Algebra Of Symmetric Functions Research Articles

Graphs and Algebras of Symmetric Functions

We describe an algebraic technique for operating with power series whose coefficients are represented by integrals of symmetric functions fn defined on the Cartesian powers Ωn of a set Ω with a measure μ. Moreover, each of the coefficient functions fn is obtained by means of a special mapping from graphs with n labeled vertices belonging to a fixed class. This technique has application to equilibrium statistical mechanics and to problems of enumeration of graphs.

A Proof of the Extended Delta Conjecture

Abstract We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $\Delta _{h_l}\Delta ' _{e_k} e_{n}$ , where $\Delta ' _{e_k}$ and $\Delta _{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $\operatorname {\mathrm {GL}}_m$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.

Schur functions in noncommuting variables

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

Transition Functions of Diffusion Processes on the Thoma Simplex

The paper deals with a three-dimensional family of diffusion processes on an infinite-dimensional simplex. These processes were constructed by Borodin and Olshanski in 2009 and 2010, and they include, as limit objects, the infinitely-many-neutral-allels diffusion model constructed by Ethier and Kurtz in 1981 and its extension found by Petrov in 2009. Each process X in our family possesses a unique symmetrizing measure M, called the z-measure. Our main result is that the transition function of X has a continuous density with respect to M. This is a generalization of earlier results due to Ethier (1992) and to Feng, Sun, Wang, and Xu (2011). Our proof substantially uses a special basis in the algebra of symmetric functions, which is related to the Laguerre polynomials.

Elements of the q-Askey Scheme in the Algebra of Symmetric Functions

The classical q-hypergeometric orthogonal polynomials are assembled into a hierarchy called the q-Askey scheme. At the top of the hierarchy, there are two closely related families, the Askey-Wilson and q-Racah polynomials. As it is well known, their construction admits a generalization leading to remarkable orthogonal symmetric polynomials in several variables. We construct an analogue of the multivariable q-Racah polynomials in the algebra of symmetric functions. Next, we show that our q-Racah symmetric functions can be degenerated into the big q-Jacobi symmetric functions, introduced in a recent paper by the second author. The latter symmetric functions admit further degenerations leading to new symmetric functions, which are analogues of q-Meixner and Al-Salam--Carlitz polynomials. Each of the four families of symmetric functions (q-Racah, big q-Jacobi, q-Meixner, and Al-Salam--Carlitz) forms an orthogonal system of functions with respect to certain measure living on a space of infinite point configurations. The orthogonality measures of the four families are of independent interest. We show that they are linked by limit transitions which are consistent with the degenerations of the corresponding symmetric functions.

Lumpings of Algebraic Markov Chains Arise from Subquotients

A function on the state space of a Markov chain is a "lumping" if observing only the function values gives a Markov chain. We give very general conditions for lumpings of a large class of algebraically-defined Markov chains, which include random walks on groups and other common constructions. We specialise these criteria to the case of descent operator chains from combinatorial Hopf algebras, and, as an example, construct a "top-to-random-with-standardisation" chain on permutations that lumps to a popular restriction-then-induction chain on partitions, using the fact that the algebra of symmetric functions is a subquotient of the Malvenuto-Reutenauer algebra.

A Hopf Algebraic Approach to Schur Function Identities

Using cocommutativity of the Hopf algebra of symmetric functions, certain skew Schur functions are proved to be equal. Some of these skew Schur function identities are new.

An analogue of the big q-Jacobi polynomials in the algebra of symmetric functions

It is well known how to construct a system of symmetric orthogonal polynomials in an arbitrary finite number of variables from an arbitrary system of orthogonal polynomials on the real line. In the special case of the big q-Jacobi polynomials, the number of variables can be made infinite. As a result, in the algebra of symmetric functions, there arises an inhomogeneous basis whose elements are orthogonal with respect to some probability measure. This measure is defined on a certain space of infinite point configurations and hence determines a random point process.

Antipode Formulas for some Combinatorial Hopf Algebras

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic analogues of the symmetric functions, quasisymmetric functions, and noncommutative symmetric functions.

Stochastic Monotonicity in Young Graph and Thoma Theorem

We show that the order on probability measures, inherited from the dominance order on the Young diagrams, is preserved under natural maps reducing the number of boxes in a diagram by $1$. As a corollary we give a new proof of the Thoma theorem on the structure of characters of the infinite symmetric group. We present several conjectures generalizing our result. One of them (if it is true) would imply the Kerov's conjecture on the classification of all homomorphisms from the algebra of symmetric functions into $\mathbb R$ which are non-negative on Hall--Littlewood polynomials.

$\mathfrak{sl}(2)$ operators and Markov processes on branching graphs

We present a unified approach to various examples of Markov dynamics on partitions studied by Borodin, Olshanski, Fulman, and the author. Our technique generalizes Kerov's operators which first appeared in Okounkov (Random Matrix Models and Their Applications, pp. 407---420, Cambridge University Press, Cambridge, 2001), and also stems from the study of duality of graded graphs in Fomin (J. Algebr. Comb., 3(4):357---404, 1994). Our main object is a countable branching graph carrying an $\mathfrak {sl}(2,\mathbb{C})$ -module of a special kind. Using this structure, we introduce distinguished probability measures on the floors of the graph, and define two related types of Markov dynamics associated with these measures. We study spectral properties of the dynamics, and our main result is the explicit description of eigenfunctions of the Markov generator of one of the processes. For the Young graph our approach reconstructs the z-measures on partitions and the associated dynamics studied by Borodin and Olshanski (Probab. Theory Relat. Fields, 135(1):84---152, 2006; Probab. Theory Relat. Fields, 144(1):281---318, 2009). The generator of the jump dynamics on the Young graph corresponding to the z-measures is diagonal in the basis of the Meixner symmetric functions introduced recently by Olshanski (Zap. Nauă?. Semin. POMI, 378:81---100, 230, 2010; Laguerre and Meixner orthogonal bases in the algebra of symmetric functions, 2011). We give new proofs to some of the results of these two papers. Other graphs to which our technique is applicable include the Pascal triangle, the Kingman graph (with the two-parameter Poisson---Dirichlet measures), the Schur graph and the general Young graph with Jack edge multiplicities.

