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# Schur Functions: Theme And Variations

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## Abstract

this article we shall survey various generalizations, analogues and deformations of Schur functions --- some old, some new --- that have been proposed at various times. We shall present these as a sequence of variations on a theme and (unlike e.g. Bourbaki) we shall proceed from the particular to the general. Thus Variations 1 and 2 are included in Variation 3 ; Variations 4 and 5 are particular cases of Variation 6 ; and in their turn Variations 6, 7 and 8 (in part) are included in Variation 9. To introduce our theme, we recall [M 1 , Ch. I, x 3] that the Schur function s (x 1 ; : : : ; xn ) (where x 1 , : : : , xn are independent indeterminates and = ( 1 ; : : : ; n ) is a partition of length n) may be defined as the quotient of two alternants : (0:1) s (x 1 ; : : : ; xn ) = det

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... As a consequence of the Jacobi-Trudi formula, the functions s λ/µ;a,b can be identified with a special case of the ninth variation introduced by Macdonald in [Mac92]. Therefore, we have the following corollary. ...
... Therefore, we have the following corollary. We refer to [Mac92] for notation. ...
... These are formal consequences of the Macdonald. These are formulas (9.6), (9.6'), (9.7), and (9.9) in [Mac92]. ...
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In this paper, we introduce a new family of Schur functions that depend on two sets of variables and two doubly infinite sequences of parameters. These functions generalize and unify various existing Schur functions, including classical Schur functions, factorial Schur functions, supersymmetric Schur functions, Frobenius-Schur functions, factorial supersymmetric Schur functions, and dual Schur functions. We prove that the new family of functions satisfies several well-known properties, such as the combinatorial description, Jacobi-Trudi identity, N\"agelsbach-Kostka formula, Giambelli formula, Ribbon formula, Weyl formula, Berele-Regev factorization, and Cauchy identity. Our approach is based on the integrable six vertex model with free fermionic weights. We show that these weights satisfy the \textit{refined Yang-Baxter equation}, which results in supersymmetry for the Schur functions. Furthermore, we derive refined operator relations for the row transfer operators and use them to find partition functions with various boundary conditions. Our results provide new proofs for known results as well as new identities for the Schur functions.
... In Section 4 we consider particular cases of the parameters x, y, r, s under which the functions F λ , G λ become either the ordinary Schur symmetric polynomials [Mac95,I.3], or their factorial or supersymmetric variations [BR87], [Mac92], [Mol09]. Here let us formulate the supersymmetric setting. ...
... where s λ (· · · / · · · ) denotes the supersymmetric Schur function [BR87], [Mac92,(6.19)]. ...
... The supersymmetric Schur functions are related to the factorial Schur polynomials which appeared in Section 4.1, but we do not need this connection here. See [BR87], [Mac92,(6.19)] for details. The next statement is independent of the explicit formulas of Theorems 3.9 and 3.10 (while may also be derived as a corollary of Theorem 3.9, see Corollary 4.11 below). ...
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Our work deals with symmetric rational functions and probabilistic models based on the fully inhomogeneous six vertex (ice type) model satisfying the free fermion condition. Two families of symmetric rational functions $F_\lambda,G_\lambda$ are defined as certain partition functions of the six vertex model, with variables corresponding to row rapidities, and the labeling signatures $\lambda=(\lambda_1\ge \ldots\ge \lambda_N)\in \mathbb{Z}^N$ encoding boundary conditions. These symmetric functions generalize Schur symmetric polynomials, as well as some of their variations, such as factorial and supersymmetric Schur polynomials. Cauchy type summation identities for $F_\lambda,G_\lambda$ and their skew counterparts follow from the Yang-Baxter equation. Using algebraic Bethe Ansatz, we obtain a double alternant type formula for $F_\lambda$ and a Sergeev-Pragacz type formula for $G_\lambda$. In the spirit of the theory of Schur processes, we define probability measures on sequences of signatures with probability weights proportional to products of our symmetric functions. We show that these measures can be viewed as determinantal point processes, and we express their correlation kernels in a double contour integral form. We present two proofs: The first is a direct computation of Eynard-Mehta type, and the second uses non-standard, inhomogeneous versions of fermionic operators in a Fock space coming from the algebraic Bethe Ansatz for the six vertex model. We also interpret our determinantal processes as random domino tilings of a half-strip with inhomogeneous domino weights. In the bulk, we show that the lattice asymptotic behavior of such domino tilings is described by a new determinantal point process on $\mathbb{Z}^{2}$, which can be viewed as an doubly-inhomogeneous generalization of the extended discrete sine process.
... In this section, we present efficient subtraction-free algorithms for computing double and supersymmetric Schur polynomials. These polynomials play important role in representation theory and other areas of mathematics, see, e.g., [12,20,22] and references therein. Our notational conventions are close to those in [20,6th Variation]; the latter differ from some other literature including [22]. ...
... These polynomials play important role in representation theory and other areas of mathematics, see, e.g., [12,20,22] and references therein. Our notational conventions are close to those in [20,6th Variation]; the latter differ from some other literature including [22]. ...
... Among many equivalent definitions of supersymmetric Schur functions (or super-Schur functions for short), we choose the one most convenient for our purposes, due to I. Goulden-C. Greene [12] and I. G. Macdonald [20]. We assume the reader's familiarity with the concepts of a Young diagram and a semistandard Young tableau (of some shape λ); see, e.g., [21,32] for precise definitions. ...
... Macdonald's ninth variation Schur function is the last type of Schur function introduced in his seminal "Schur Functions: Theme and Variations" paper [Mac1]. In some sense it is the ultimate generalization as it includes as special cases five of the other generalizations listed in that 1992 paper. ...
... Although our motivation is the ninth variation Q-functions, we reverse the exposition to deal with the more basic ninth variation Schur functions first. Our tableau version of Macdonald's ninth variation of Schur functions [Mac1], proceeds by introducing weight parameters X = (x kc ) for k ∈ {1, 2, . . . , n} and c ∈ Z so as to define s λ/µ (X) = T ∈T λ/µ (i,j)∈F λ/µ wgt(t ij ), (1.1) ...
... x k Schur functions [Sch,Mac1] ...
