Terms Of Schur Functions Research Articles

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.

Thom polynomials and Schur functions: the singularities III2,3(-)

We give a closed formula for the Thom polynomials of the singularities $III_{2,3}(-)$ in terms of Schur functions. Our computations combine the characterization of the Thom polynomials via the “method of restriction equations” of Rimányi et al

Quasisymmetric Schur functions

We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. From here we derive a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions. After surveying combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions, we show how some of our results can be extended to include the t parameter from Hall–Littlewood theory.

The q-log-convexity of the Narayana polynomials of type B

We prove a conjecture of Liu and Wang on the q-log-convexity of the Narayana polynomials of type B. By using Pieri's rule and the Jacobi–Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. By the principal specialization, this leads to q-log-convexity. We also show that the linear transformation with respect to the triangular array of Narayana numbers of type B is log-convexity preserving.

The Symmetric Fourth Painlevé Hierarchy and Associated Special Polynomials

In this paper two families of rational solutions and associated special polynomials for the equations in the symmetric fourth Painlevé hierarchy are studied. The structure of the roots of these polynomials is shown to be highly regular in the complex plane. Further representations are given of the associated special polynomials in terms of Schur functions. The properties of these polynomials are compared and contrasted with the special polynomials associated with rational solutions of the fourth Painlevé equation.

On a boundary analogue of the Carathéodory-Schur interpolation problem

Characterization of Schur functions in terms of their Taylor coefficients is due to C. Carathéodory and I. Schur. We discuss the boundary analogue of this problem.

Hopf algebras and characters of classical groups

Schur functions provide an integral basis of the ring of symmetric functions. It is shown that this ring has a natural Hopf algebra structure by identifying the appropriate product, coproduct, unit, counit and antipode, and their properties. Characters of covariant tensor irreducible representations of the classical groups GL(n), O(n) and Sp(n) are then expressed in terms of Schur functions, and the Hopf algebra is exploited in the determination of group-subgroup branching rules and the decomposition of tensor products. The analysis is carried out in terms of n-independent, universal characters. The corresponding rings, CharGL, CharO and CharSp, of universal characters each have their own natural Hopf algebra structure. The appropriate product, coproduct, unit, counit and antipode are identified in each case.

Necessary conditions for Schur-positivity

In recent years, there has been considerable interest in showing that certain conditions on skew shapes A and B are sufficient for the difference s A −s B of their skew Schur functions to be Schur-positive. We determine necessary conditions for the difference to be Schur-positive. Specifically, we prove that if s A −s B is Schur-positive, then certain row overlap partitions for A are dominated by those for B. In fact, our necessary conditions require a weaker condition than the Schur-positivity of s A −s B ; we require only that, when expanded in terms of Schur functions, the support of s A contains that of s B . In addition, we show that the row overlap condition is equivalent to a column overlap condition and to a condition on counts of rectangles fitting inside A and B. Our necessary conditions are motivated by those of Reiner, Shaw and van Willigenburg that are necessary for s A =s B , and we deduce a strengthening of their result as a special case.

Vanishing Integrals of Macdonald and Koornwinder polynomials

When one expands a Schur function in terms of the irreducible characters of the symplectic (or orthogonal) group, the coefficient of the trivial character is 0 unless the indexing partition has an appropriate form. A number of q,t-analogues of this fact were conjectured in [10]; the present paper proves most of those conjectures, as well as some new identities suggested by the proof technique. The proof involves showing that a nonsymmetric version of the relevant integral is annihilated by a suitable ideal of the affine Hecke algebra, and that any such annihilated functional satisfies the desired vanishing property. This does not, however, give rise to vanishing identities for the standard nonsymmetric Macdonald and Koornwinder polynomials; we discuss the required modification to these polynomials to support such results.

Multiplicity Free Expansions of Schur P-Functions

After deriving inequalities on coefficients arising in the expansion of a Schur P-function in terms of Schur functions we give criteria for when such expansions are multiplicity free. From here we study the multiplicity of an irreducible spin character of the twisted symmetric group in the product of a basic spin character with an irreducible character of the symmetric group, and determine when it is multiplicity free.

Cylindric skew Schur functions

Cylindric skew Schur functions, which are a generalisation of skew Schur functions, arise naturally in the study of P-partitions. Also, recent work of A. Postnikov shows they have a strong connection with a problem of considerable current interest: that of finding a combinatorial proof of the non-negativity of the 3-point Gromov–Witten invariants. After explaining these motivations, we study cylindric skew Schur functions from the point of view of Schur-positivity. Using a result of I. Gessel and C. Krattenthaler, we generalise a formula of A. Bertram, I. Ciocan-Fontanine and W. Fulton, thus giving an expansion of an arbitrary cylindric skew Schur function in terms of skew Schur functions. While we show that no non-trivial cylindric skew Schur functions are Schur-positive, we conjecture that this can be reconciled using the new concept of cylindric Schur-positivity.

The Flagged Cauchy Determinant

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.

Partition analysis and symmetrizing operators

Using a symmetrizing operator, we give a new expression for the Omega operator used by MacMahon in Partition Analysis, and given a new life by Andrews, Paule and Riese. Our result is stated in terms of Schur functions.

