The double Schur functions form a distinguished basis of the ring Λ(x||a) which is a multiparameter generalization of the ring of symmetric functions Λ(x). The canonical comultiplication on Λ(x) is extended to Λ(x||a) in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood-Richardson coefficients in two different ways thus providing comultiplication rules for the double Schur functions. We also prove multiparameter analogues of the Cauchy identity. A new family of Schur type functions plays the role of a dual object in the identities. We describe some properties of these dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions. The dual Littlewood-Richardson coefficients provide a multiplication rule for the dual Schur functions.
We present several generalizations of Cauchy's determinant and Schur's Pfaffian by considering matrices whose entries involve some generalized Vandermonde determinants. Special cases of our formulae include previuos formulae due to S.Okada and T. Sundquist. As an application, we give a relation for the Littlewood--Richardson coefficients involving a rectangular partition.
We give bijective proofs of orthosymplectic analogues of the Cauchy identity and dual Cauchy identity for orthosymplectic Schur functions. To do so, we present two insertion algorithms; these are orthosymplectic versions of Berele's symplectic insertion algorithms, which were used by Sundaram to give bijective proofs of Cauchy identities for symplectic Schur functions.
In this article, we use Lindström Gessel Viennot Lemma to give a short, combinatorial, visualizable proof of the identity of Schur polynomials -- the sum of monomials of Young tableaux equals to the quotient of determinants. As a by-product, we have a proof of Vandermonde determinant without words. We also prove the cauchy identity. In the remarks, we discuss factorial Schur polynomials, dual Cauchy identity and the relation beteen Newton interpolation formula.
Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. With this approach we prove skew Cauchy identity and then derive various applications: skew Pieri rules, dual filtrations of Young's lattice, generating series and enumerative identities. We also give a new explanation of the finite expansion property for products of Grothendieck polynomials.
We establish Pfaffian analogues of the Cauchy--Binet formula and the Ishikawa--Wakayama minor-summation formula. Each of these Pfaffian analogues expresses a sum of products of subpfaffians of two skew-symmetric matrices in terms of a single Pfaffian. By using these Pfaffian formulas we give new transparent proofs to several identities for Schur Q23 pa-functions.
The Kronecker product of two Schur functions $s_μ$ and $s_ν$, denoted by $s_μ*s_ν$, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions $μ$ and $ν$. The coefficient of $s_λ$ in this product is denoted by $γ^λ_{μν}$, and corresponds to the multiplicity of the irreducible character $χ^λ$ in $χ^μχ^ν.$ We use Sergeev's Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for $s_λ[XY]$ to find closed formulas for the Kronecker coefficients $γ^λ_{μν}$ when $λ$ is an…
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, Józefiak--Pragacz--Nimmo formula for skew $Q$-functions…
The classical Peter-Weyl theorem describes the structure of the space of functions on a semi-simple algebraic group. On the level of characters (in type A) this boils down to the Cauchy identity for the products of Schur polynomials. We formulate and prove the analogue of the Peter-Weyl theorem for the current groups. In particular, in type A the corresponding characters identity is governed by the Cauchy identity for the products of q-Whittaker functions. We also formulate and prove a version of the Schur-Weyl theorem for current groups. The link between the Peter-Weyl and Schur-Weyl theorems…
We present a Pfaffian identity involving elliptic functions, whose rational limit gives a generalization of Schur's Pfaffian identity for Pf ((x_j - x_i)/(x_j + x_i)). This identity is regarded as a Pfaffian counterpart of Frobenius identity, which is an elliptic generalization of Cauchy's determinant identity for det (1/ (x_i + x_j)).
This paper introduces noncommutative analogs of monomial symmetric functions and fundamental noncommutative symmetric functions. The expansion of ribbon Schur functions in both of these basis is nonnegative. With these functions at hand, one can derive a noncommutative Cauchy identity as well as study a noncommutative scalar product implied by Cauchy identity. This scalar product seems be the noncommutative analog of Hall scalar product in the commutative theory.
We introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler-Zagier type combinatorially. We first study their basic properties including a region of absolute convergence and the case where all variables are the same. Then, under an assumption on variables, some determinant formulas coming from theory of Schur functions such as the Jacobi-Trudi, Giambelli and dual Cauchy formula are established with the help of Macdonald's ninth variation of Schur functions. Moreover, we investigate the quasi-symmetric functions corresponding…
Lascoux has given a triangular version of the Cauchy identity where Schur polynomials are replaced by Demazure characters and Demazure atoms. He has then used the staircase expansion to recover expansions for all Ferrers shapes, where the Demazure characters and Demazure atoms are under the action of Demazure operators specified by the cells above the staircase. The characterisation of the tableau-pairs in these last expansions is less explicit. We give here a bijective proof for expansions over near staircases, where the tableau-pairs are made explicit. Our analysis formulates Mason's RSK ana…
Factorial Schur functions are generalizations of Schur functions that have, in addition to the usual variables, a second family of "shift" parameters. We show that a factorial Schur function times a deformation of the Weyl denominator may be expressed as the partition function of a particular statistical-mechanical system (six vertex model). The proof is based on the Yang-Baxter equation. There is a deformation parameter $t$ which may be specialized in different ways. If $t=-1$, then we recover the expression of the factorial Schur function as a ratio of alternating polynomials. If $t=0$, we r…
We construct a vertex model whose partition function is a refined dual Grothendieck polynomial, where the states are interpreted as nonintersecting lattice paths. Using this, we show refined dual Grothendieck polynomials are multi-Schur functions and give a number of identities, including a Littlewood and Cauchy(-Littlewood) identity. We then refine Yeliussizov's connection between dual Grothendieck polynomials and the last passage percolation (LPP) stochastic process discussed by Johansson. By refining algebraic techniques of Johansson, we show Jacobi-Trudi formulas for skew refined dual Grot…
Fully inhomogeneous spin Hall-Littlewood symmetric rational functions $\mathsf{F}_λ$ arise in the context of $\mathfrak{sl}(2)$ higher spin six vertex models, and are multiparameter deformations of the classical Hall-Littlewood symmetric polynomials. We obtain a refined Cauchy identity expressing a weighted sum of the product of two $\mathsf{F}_λ$'s as a determinant. The determinant is of Izergin-Korepin type: it is the partition function of the six vertex model with suitably decorated domain wall boundary conditions. The proof of equality of two partition functions is based on the Yang-Baxter…
There exist a number of well known multiplicative generating functions for series of Schur functions. Amongst these are some related to the dual Cauchy identity whose expansion coefficients are rather simple, and in some cases periodic in parameters specifying the Schur functions. More recently similar identities have been found involving expansions in terms of characters of the symplectic group. Here these results are extended and generalised to all classical Lie groups. This is done through the derivation of explicit recurrence relations for the expansion coefficients based on the action of …
We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the set of sentences over alphabet $A$ (the set of colours). We present also its graded dual algebra $\mathrm{QSym}_A$ of coloured quasi-symmetric functions together with its realization in terms of power series in partially commutative variables. We provide formulas expressing multiplication, comultiplication and the antipode for these Hopf algebras in various…
The representation of the Bethe wave functions of certain integrable models via the Schur functions allows to apply the well-developed theory of the symmetric functions to the calculation of the thermal correlation functions. The algebraic relations arising in the calculation of the scalar products and the correlation functions are based on the Binet-Cauchy formula for the Schur functions. We provide a combinatorial interpretation of the formula for the scalar products of the Bethe state-vectors in terms of nests of the self-avoiding lattice paths constituting the so-called watermelon configur…
We propose periodic Macdonald processes as a $(q,t)$-deformation of periodic Schur processes and a periodic analogue of Macdonald processes. It is known that, in the theory of stochastic processes related to a family of symmetric functions, the Cauchy-like identity gives an explicit expression of a partition function. We compute the partition functions of periodic Macdonald processes relying on the free field realization of the Macdonald theory. We also study several families of observables for periodic Macdonald processes and give formulas of their moments. The technical tool is the free fiel…