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Recently Okada defined algebraically ninth variation skew Q-functions, in parallel to Macdonald's ninth variation skew Schur functions. Here we introduce a skew shifted tableaux definition of these ninth variation skew Q-functions, and prove by means of a non-intersecting lattice path model a Pfaffian outside decomposition result in the form of a ninth variation version of Hamel's Pfaffian outside decomposition identity. As corollaries to this we derive Pfaffian identities generalizing those of Josefiak-Pragacz, Nimmo, and most recently Okada. As a preamble to this we present a parallel development based on (unshifted) semistandard tableaux that leads to a ninth variation version of the outside decomposition determinantal identity of Hamel and Goulden. In this case the corollaries we offer include determinantal identities generalizing the Schur and skew Schur function identities of Jacobi-Trudi, Giambelli, Lascoux-Pragacz, Stembridge, and Okada.
... The presentation is structured as follows. After setting up the notation for the Gauss quadrature rule in Section 2, we emphasize in Section 3 the effectiveness of the Cauchy-Binet-Andréief formulas when extending the underlying family of orthogonal polynomials to the multivariate level via associated generalized Schur polynomials [M92,BSX95,NNSY00,SV14]. This readily allows to recover a Gaussian cubature rule for the integrations of symmetric functions from [BSX95,Equation (8)] (with ρ = 0) in Section 4. In Section 5 we highlight the explicit cubature rules stemming the classical Hermite, Laguerre and Jacobi families, which permit the exact integration of symmetric polynomials with respect to the densities of the corresponding unitary ensembles. ...
... the generalized Schur polynomial P λ (x) associated with the orthonormal system p 0 (x), p 1 (x), p 2 (x), . . . is defined via the determinantal formula (cf. [M92,BSX95,NNSY00,SV14]) ...
... e.g. [M92,Equation (0.1)]). Hence-up to normalization-the top-degree terms of P λ (x) are given by S λ (x). ...
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We employ a multivariate extension of the Gauss quadrature formula, originally due to Berens, Schmid and Xu, so as to derive cubature rules for the integration of symmetric functions over hypercubes (or infinite limiting degenerations thereof) with respect to the densities of unitary random matrix ensembles. Our main application concerns the explicit implementation of a class of cubature rules associated with the Bernstein-Szegö polynomials, which permit the exact integration of symmetric rational functions with prescribed poles at coordinate hyperplanes against unitary circular Jacobi distributions stemming from the Haar measures on the symplectic and the orthogonal groups.
... This impression is incardinated by Macdonald. In [3,4] in discussing generalizations of the Schur polynomial, in the final generalization, which combines many generalizations given previously, Macdonald regards the Jacobi-Trudi identity as the definition of the Schur functions in arbitrary representations out of the complete symmetric functions (the Schur functions in the totally symmetric representations). Namely, we prepare a set of functions as the complete symmetric functions and define the generalized Schur functions in any representation out of them using the Jacobi-Trudi identity (possibly along with an automorphism). ...
... From the viewpoint of the Macdonald's generalization of the Schur polynomials [3], it is natural to ask whether the Giambelli identity which is satisfied by the ABJM matrix model is lifted to the Jacobi-Trudi identity, and if yes, what the automorphism is. ...
... Note that for the case of R ′ = 0 where |M| is large enough so that the shifted diagonal does not intersect with the Young diagram, the block of S ′ (α ′ |β ′ ) is missing and the shifted Giambelli compatibility resembles the Weyl formula. Also note that S ′ (α ′ |β ′ ) and H ℓ appearing in the definitions coincide with S ′ (α ′ |β ′ ) = S 0 (α ′ =α|β ′ =β) and H ℓ = S [ℓ] respectively as obtained from an appropriate choice of λ and M. The Jacobi-Trudi compatibility is utilized in [3] to define an ultimate variation of the Schur functions in arbitrary representations from the complete symmetric functions H ℓ = S [ℓ] , containing many generalizations by specific choices of H M ℓ . The defining equation (1.11) can alternatively be expressed as ...
Article
It was proved by Macdonald that the Giambelli identity holds if we define the Schur functions using the Jacobi-Trudi identity. Previously for the super Chern-Simons matrix model (the spherical one-point function of the superconformal Chern-Simons theory describing the worldvolume of the M2-branes) the Giambelli identity was proved from a shifted version of it. With the same shifted Giambelli identity we can further prove the Jacobi-Trudi identity, which strongly suggests an integrable structure for this matrix model.
... Beidenharn and Louck [1] introduced the notion of a factorial Schur function defined in terms of the Gelfand-Tsetlin patterns associated with GL(n, C). This was further studied by Chen and Louck [3] before it was given its more general form by Goulden and Greene [8] and Macdonald [19], expressed this time in terms of semistandard tableaux, with Macdonald also giving an alternative definition as a ratio of alternants. It is this latter form that we wish to generalise to the case of the other classical Lie groups G. ...
... This results in our proposed formulae for factorial characters g λ (x|a) given in Definition 1. Following Macdonald [19], it is useful to introduce a shift operator τ defined in such a way that τ r a = (a r+1 , a r+2 , . . .) for any integer r and any a = (a 1 , a 2 , . . ...
... That the definitions are appropriate depends to what extent the properties of factorial characters are analogous to those of factorial Schur functions. We have in mind things like deriving for each of our factorial characters a factorial Jacobi-Truditype identity and a combinatorial interpretation in terms of tableau, both of which were established in the case of factorial Schur functions by Macdonald [19,20], and perhaps more ambitiously, the derivation of Tokuyama type identities [27] as recently derived in the factorial Schur function case by Bump, McNamara and Nakasuji [2], with an alternative derivation appearing in [9]. This latter step necessarily requires generalisations of the classical Schur Q-functions [23,25] first from the general linear case to that of the other classical Lie groups, and then to factorial versions of these. ...
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Just as the definition of factorial Schur functions as a ratio of determinants allows one to show that they satisfy a Jacobi-Trudi-type identity and have an explicit combinatorial realisation in terms of semistandard tableaux, so we offer here definitions of factorial irreducible characters of the classical Lie groups as ratios of determinants that share these two features. These factorial characters are each specified by a partition, $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_n)$, and in each case a flagged Jacobi-Trudi identity is derived that expresses the factorial character as a determinant of corresponding factorial characters specified by one-part partitions, $(m)$, for which we supply generating functions. These identities are established by manipulating determinants through the use of certain recurrence relations derived from these generating functions. The transitions to combinatorial realisations of the factorial characters in terms of tableaux are then established by means of non-intersecting lattice path models. The results apply to $gl(n)$, $so(2n+1)$, $sp(2n)$ and $o(2n)$, and are extended to the case of $so(2n)$ by making use of newly defined factorial difference characters.
