Skew Symplectic and Orthogonal Schur Functions

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.

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.

We present a realization of untwisted vertex operators in terms of operations on Schur functions. Calculations of matrix elements and traces of products of vertex operators are performed using results from the classical theory of symmetric functions. The concepts of compound, composite and supersymmetric Schur functions naturally appear in this context. Furthermore, a trace formula for a product of vertex operators turns out to be a generalization of a MacDonald identity reformulated in terms of Schur functions.

Toroidal Lie algebras and their vertex operator representations were introduced in [MEY] and a class of indecomposable modules were investigated. In this work, we extend the toroidal algebra by the Virasoro algebra thus constructing a semi-direct product algebra containing the toroidal algebra as an ideal and the Virasoro algebra as a subalgebra. With the use of vertex operators and certain oscillator representations of the Virasoro algebra it is proved that the corresponding Fock space gives rise to a class of irreducible modules for the Virasoro-toroidal algebra.

Generalized Heisenberg algebras and their vertex operator representations

A regular gradient-holonomic approach to studying the Lax type integrability of the Ablowitz–Ladik hierarchy of nonlinear Lax type integrable discrete dynamical systems in the vertex operator representation is presented. The relationship to the Lie-algebraic integrability scheme is analyzed and the connection with the τ-function representation is discussed.

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.

A new family of symmetric functions is considered. These functions are analogous to the classical Schur functions, but depend on an integer modulus p ⩾ 2 p \geqslant 2 , as well as on a partition λ \lambda . In the case where p p is prime, certain of these functions are shown to be irreducible characters of the general linear group G L ( n , K ) GL(n,K) in the natural characteristic p p of the field K K . This dualises a wellknown criterion of G. D. James for such characters to be given by classical Schur functions.

A relationship between orthogonal rational functions and discrete integrable systems is studied by an approach based on Schur-type symmetric functions. A system of orthogonal rational functions is constructed using a multiparameter deformation of the Schur functions. Spectral equations for the orthogonal rational functions are derived by using properties of the Schur-type symmetric functions. The compatibility condition of the spectral equations induces a discrete dressing chain which is a Toda-type discrete integrable system describing dressing transformations for orthogonal rational functions.

Lam and Pylyavskyy introduced loop symmetric functions as a generalization of symmetric functions. They defined loop Schur functions as generating functions over semistandard tableaux with respect to a `colored weight,' and they proved a Jacobi--Trudi-style determinantal formula for these generating functions. We prove that loop Schur functions can be expressed as a ratio of `loop alternants,' extending the analogy with Schur functions. As an application, we give a new proof of the loop version of the Murnaghan--Nakayama rule.

We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials, $k$-Schur functions, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $GL_n$, Grothendieck functions $\{G_\lambda\}$ represent the $K$-theory of the same space. In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$.

Green Functions Associated to Complex Reflection Groups

We establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by permutations into Grassmannian stable Grothendieck polynomials. Our expansion is the K K -theoretic analogue of that of a Schubert polynomial into Demazure characters, whose symmetric analogue is the expansion of a Stanley symmetric function into Schur functions. Our expansions extend to flagged Grothendieck polynomials.

The Jack polynomials J (x; ) form a remarkable class of symmetric polynomials. They are indexed by a partition and depend on a parameter . One of their properties is that several classical families of symmetric functions can be obtained by specializing , e.g., the monomial symmetric functions m ( =∞), the elementary functions e ′ ( = 0), the Schur functions s ( = 1) and nally the two classes of zonal polynomials ( = 2, = 1=2). The Jack polynomials can be de ned in various ways, e.g.: a) as an orthogonal family of functions which is compatible with the canonical ltration of the ring of symmetric functions or b) as simultaneous eigenfunctions of certain di erential operators (the Sekiguchi–Debiard operators). Recently Opdam, [O], constructed a similar family F (x; ) of nonsymmetric polynomials. The index runs now through all compositions ∈ Nn. They are de ned in a completely similar fashion, e.g., the Sekiguchi–Debiard operators are being replaced by the Cherednik di erential-re ection operators (see Sect. 3). It is becoming more and more clear that these polynomials are as important as their symmetric counterparts. The purpose of this paper is to add to the existing characterizations of Jack polynomials two further ones: c) a recursion formula among the F together with two formulas to obtain J from them. d) combinatorial formulas of both J and F in terms of certain generalized tableaux.

Hall–Littlewood Vertex Operators and Generalized Kostka Polynomials