We establish a fundamental connection between the geometric RSK correspondence and GL(N,R)-Whittaker functions, analogous to the well known relationship between the RSK correspondence and Schur functions. This gives rise to a natural family of measures associated with GL(N,R)-Whittaker functions which are the analogues in this setting of the Schur measures on integer partitions. The corresponding analogue of the Cauchy-Littlewood identity can be seen as a generalisation of an integral identity for GL(N,R)-Whittaker functions due to Bump and Stade. As an application, we obtain an explicit integ…
We attempt to explain the ubiquity of tableaux and of Pieri and Cauchy formulae for combinatorially defined families of symmetric functions. We show that such formulae are to be expected from symmetric functions arising from representations of Heisenberg algebras. The resulting framework that we describe is a generalization of the classical Boson-Fermion correspondence, from which Schur functions arise. Our work can be used to understand Hall-Littlewood polynomials, Macdonald polynomials and Lascoux, Leclerc and Thibon's ribbon functions, together with other new families of symmetric functions…
Using symmetric function theory, we study the cycle structure and increasing subsequence structure of permutations after various shuffling methods, emphasizing the role of Cauchy type identities and the Robinson-Schensted-Knuth correspondence. One consequence is a cycle index for riffle shuffles when one deals from the bottom of the deck; conclusions are drawn concerning the distribution of fixed points and the asymptotic distribution of cycle structure. In studying cycle structure of unimodal permutations and their generalizations we are led to an interesting variation of the RSK corresponden…
This is an overview of recent results on the use of 2D Toda $τ$-functions as generating functions for multiparametric families of weighted Hurwitz numbers. The Bose-Fermi equivalence composed with the characteristic map provides an isomorphism between the zero charge sector of the Fermionic Fock space and the direct sum of the centers of the group algebra of the symmetric groups $S_n$. Specializing the fermionic formula to the case of diagonal group elements gives $τ$-functions of hypergeometric type, for which the expansion over products of Schur functions is diagonal, with coefficients of {\…
A special case of a fundamental result of Loewner and Horn [Trans. Amer. Math. Soc. 1969] says that given an integer $n \geq 1$, if the entrywise application of a smooth function $f : (0,\infty) \to \mathbb{R}$ preserves the set of $n \times n$ positive semidefinite matrices with positive entries, then $f$ and its first $n-1$ derivatives are non-negative on $(0,\infty)$. In a recent joint work with Belton-Guillot-Putinar [J. Eur. Math. Soc., in press], we proved a stronger version, and used it to strengthen the Schoenberg-Rudin characterization of dimension-free positivity preservers [Duke Mat…
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 $F_λ,G_λ$ are defined as certain partition functions of the six vertex model, with variables corresponding to row rapidities, and the labeling signatures $λ=(λ_1\ge \ldots\ge λ_N)\in \mathbb{Z}^N$ encoding boundary conditions. These symmetric functions generalize Schur symmetric polynomials, as well as some of their variations, such as factorial and supersymmetric Schur polynomi…
In this text we introduce and analyze families of symmetric functions arising as partition functions for colored fermionic vertex models associated with the quantized affine Lie superalgebra $U_q \big( \widehat{\mathfrak{sl}} (1 | n) \big)$. We establish various combinatorial results for these vertex models and symmetric functions, which include the following. (1) We apply the fusion procedure to the fundamental $R$-matrix for $U_q \big( \widehat{\mathfrak{sl}} (1 | n) \big)$ to obtain an explicit family of vertex weights satisfying the Yang-Baxter equation. (2) We define families of symmetric…
We consider a fully inhomogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the q-correlation functions. At least in the case of the step initial condition, our formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six vertex model, ASEP, various q-TASEPs, and associated zero range processes. Our arguments are largely based on properties of a fam…
We consider a homogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the q-correlation functions. At least in the case of the step initial condition, our formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six vertex model, ASEP, various q-TASEPs, and associated zero range processes. Our arguments are largely based on properties of a family of s…