... The algebra Λ * (n) of shifted symmetric polynomials is a remarkable deformation of the algebra Λ(n) of symmetric polynomials and its study fits into the mainstream of generalizations of the classical theory (see, e.g. factorial symmetric functions, [6,7,21,30,31,40,41]). ...
... We call the decomposition (40) the Olshanski decomposition of the element ∈ ζ(n + 1). In this notation, the projection μ n,n+1 : ζ(n + 1) ζ(n), μ n+1,n ( ) = , ∈ ζ(n + 1) is defined. ...
Article
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In this paper, we introduced two classes of elements in the enveloping algebra [Formula: see text]: the double Young–Capelli bitableaux [Formula: see text] and the central Schur elements [Formula: see text], that act in a remarkable way on the highest weight vectors of irreducible Schur modules. Any element [Formula: see text] is the sum of all double Young–Capelli bitableaux [Formula: see text], [Formula: see text] row (strictly) increasing Young tableaux of shape [Formula: see text]. The Schur elements [Formula: see text] are proved to be the preimages — with respect to the Harish-Chandra isomorphism — of the shifted Schur polynomials [Formula: see text]. Hence, the Schur elements are the same as the Okounkov quantum immanants, recently described by the present authors as linear combinations of Capelli immanants. This new presentation of Schur elements/quantum immanants does not involve the irreducible characters of symmetric groups. The Capelli elements [Formula: see text] are column Schur elements and the Nazarov elements [Formula: see text] are row Schur elements. The duality in [Formula: see text] follows from a combinatorial description of the eigenvalues of the [Formula: see text] on irreducible modules that is dual (in the sense of shapes/partitions) to the combinatorial description of the eigenvalues of the [Formula: see text]. The passage [Formula: see text] for the algebras [Formula: see text] is obtained both as direct and inverse limit in the category of filtered algebras, via the Olshanski decomposition/projection.
... In the paper "Schur functions: theme and variations" I. G. Macdonald [Mac92] introduced nine analogues of Schur functions, which he called variations. The 6th variation is the factorial Schur function. ...
... Let c " pc n q ně0 be a sequence of indeterminates. Taking f k ptq " pt | cq k " pt´c 0 q¨¨¨pt´c k´1 q, we obtain the factorial Schur functions s λ px | cq (see Macdonald [Mac92], Molev [Mol09]). ...
Preprint
We introduce certain generalisations of the characters of the classical Lie groups, extending the recently defined factorial characters of Foley and King. This is done by replacing the factorial powers with a sequence of polynomials. In particular, we offer a ninth variation generalisation for the rational Schur functions. We derive Littlewood-type identities for our generalisations.
... Proof. It is enough to check (26) for the basis elements x = V (ν). We compute ...
... It is easy to see that F λ (x) is a non-homogeneous polynomial of degree |λ|, and the top degree component equals (−1) |λ|+( N 2 ) q D N (λ) s λ where s λ is the Schur function and D N (λ) is defined by (6). The function F λ (x) is known as a special case of a factorial Schur function [26,27,28], it is also a specialization of nonsymmetric Macdonald polynomials described below. ...
Article
In this paper we construct a new basis for the cyclotomic completion of the center of the quantum glN in terms of the interpolation Macdonald polynomials. Then we use a result of Okounkov to provide a dual basis with respect to the quantum Killing form (or Hopf pairing). The main applications are: 1) cyclotomic expansions for the glN Reshetikhin--Turaev link invariants and the universal glN knot invariant; 2) an explicit construction of the unified glN invariants for integral homology 3-spheres using universal Kirby colors. These results generalize those of Habiro for sl2. In addition, we give a simple proof of the fact that the universal glN invariant of any evenly framed link and the universal slN invariant of any 0-framed algebraically split link are Γ-invariant, where Γ=Y/2Y with the root lattice Y.
... Factorial Schur functions-introduced by Biedenharn and Louck [1], extended by Goulden and Greene [6] and Macdonald [12]-have a fundamental role to play in symmetric function theory, and are intimately connected to Schubert calculus through the double Schubert polynomials. As such, they have been the object of much active research, and numerous results and applications have been established for them [9]. ...
... using (12). Applying this to the numerator and denominator of (4) with m = λ j + n − j and m = n − j, the result follows from cancellation of factors x i − x i . ...
Article
International audience In this paper we introduce factorial characters for the classical groups and derive a number of central results. Classically, the factorial Schur function plays a fundamental role in traditional symmetric function theory and also in Schubert polynomial theory. Here we develop a parallel theory for the classical groups, offering combinatorial definitions of the factorial characters for the symplectic and orthogonal groups, and further establish flagged factorial Jacobi-Trudi identities and factorial Tokuyama identities, providing proofs in the symplectic case. These identities are established by manipulating determinants through the use of certain recurrence relations and by using lattice paths.
... The next step is to use the dual Cauchy identity [55,56] ...
... The dual Cauchy identity (2.2) yields the expansion in Schur polynomials of the correlation function of characteristic polynomials in any beta-ensemble. However, denoting the beta-parameter by 2γ, it is more convenient in the γ = 1 setup to use the alternative dual Cauchy identity [56] (2.5) ...
Preprint
We give expansions of reproducing kernels of the Christoffel-Darboux type in terms of Schur polynomials. For this, we use evaluations of averages of characteristic polynomials and Schur polynomials in random matrix ensembles. We explicitly compute new Schur averages, such as the Schur average in a $q$-Laguerre ensemble, and the ensuing expansions of random matrix kernels. In addition to classical and $q$-deformed cases on the real line, we use extensions of Dotsenko-Fateev integrals to obtain expressions for kernels on the complex plane. Moreover, a known interplay between Wronskians of Laguerre polynomials, Painlev\'e tau functions and conformal block expansions is discussed in relationship to the Schur expansion obtained.
... Combining the previous results we obtain an explicit expression for the unified invariant using the universal Kirby color (25) I ...
... It is easy to see that F λ (x) is a non-homogeneous polynomial of degree |λ|, and the top degree component equals (−1) |λ|+( N 2 ) q D N (λ) s λ where s λ is the Schur function and D N (λ) is defined by (5). The function F λ (x) is known as a special case of a factorial Schur function [25,26,27], it is also a specialization of nonsymmetric Macdonald polynomials described below. ...