The skew Schubert polynomials

We obtain a tableau definition of the skew Schubert polynomials named by Lascoux, which are defined as flagged double skew Schur functions. These polynomials are in fact Schubert polynomials in two sets of variables indexed by 321-avoiding permutations. From the divided difference definition of the skew Schubert polynomials, we construct a lattice path interpretation based on the Chen–Li–Louck pairing lemma. The lattice path explanation immediately leads to the determinantal definition and the tableau definition of the skew Schubert polynomials. For the case of a single variable set, the skew Schubert polynomials reduce to flagged skew Schur functions as studied by Wachs and by Billey, Jockusch, and Stanley. We also present a lattice path interpretation for the isobaric divided difference operators, and derive an expression of the flagged Schur function in terms of isobaric operators acting on a monomial. Moreover, we find lattice path interpretations for the Giambelli identity and the Lascoux–Pragacz identity for super-Schur functions. For the super-Lascoux–Pragacz identity, the lattice path construction is related to the code of the partition which determines the directions of the lines parallel to the y-axis in the lattice.

The square of the Vandermonde determinant and itsq-generalization

The Vandermonde determinant plays a crucial role in the quantum Hall effect via Laughlin's wavefunction ansatz. Herein the properties of the square of the Vandermonde determinant as a symmetric function are explored in detail. Important properties satisfied by the coefficients arising in the expansion of the square of the Vandermonde determinant in terms of Schur functions are developed and generalized to q-dependent coefficients via the q-discriminant. Algorithms for the efficient calculation of the q-dependent coefficients as finite polynomials in q are developed. The properties, such as the factorization of the q-dependent coefficients, are exposed. Further light is shed upon the vanishing of certain expansion coefficients at q = 1. The q-generalization of the sum rule for the squares of the coefficients is derived. A number of compelling conjectures are stated.

Q-Narayana numbers and the flag h-vector of [formula omitted

The Narayana numbers are N(n,k)=(1/n) n k n k+1 . There are several natural statistics on Dyck paths with a distribution given by N( n, k). We show the equidistribution of Narayana statistics by computing the flag h-vector of J( 2 × n ) in different ways. In the process we discover new Narayana statistics and provide co-statistics for the Narayana statistics so that the bi-statistics have a distribution given by Fürlinger and Hofbauer's q-Narayana numbers. We interpret the flag h-vector in terms of semi-standard Young tableaux, which enables us to express the q-Narayana numbers in terms of Schur functions. We also introduce what we call pre-shellings of simplicial complexes.

Scalar Products of Symmetric Functions and Matrix Integrals

We present relations between Hirota-type bilinear operators, scalar products on spaces of symmetric functions and integrals defining matrix model partition functions. Using the fermionic Fock space representation, a proof of the expansion of an associated class of KP and 2-Toda tau functions $\tau_{r,n}$ in a series of Schur functions generalizing the hypergeometric series is given and related to the scalar product formulae. It is shown how special cases of such $\tau$-functions may be identified as formal series expansions of partition functions. A closed form exapnsion of $\log \tau_{r,n}$ in terms of Schur functions is derived.

Finite Chern-Simons matrix model - algebraic approach

We analyze the algebra of observables and the physical Fock space of the finite Chern-Simons matrix model. We observe that the minimal algebra of observables acting on that Fock space is identical to that of the Calogero model. Our main result is the identification of the states in the l-th tower of the Chern-Simons matrix model Fock space and the states of the Calogero model with the interaction parameter nu=l+1. We describe quasiparticle and quasihole states in the both models in terms of Schur functions, and discuss some nontrivial consequences of our algebraic approach.

The Flagged Double Schur Function

The double Schur function is a natural generalization of the factorial Schur function introduced by Biedenharn and Louck. It also arises as the symmetric double Schubert polynomial corresponding to a class of permutations called Grassmannian permutations introduced by A. Lascoux. We present a lattice path interpretation of the double Schur function based on a flagged determinantal definition, which readily leads to a tableau interpretation similar to the original tableau definition of the factorial Schur function. The main result of this paper is a combinatorial treatment of the flagged double Schur function in terms of the lattice path interpretations of divided difference operators. Finally, we find lattice path representations of formulas for the symplectic and orthogonal characters for i>sp(2i>n) and i>so(2i>n + 1) based on the tableau representations due to King and El-Shakaway, and Sundaram. Based on the lattice path interpretations, we obtain flagged determinantal formulas for these characters.

A MAPLE program for calculations with Schur functions

We describe an interactive program, to be used in the MAPLE environment, for calculations involving Schur functions. The program performs Schur function operations of interest to applications in many-body physics: outer product, internal product, division and outer plethysm. Essential to the program is a new algorithm for the multiplication of monomial symmetric functions, which is used to implement the evaluation of outer product and outer plethysm. The same algorithm is extended to apply to the division of monomial symmetric functions, with a novel use of partitions containing negative integers, useful for implementing the evaluation of division. In particular, the evaluation of outer plethysms involving sums, differences, and products of Schur functions with this program is as simple as those involving only single Schur functions. The program can also be used to expand a Schur function in terms of monomial symmetric functions or expand a monomial symmetric function in terms of Schur functions, i.e. to calculate the Kostka and inverse Kostka numbers. The calculation of Kostka numbers makes substantial use of Gel'fand patterns. The number of variables in which Schur functions or monomial symmetric functions are expressed can be specified in advance by the user, which results in significant reduction of calculation time.