the Donaldson–Thomas partition function for some simple orbifold geometries: the local football P a , b 1 and the local B Z 2 gerbe.

The evaluation of a quartic integral via Schwinger, Schur and Bessel

We provide additional methods for the evaluation of the integral $$N_{0,4}(a;m) := \int_{0}^{\infty} \frac{dx}{( x^{4} + 2ax^{2} + 1 )^{m+1}},$$ where m∈ℕ and a∈(−1,∞) in the form $$N_{0,4}(a;m) = \frac{\pi}{2^{m+3/2} (a+1)^{m+1/2} }P_{m}(a),$$ where Pm(a) is a polynomial in a. The first one is based on a method of Schwinger to evaluate integrals appearing in Feynman diagrams, the second one is a byproduct of an expression for a rational integral in terms of Schur functions. Finally, the third proof is obtained from an integral representation involving modified Bessel functions.

A Littlewood-Richardson type rule for row-strict quasisymmetric Schur functions

We establish several properties of an algorithm defined by Mason and Remmel (2010) which inserts a positive integer into a row-strict composition tableau. These properties lead to a Littlewood-Richardson type rule for expanding the product of a row-strict quasisymmetric Schur function and a symmetric Schur function in terms of row-strict quasisymmetric Schur functions. Nous établissons plusieurs propriétés d'un algorithme défini par Mason et Remmel (2010), qui insère un entier positif dans un tableau dont la forme est une composition, avec ordre strict sur les lignes (row-strict). Ces propriétés conduisent à une règle de type Littlewood-Richardson pour étendre le produit d'une fonction de Schur quasi-symétrique "row-strict'' et d'une fonction de Schur symétrique en termes de fonctions de Schur quasi-symétriques "row-strict''.

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 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.