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In this paper we construct a new basis for the cyclotomic completion of the center of the quantum $\mathfrak{gl}_N$ in terms of the interpolation Macdonald polynomials. Then we use a result of Okounkov to provide a dual basis with respect to the quantum Killing form (or Hopf pairing). Two main applications are: 1) a cyclotomic expansion of the universal $\mathfrak{gl}_N$ knot invariant and 2) an explicit construction of the unified $\mathfrak{gl}_N$ invariants for integer homology 3-spheres obtained by knot surgeries. These results generalize those of Habiro for $\mathfrak{sl}_2$. In addition, we give a simple proof of the fact that the universal $\mathfrak{gl}_N$ invariant of any evenly framed link and the universal $\mathfrak{sl}_N$ invariant of any $0$-framed algebraically split link are $\Gamma$-invariant, where $\Gamma=Y/2Y$ with the root lattice $Y$.
... Schur functions have been extensively studied and there are numerous generalizations and variations of them in the literature. In particular, Macdonald [10] introduced nine variations of Schur functions. Macdonald's 9th variation of Schur functions generalize Schur functions and many of their variations. ...
... Macdonald's 6th variation s λ (x|a) of Schur functions [10], also known as factorial Schur functions, are defined by ...
Preprint
In the literature there are several determinant formulas for Schur functions: the Jacobi-Trudi formula, the dual Jacobi-Trudi formula, the Giambelli formula, the Lascoux-Pragacz formula, and the Hamel-Goulden formula, where the Hamel-Goulden formula implies the others. In this paper we use an identity proved by Bazin in 1851 to derive determinant identities involving Macdonald's 9th variation of Schur functions. As an application we prove a determinant identity for factorial Schur functions conjectured by Morales, Pak, and Panova. We also obtain a generalization of the Hamel-Goulden formula, which contains a result of Jin, and prove a converse of the Hamel-Goulden theorem and its generalization.
... A large number of examples of lattices of polynomial KP Schur functions, labelled by partitions λ, that can be expressed in terms of systems of monic polynomials {p j (x|Ã)} j∈N + via formula (3.5) are detailed in [7,17,26]. These include: several of the various types of generalized Schur functions studied in [17], the shifted Schur functions [20], and orthogonal and symplectic group characters [7,26]. ...
... A large number of examples of lattices of polynomial KP Schur functions, labelled by partitions λ, that can be expressed in terms of systems of monic polynomials {p j (x|Ã)} j∈N + via formula (3.5) are detailed in [7,17,26]. These include: several of the various types of generalized Schur functions studied in [17], the shifted Schur functions [20], and orthogonal and symplectic group characters [7,26]. ...
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Lattices of polynomial KP and BKP $\tau$-functions labelled by partitions, with the flow variables equated to finite power sums, as well as associated multipair KP and multipoint BKP correlation functions are expressed via generalizations of Jacobi's bialternant formula for Schur functions and Nimmo's Pfaffian ratio formula for Schur $Q$-functions. These are obtained by applying Wick's theorem to fermionic vacuum expectation value representations in which the infinite group element acting on the lattice of basis states stabilizes the vacuum.
... i , which is a contradiction to the assumption i < N . We now prove (17). By definition, ...
... By the above claim, as n → ∞, the condition (21) is equivalent to α 1 ≥ · · · ≥ α k ≥ 0, which implies (17). ...
Article
We introduce lecture hall tableaux, which are fillings of a skew Young diagram satisfying certain conditions. Lecture hall tableaux generalize both lecture hall partitions and anti-lecture hall compositions, and also contain reverse semistandard Young tableaux as a limit case. We show that the coefficients in the Schur expansion of multivariate little $q$-Jacobi polynomials are generating functions for lecture hall tableaux. Using a Selberg-type integral we show that moments of multivariate little $q$-Jacobi polynomials, which are equal to generating functions for lecture hall tableaux of a Young diagram, have a product formula. We also explore various combinatorial properties of lecture hall tableaux.
... Example 4.2 (Multiparameter Schur polynomials, cf. Macdonald [16], Molev [18]). Let (c 0 , c 1 , c 2 , . . . ) be an infinite sequence of parameters and ...
... If c 0 = c 1 = · · · = 0, they turn into the ordinary Schur polynomials, and in the case c i = i they give the factorial Schur polynomials introduced by Biedenharn and Louck [1], [2] and further studied in a number of works, see, e.g., Chen and Louck [8], Goulden and Hamel [12], Macdonald [16]. (Note that in some recent works, the polynomials s µ|N (x 1 , . . . ...
Article
Let Sym denote the algebra of symmetric functions and $P_\mu(\,\cdot\,;q,t)$ and $Q_\mu(\,\cdot\,;q,t)$ be the Macdonald symmetric functions (recall that they differ by scalar factors only). The $(q,t)$-Cauchy identity $$\sum_\mu P_\mu(x_1,x_2,\dots;q,t)Q_\mu(y_1,y_2,\dots;q,t)=\prod_{i,j=1}^\infty\frac{(x_iy_jt;q)_\infty}{(x_iy_j;q)_\infty}$$ expresses the fact that the $P_\mu(\,\cdot\,;q,t)$'s form an orthogonal basis in Sym with respect to a special scalar product $\langle\,\cdot\,,\,\cdot\,\rangle_{q,t}$. The present paper deals with the inhomogeneous \emph{interpolation} Macdonald symmetric functions $$I_\mu(x_1,x_2,\dots;q,t)=P_\mu(x_1,x_2,\dots;q,t)+\text{lower degree terms}.$$ These functions come from the $N$-variate interpolation Macdonald polynomials, extensively studied in the 90's by Knop, Okounkov, and Sahi. The goal of the paper is to construct symmetric functions $J_\mu(\,\cdot\,;q,t)$ with the biorthogonality property $$\langle I_\mu(\,\cdot\,;q,t), J_\nu(\,\cdot\,;q,t)\rangle_{q,t}=\delta_{\mu\nu}.$$ These new functions live in a natural completion of the algebra Sym. As a corollary one obtains a new Cauchy-type identity in which the interpolation Macdonald polynomials are paired with certain multivariate rational symmetric functions. The degeneration of this identity in the Jack limit is also described.
... i, j ≤ d) be the Schur polynomial associated to c = (c 1 , . . . , c d ) ([Mac92]). It is a symmetric polynomial of degree |c|. WriteS := c S c (qx 1 , . . . ...
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We prove an explicit degree formula for certain unitary Deligne-Lusztig varieties. Combining with an alternative degree formula in terms of Schubert calculus, we deduce several algebraic combinatorial identities which may be of independent interest.
... In this paper, the key role is played by the factorial Grothendieck polynomials [28,37], which generalize both the well-studied Grothendieck polynomials and factorial symmetric functions. The latter were also first introduced by Lascoux and Schützenberger [31] in the guise of double 634 Schubert polynomials for Grassmannian permutations, and have been systematically studied by Macdonald [35], see also [7] for further background. ...
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We present a new family of hook-length formulas for the number of standard increasing tableaux which arise in the study of factorial Grothendieck polynomials. In the case of straight shapes, our formulas generalize the classical hook-length formula and the Littlewood formula. For skew shapes, our formulas generalize the Naruse hook-length formula and its q-analogs, which were studied in previous papers of the series.
... When k < n, one can prove (5.19) by a method similar to that of the proof of [13, Theorem 1.2], i.e., by using the induction on m + k and appealing to Ciucu's generalization [7, Corollary 1] of Kuo condensation.A. More determinant formulas for spIn this appendix, we apply the theory of Macdonald's ninth variation of Schur functions[20] to derive dual Jacobi-Trudi and Giambelli formulas for intermediate symplectic characters. Recall Macdonald's ninth variation of Schur functions. ...
... Factorial Schur functions were first introduced by Biedenharn and Louck in [BL89]. Macdonald generalized those functions into the form used in this paper, in [Mac92, 6th Variation]: ...
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We use Bott-Samelson resolutions of Schubert varieties in Grassmannians along with equiariant localization techniques to show that the factorial Schur functions and the factorial Grothendieck polynomials represent Schubert classes in equivariant cohomology and equivariant K-theory respectively.
... The algebra Λ * (n) of shifted symmetric polynomials is a remarkable deformation of the algebra Λ(n) of symmetric polynomials and its study fits into the mainstream of generalizations of the classical theory (see, e.g. factorial symmetric functions, [6], [7], [23], [30], [31], [40], [41]). ...
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In this paper are introduced two classes of elements in the enveloping algebra $\mathbf{U}(gl(n))$: the \emph{double Young-Capelli bitableaux} $[\ \fbox{$S \ | \ T$}\ ]$ and the \emph{central} \emph{Schur elements} $\mathbf{S}_{\lambda}(n)$, that act in a remarkable way on the highest weight vectors of irreducible Schur modules. Any element $\mathbf{S}_{\lambda}(n)$ is the sum of all double Young-Capelli bitableaux $[\ \fbox{$S \ | \ S$}\ ]$, $S$ row (strictly) increasing Young tableaux of shape $\widetilde{\lambda}$. The Schur elements $\mathbf{S}_\lambda(n)$ are proved to be the preimages - with respect to the Harish-Chandra isomorphism - of the \emph{shifted Schur polynomials} $s_{\lambda|n}^* \in \Lambda^*(n)$. Hence, the Schur elements are the same as the Okounkov \textit{quantum immanants}, recently described by the present authors as linear combinations of \emph{Capelli immanants}. This new presentation of Schur elements/quantum immanants doesn't involve the irreducible characters of symmetric groups. The Capelli elements $\mathbf{H}_k(n)$ are column Schur elements and the Nazarov-Umeda elements $\mathbf{I}_k(n)$ are row Schur elements. The duality in $\boldsymbol{\zeta}(n)$ follows from a combinatorial description of the eigenvalues of the $\mathbf{H}_k(n)$ on irreducible modules that is {\it{dual}} (in the sense of shapes/partitions) to the combinatorial description of the eigenvalues of the $\mathbf{I}_k(n)$. The passage $n \rightarrow \infty$ for the algebras $\boldsymbol{\zeta}(n)$ is obtained both as direct and inverse limit in the category of filtered algebras, via the \emph{Olshanski decomposition/projection}.
... They are particular cases of the factorial or shifted Schur polynomials; see [9] and [15]. and . . . ...
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We consider the centers of the affine vertex algebras at the critical level associated with simple Lie algebras. We derive new formulas for generators of the centers in the classical types. We also give a new formula for the Capelli-type determinant for the symplectic Lie algebras and calculate the Harish-Chandra images of the Casimir elements arising from the characteristic polynomial of the matrix of generators of each classical Lie algebra.
... These polynomials P ip m z; d 2 were introduced by Sahi [13], Knop-Sahi [3] and Okounkov-Olshanski [10] as a continuous deformation of shifted Schur polynomials P ip m (z; 1) [6,11]. The interpolation Jack polynomials appear as a multivariate analogue of the falling factorials ...
Article
We present new Pieri type formulas for Jack polynomials. As an application, we give a new derivation of higher order difference equations for interpolation Jack polynomials originally found by Knop and Sahi. We also propose Pieri formulas for interpolation Jack polynomials and intertwining relations for a kernel function for Jack polynomials.
... In this appendix, we apply the theory of Macdonald's ninth variation of Schur functions [15] to derive Giambelli and dual Jacobi- ...
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We use intermediate symplectic characters to give a proof and variations of Hopkins' conjecture, now proved by Hopkins and Lai, on the number of shifted plane partitions of shifted double staircase shape with bounded entries. In fact, we prove some character identities involving intermediate symplectic characters, and find generating functions for such shifted plane partitions. The key ingredients of the proof are a bialternant formula for intermediate symplectic characters, which interpolates between those for Schur functions and symplectic characters, and the Ishikawa-Wakayama minor-summation formula.
... Note that from this analogue of the bialternant definition of Schur functions we conclude that s λ (x | a) is symmetric in x. Given a skew shape θ and a two-dimensional array of variables z = (z i,j ) i≥1,j∈Z , define the ninth variation Schur function defined by Macdonald [Mac92] and recently studied in [Nak+01; BC20; FK20]. ...
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The classical hook length formula counts the number of standard tableaux of straight shapes. In 1996, Okounkov and Olshanski found a positive formula for the number of standard Young tableaux of a skew shape. We prove various properties of this formula, including three determinantal formulas for the number of nonzero terms, an equivalence between the Okounkov-Olshanski formula and another skew tableaux formula involving Knutson-Tao puzzles, and two $q$-analogues for reverse plane partitions, which complements work by Stanley and Chen for semistandard tableaux. We also give several reformulations of the formula, including two in terms of the excited diagrams appearing in a more recent skew tableaux formula by Naruse. Lastly, for thick zigzag shapes we show that the number of nonzero terms is given by a determinant of the Genocchi numbers and improve on known upper bounds by Morales-Pak-Panova on the number of standard tableaux of these shapes.
... 2π Ω m 1 ,n 1w e use the following Cauchy formula (see [19]) ...
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In this article, we obtain a recursive description of the Horn cone Horn(p,q) with respect to the integers p and q, as in the classical Horn's conjecture.
... For large skew shapes, Okounkov-Olshanski [OO98] and Stanley [Sta03] computed the asymptotics of f (λ/µ) for fixed µ, as |λ| → ∞. Both papers rely on the factorial Schur functions introduced by Macdonald in [Mac92,§6]. The Naruse hook-length formula (NHLF) was introduced by Hiroshi Naruse in a talk in 2014, and given multiple proofs and generalizations in [MPP1,MPP2]. ...
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For a finite poset $P=(X,\prec)$, let $\mathcal{L}_P$ denote the set of linear extensions of $P$. The sorting probability $\delta(P)$ is defined as $\delta(P) \, := \, \min_{x,y\in X} \, \bigl| \mathbf{P} \, [L(x)\leq L(y) ] \ - \ \mathbf{P} \, [L(y)\leq L(x) ] \bigr|\,,$ where $L \in \mathcal{L}_P$ is a uniform linear extension of $P$. We give asymptotic upper bounds on sorting probabilities for posets associated with large Young diagrams and large skew Young diagrams, with bounded number of rows.
... In notation of §2.3, when a = 0 as in Figure 4, one can think of F A,B (x , |y ) in Theorem 2.4 the multivariate deformation of N (λ). This deformation is different, but curiously similar to the x -symmetric and y -parametrized factorial Schur functions s λ (x | − y ), which forms a basis in symmetric polynomials of x , see [Mac,§6]. This should not come as a surprise as the proof in [MPP3] is based on combinatorics and algebra of factorial Schur functions. ...
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We study the weighted partition function for lozenge tilings, with weights given by multivariate rational functions originally defined by Morales, Pak and Panova (2019) in the context of the factorial Schur functions. We prove that this partition function is symmetric for large families of regions. We employ both combinatorial and algebraic proofs.
... For the basic properties of the symmetric polynomials, see Macdonald. 5 Some recent studies on symmetric polynomials include, for example, literature [17][18][19][20][21][22][23][24][25] (see also previous works [26][27][28][29][30][31][32]. ...
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This paper provides some characteristic properties of the weighted particular Schur polynomial mean of several variables. In addition, an elementary proof of an important inequality involving the weighted particular Schur polynomial mean is given. Various related results involving a family of the Schur polynomials, symmetric polynomials, and other associated polynomials, together with the potential for their applications, are also considered.
... There are several generalizations, variations or deformations of Schur functions, such as Hall-Littlewood functions, Macdonald functions and factorial Schur functions. The generalization relevant to this paper is Macdonald's ninth variation ( [10], see also [15]) associated to a polynomial sequence, which is defined as follows. ...
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We introduce and study a generalization of Schur's $P$-/$Q$-functions associated to a polynomial sequence, which can be viewed as Macdonald's ninth variation'' for $P$-/$Q$-functions. This variation includes as special cases Schur's $P$-/$Q$-functions, Ivanov's factorial $P$-/$Q$-functions and the $t=-1$ specialization of Hall--Littlewood functions associated to the classical root systems. We establish several identities and properties such as generalizations of Schur's original definition of Schur's $Q$-functions, Cauchy-type identity, J\'ozefiak--Pragacz--Nimmo formula for skew $Q$-functions, and Pieri-type rule for multiplication.
... In fact, in the study of soliton systems or solvable lattice systems, similar formulas appear (see, for example, [17][18][19][20]). Also, in [21,22] in a series of variations of the Schur function, as a final one called the ninth variation, the Jacobi-Trudi formula was regarded as the definition of the Schur function and many interesting properties such as the Giambelli formula can still be derived from this simple setup. Hence, the proof of the Giambelli formula or the Jacobi-Trudi formula in the ABJM matrix model strongly suggests the integrable structure in behind. ...
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A bstract It was known that one-point functions in the ABJM matrix model (obtained by applying the localization technique to one-point functions of the half-BPS Wilson loop operator in the ABJM theory) satisfy the Jacobi-Trudi formula, which strongly indicates the integrable structure of the system. In this paper, we identify the integrable structure of two-point functions in the ABJM matrix model as the two-dimensional Toda lattice hierarchy. The identification implies infinitely many non-linear differential equations for the generating function of the two-point functions.
... In fact, in the study of soliton systems or solvable lattice systems, similar formulas appear (see, for example, [17][18][19][20]). Also, in [21,22] in a series of variations of the Schur function, as a final one called the ninth variation, the Jacobi-Trudi formula was regarded as the definition of the Schur function and many interesting properties such as the Giambelli formula can still be derived from this simple setup. Hence, the proof of the Giambelli formula or the Jacobi-Trudi formula in the ABJM matrix model strongly suggests the integrable structure in behind. ...
Preprint
It was known that one-point functions in the ABJM matrix model (obtained by applying the localization technique to one-point functions of the half-BPS Wilson loop operator in the ABJM theory) satisfy the Jacobi-Trudi formula, which strongly indicates the integrable structure of the system. In this paper, we identify the integrable structure of two-point functions in the ABJM matrix model as the two-dimensional Toda lattice hierarchy. The identification implies infinitely many non-linear differential equations for the generating function of the two-point functions.
... These polynomials can also be interpreted in terms of set-valued tableaux [3,15,16,25,26], or expressed as the quotient of determinants [12][13][14]24]. The lowest degree homogeneous component of G λ (x|y) is equal to the factorial Schur function s λ (x|y), which is the double Schubert polynomial of a Grassmannian permutation and has received extensive attention, see, for example, [1,2,5,10,17,23,27,28]. Restricting a factorial Grothendieck polynomial to a Schur function, we are led to a combinatorial proof of the identity (1.4). ...
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Gustafson and Milne proved an identity on the Schur function indexed by a partition of the form $(\lambda_1-n+k,\lambda_2-n+k,\ldots,\lambda_k-n+k)$. On the other hand, Feh\'{e}r, N\'{e}methi and Rim\'{a}nyi found an identity on the Schur function indexed by a partition of the form $(m-k,\ldots,m-k, \lambda_1,\ldots,\lambda_k)$. Feh\'{e}r, N\'{e}methi and Rim\'{a}nyi gave a geometric explanation of their identity, and they raised the question of finding a combinatorial proof. In this paper, we establish a Gustafson-Milne type identity as well as a Feh\'{e}r-N\'{e}methi-Rim\'{a}nyi type identity for factorial Grothendieck polynomials. Specializing a factorial Grothendieck polynomial to a Schur function, we obtain a combinatorial proof of the Feh\'{e}r-N\'{e}methi-Rim\'{a}nyi identity.
... The ring Λ˚posses a basis of shifted Schur functions also indexed by partitions. Their definition is closely related to the factorial Schur functions due to [BL89,BL90], or the more general multiparameter (or double) Schur functions of [Mac92,GG94,Mac95]. Note also their connection to flagged double Schur functions [CLL02,Las03]. ...
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We introduce a ring of noncommutative shifted symmetric functions based on an integer-indexed sequence of shift parameters. Using generating series and quasideterminants, this multiparameter approach produces deformations of the ring of noncommutative symmetric functions. Shifted versions of ribbon Schur functions are defined and form a basis for the ring. Further, we produce analogues of Jacobi--Trudi and N\"agelsbach--Kostka formulas, a duality anti-algebra isomorphism, shifted quasi-Schur functions, and Giambelli's formula in this setup. In addition, an analogue of power sums is provided, satisfying versions of Wronski and Newton formulas. Finally, a realization of these shifted noncommutative symmetric functions as rational functions in noncommuting variables is given. These realizations have a shifted symmetry under exchange of the variables, and are well-behaved under extension of the list of variables.
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We introduce an intermediate family of Laurent polynomials between Schur's $Q$-functions and S. Okada's symplectic $Q$-functions. It can also be regarded as a $Q$-function analogue of Proctor's intermediate symplectic characters, and is named the family of intermediate symplectic $Q$-functions. We also derive a tableau-sum formula and a J\'ozefiak-Pragacz-type Pfaffian formula of the Laurent polynomials.
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We propose a unified method for constructing higher Capelli elements for the classical Lie algebras. The higher Capelli elements are obtained as the Jacobi-Trudi determinants of central elements attached to partitions of single columns.
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We show that for a domain of parameter values subject to a truncation condition, a previously introduced elliptic Ruijsenaars type quantum particle hamiltonian with hyperoctahedral symmetry restricts to a self-adjoint discrete difference operator in a finite-dimensional Hilbert space of functions supported on bounded partitions; the construction of an orthogonal eigenbasis diagonalizing the corresponding discrete quantum model hinges in turn on the spectral theorem for self-adjoint operators in finite dimension. We verify that in the trigonometric limit the eigenfunctions in question recover a previously studied q-Racah type reduction of the Koornwinder–Macdonald polynomials. When the interaction between the particles degenerates to a Pauli repulsion of free fermions, the orthogonal eigenbasis can be expressed in terms of generalized Schur polynomials on the spectrum that are associated with recently found elliptic Racah polynomials.
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We present a new family of hook-length formulas for the number of standard increasing tableaux which arise in the study of factorial Grothendieck polynomials. In the case of straight shapes our formulas generalize the classical hook-length formula and Stanley's formula. For skew shapes, our formulas generalize the Naruse hook-length formula and its $q$-analogues, which were studied in previous papers of the series.
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We construct the orthogonal eigenbasis for a discrete elliptic Ruijsenaars type quantum particle Hamiltonian with hyperoctahedral symmetry. In the trigonometric limit the eigenfunctions in question recover a previously studied $q$-Racah type reduction of the Koornwinder-Macdonald polynomials. When the inter-particle interaction degenerates to that of impenetrable bosons, the orthogonal eigenbasis simplifies in terms of generalized Schur polynomials on the spectrum associated with recently found elliptic Racah polynomials.
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Lattices of polynomial KP and BKP $$\tau$$-functions labelled by partitions, with the flow variables equated to finite power sums, as well as associated multipair KP and multipoint BKP correlation functions, are expressed via generalizations of Jacobi’s bialternant formula for Schur functions and Nimmo’s Pfaffian ratio formula for Schur Q-functions. These are obtained by applying Wick’s theorem to fermionic vacuum expectation value representations in which the infinite group element acting on the lattice of basis states stabilizes the vacuum.
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The determinantal identities of Hamel and Goulden have recently been shown to apply to a tableau-based ninth variation of skew Schur functions. Here we extend this approach and its results to the analogous tableau-based ninth variation of supersymmetric skew Schur functions. These tableaux are built on entries taken from an alphabet of unprimed and primed numbers and that may be ordered in a myriad of different ways, each leading to a determinantal identity. At the level of the ninth variation the corresponding determinantal identities are all distinct but the original notion of supersymmetry is lost. It is shown that this can be remedied at the level of the sixth variation involving a doubly infinite sequence of factorial parameters. Moreover, it is shown that the resulting factorial supersymmetric skew Schur functions are independent of the ordering of the unprimed and primed entries in the alphabet.
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Recently Okada defined algebraically ninth variation skew Q-functions, in parallel to Macdonald’s ninth variation skew Schur functions. Here we introduce a skew shifted tableaux definition of these ninth variation skew Q-functions, and prove by means of a non-intersecting lattice path model a Pfaffian outside decomposition result in the form of a ninth variation version of Hamel’s Pfaffian outside decomposition identity. As corollaries to this we derive Pfaffian identities generalising those of Józefiak–Pragacz, Nimmo, and most recently Okada. As a preamble to this we present a parallel development based on (unshifted) semistandard tableaux that leads to a ninth variation version of the outside decomposition determinantal identity of Hamel and Goulden. In this case the corollaries we offer include determinantal identities generalising the Schur and skew Schur function identities of Jacobi–Trudi, Giambelli, Lascoux–Pragacz, Stembridge, and Okada.
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We study polynomial interpolation of Hermite type of holomorphic functions based on Radon projections. We give two kinds of interpolation schemes and show that the interpolation polynomials are continuous with respect to the angles and the distances. When the chords are suitably distributed, we prove that the interpolation polynomials converge geometrically on the closed unit disk to the functions.
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Gustafson and Milne proved an identity which can be used to express a Schur function sμ(x1,x2,…,xn) with μ=(μ1,μ2,…,μk) in terms of the Schur function sλ(x1,x2,…,xn), where λ=(λ1,λ2,…,λk) is a partition such that λi=μi+n−k for 1≤i≤k. On the other hand, Fehér, Némethi and Rimányi found an identity which relates sμ(x1,x2,…,xn) to the Schur function sλ(x1,x2,…,xℓ), where λ=(λ1,λ2,…,λℓ) is a partition obtained from μ by removing some of the largest parts of μ. Fehér, Némethi and Rimányi gave a geometric explanation of their identity, and they raised the question of finding a combinatorial proof. In this paper, we establish a Gustafson-Milne type identity as well as a Fehér-Némethi-Rimányi type identity for factorial Grothendieck polynomials. Specializing a factorial Grothendieck polynomial to a Schur function, we obtain a combinatorial proof of the Fehér-Némethi-Rimányi identity.
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p>Factorial characters of each of the classical Lie groups have recently been defined algebraically as rather simple deformations of irreducible characters. Each such factorial character has been shown to satisfy a flagged Jacobi–Trudi identity, thereby allowing for its combinatorial realisation in terms of first a non-intersecting lattice path model and then a tableau model. Here we propose algebraic definitions of factorial Q-functions of the classical Lie groups and translate these definitions into combinatorial realisations in terms of non-intersecting lattice path and primed shifted tableaux models. By way of some justification of our chosen definitions, it is then shown that our factorial Q-functions satisfy Tokuyama-type identities and relate some special case of these to other identities that have appeared in the literature.</p
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p>Just as the definition of factorial Schur functions as a ratio of determinants allows one to show that they satisfy a Jacobi–Trudi-type identity and have an explicit combinatorial realisation in terms of semistandard tableaux, so we offer here definitions of factorial irreducible characters of the classical Lie groups as ratios of determinants that share these two features. These factorial characters are each specified by a partition, λ=(λ<sub>1</sub>,λ<sub>2</sub>,…,λ<sub>n</sub>), and in each case a flagged Jacobi–Trudi identity is derived that expresses the factorial character as a determinant of corresponding factorial characters specified by one-part partitions, (m), for which we supply generating functions. These identities are established by manipulating determinants through the use of certain recurrence relations derived from these generating functions. The transitions to combinatorial realisations of the factorial characters in terms of tableaux are then established by means of non-intersecting lattice path models. The results apply to gl(n), so(2n+1), sp(2n) and o(2n), and are extended to the case of so(2n) by making use of newly defined factorial difference characters.</p
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The main result of the paper is a construction of a five-parameter family of new bases in the algebra of symmetric functions. These bases are inhomogeneous and share many properties of systems of orthogonal polynomials on an interval of the real line. This means, in particular, that the algebra of symmetric functions is embedded into the algebra of continuous functions on a certain compact space Omega, and under this realization, our bases turn into orthogonal bases of weighted Hilbert spaces corresponding to certain probability measures on Omega. These measures are of independent interest --- they are an infinite-dimensional analogue of the multidimensional q-Beta distributions. Our construction uses the big q-Jacobi polynomials and an extension of the Knop-Okounkov-Sahi multivariate interpolation polynomials to the case of infinite number of variables.
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We present a new determinantal expressin for Schur functions. Previous expressions were due to Jacobi, Trudi, Giambelli and others [see I. G. Macdonald, Symmetric functions and Hall polynomials (1979; Zbl 0487.20007)] and involved elementary symmetric functions or hook functions. We give, in Theorem 1.1, a decomposition of a Schur function into ribbon functions (also called skew hook functions, new functions by MacMahon, and MacMahon functions by others). We provide two different proofs of this result in Sections 2 and 3. In Section 2, we use Bazin’s formula for the minors of a general matrix, as we already did in C. R. Acad. Sci., Paris, Ser. I 299, 955-958 (1984; Zbl 0579.05012), to decompose a skew Schur function into hooks. In Section 3, we show how to pass from hooks to ribbons and conversely. In Section 4, we generalize to skew Schur functions. In Section 5, we give some applications, and show how such constructions, in the case of staircase partitions, generalize the classical continued fraction for the tangent function due to Euler.
Article
A new class of symmetric polynomials in n variables z = (z1,…, zn), denoted tλ(z), and labelled by partitions λ = [λ1 … λn] is defined in terms of standard tableaux (equivalently, in terms of Gel'fand-Weyl patterns of the general linear group GL(n,C)). The tλ(z) are shown to be a -basis of the ring of all symmetric polynomials in n variables. In contrast to the usual basis sets such as the Schur functions eλ(z), which are homogeneous polynomials in the zi, the tλ(z) are inhomogeneous. This property is reflected in the fact that the tλ(z) are a natural basis for the expansion of certain (inhomogeneous) symmetric polynomials constructed from rising factorials. This and several other properties of the tλ(z) are proved. Two generalizations of the tλ(z) are also given. The first generalizes the tλ(z) to a 1-parameter family of symmetric polynomials, Tλ(α; z), where α is an arbitrary parameter. The Tλ(α; z) are shown to possess properties similar to those of the tλ(z). The second generalizes the tλ(z) to a class of skew-tableau symmetric polynomials, , for which only a few preliminary results are given.
Article
Flagged Schur functions are generalizations of Schur functions. They appear in the work of Lascoux and Schutzenberger [2] in their study of Schubert polynomials. Gessel [ 1 ] has shown that flagged Schur functions can be expressed both as a determinant in the complete homogeneous symmetric functions and in terms of column-strict tableaux just as can ordinary Schur functions (Jacobi-Trudi identity). For each row of these tableaux there is an upper bound (flag) on the entries. The Schubert polynomials are obtained by applying certain symmetrizing operators to a monomial. In Section 1 we study the effect of applying these symmetrizing operators to flagged Schur functions. Although it is trivial to do this for the determinantal expression, we show, by direct means, how to apply the symmetrizing operators to the tableau expression (without the use of determinants). This produces another proof of Gessel’s result and hence a new inductive proof of the Jacobi-Trudi identity. Each Schubert polynomial is determined by some permutation. Lascoux and Schutzenberger [Z] state a result which enables one to identify those permutations whose Schubert polynomial is a flagged Schur function. In Section 2 we present an explicit expression for the shape and flags (row bounds) in terms of the permutation. We do this by applying the symmetrizing operators to flagged Schur functions. We also show that any flagged Schur function can be obtained by applying a sequence of symmetrizing operators to some monomial. In Section 4 we consider row (column) flagged skew Schur functions. Here, for each row (column) of the skew tableaux there is an upper and lower bound on the entries. In fact the above cited work of Gessel actually
| A new class of symmetric functions, Actes 20 e S eminaire Lotharingien
MACDONALD (I.G.). | A new class of symmetric functions, Actes 20 e S eminaire Lotharingien (Publ. I.R.M.A. Strasbourg, 1988, 372/S{20), p. 131{171.
| Inhomogeneous basis set of symmetric polynomials de ned by tableaux
BL 2 ] BIEDENHARN (L.C) and LOUCK (J.D.). | Inhomogeneous basis set of symmetric polynomials de ned by tableaux, Proc. Nat. Acad. Sci. U.S.A., t. 87, 1990, p. 1441{1445.