Laguerre and Meixner Orthogonal Bases in the Algebra of Symmetric Functions

Analogs of Laguerre and Meixner orthogonal polynomials in the algebra of symmetric functions are studied. This is a detailed exposition of part of the results announced in arXiv:1009.2037. The work is motivated by a connection with a model of infinite-dimensional Markov dynamics.

HOMFLY-PT polynomial and normal rulings of Legendrian solid torus links

We show that for any Legendrian link $L$ in the $1$-jet space of $S^1$ the $2$-graded ruling polynomial, $R^2_L(z)$, is determined by the Thurston-Bennequin number and the HOMFLY-PT polynomial. Specifically, we recover $R^2_L(z)$ as a coefficient of a particular specialization of the HOMFLY-PT polynomial. Furthermore, we show that this specialization may be interpreted as the standard inner product on the algebra of symmetric functions that is often identified with a certain subalgebra of the HOMFLY-PT skein module of the solid torus. In contrast to the $2$-graded case, we are able to use $0$-graded ruling polynomials to distinguish many homotopically non-trivial Legendrian links with identical classical invariants.

Laguerre and Meixner symmetric functions, and infinite-dimensional diffusion processes

The Laguerre symmetric functions introduced in the note are indexed by arbitrary partitions and depend on two continuous parameters. The top degree homogeneous component of every Laguerre symmetric function coincides with the Schur function with the same index. Thus, the Laguerre symmetric functions form a two-parameter family of inhomogeneous bases in the algebra of symmetric functions. These new symmetric functions are obtained from the N-variate symmetric polynomials of the same name by a procedure of analytic continuation. The Laguerre symmetric functions are eigenvectors of a second-order differential operator, which depends on the same two parameters and serves as the infinitesimal generator of an infinite-dimensional diffusion process X(t). The process X(t) admits approximation by some jump processes related to one more new family of symmetric functions, the Meixner symmetric functions. In equilibrium, the process X(t) can be interpreted as a time-dependent point process on the punctured real line $\mathbb {R}$ {0}, and the point configurations may be interpreted as doubly infinite collections of particles of two opposite charges and with log-gas-type interaction. The dynamical correlation functions of the equilibrium process have determinantal form: they are given by minors of the so-called extended Whittaker kernel, introduced earlier in a paper by Borodin and the author. Bibliography: 28 titles.

Differential operators and crystals of extremal weight modules

We give a combinatorial realization of extremal weight crystals over the quantum group of type A + ∞ and their Littlewood–Richardson rule. Based on this description, we show that the Grothendieck ring generated by the isomorphism classes of extremal weight A + ∞ -crystals is isomorphic to the Weyl algebra of infinite rank, and hence each isomorphism class is realized as a differential operator or non-commutative Schur function acting on the algebra of symmetric functions. We also find a duality between extremal weight A + ∞ -crystals and generalized Verma A ∞ -crystals appearing in the crystal of the Fock space with infinite level, which recovers the generalized Cauchy identity for Schur operators in a bijective and crystal theoretic way.

Implementing Algebraic Combinatorics Some feedback from the development of MuPAD-Combinat (abstract only)

MuPAD-Combinat is an open-source algebraic combinatorics package for the computer algebra system MuPAD. Its main purpose is to provide an extensible toolbox for computer exploration, and to foster code sharing between researchers in this area. The development started in 2001, and it now contains functions to deal with most usual combinatorial classes (partitions, tableaux, decomposable classes, ...). It also supplies the user with tools for constructing new combinatorial classes and combinatorial (Hopf) algebras. As an application, it provides well-known combinatorial Hopf algebras like the algebra of symmetric functions and many generalizations. There is also experimental support for combinatorial Lie algebras, operads, (affine) Weyl groups, crystals, Schubert polynomials, invariant rings, ... Altogether, MuPAD-Combinat played a central role in 25+ publications with more than a dozen contributors. In this talk, we first demonstrate the main functionalities of the package and present some typical applications. Based on that, we discuss its design, successes and flaws, and its future. Feel free to contact the speaker if you would like to get some first hand experience through an informal tutorial, and bring your laptop if you have one.

Eine Symmetrieeigenschaft von Solomons Algebra und der höheren Lie-Charaktere

We prove here three results in chain: the result of Section 2 is a symmetry property of the higher Lie characters ofS n (which are indexed by partitions) : their character table is essentially symmetric, up to well-known factors. This is established using plethystic methods in the algebra of symmetric functions. In Section 3, we show that for any elements ϕ,ωof the Solomon descent algebra ofS n , one hasc( ϕ)(ω) =c(ω ϕ), wherec is the Solomon mapping from this algebra to the space of central functions onS n (implicitly extended to its group algebra). We address also the question whether this is true for each finite Coxeter group. Then in the last section, we deduce a new proof of a result of Gessel and the second author that gives the number of permutations with given cycle type and descent set as scalar product of two special characters.

Canonical Basis and Macdonald Polynomials

In the basic representation of[formula]realized via the algebra of symmetric functions, we compare the canonical basis with the basis of Macdonald polynomials wheret=q2. We show that the Macdonald polynomials are invariant with respect to the bar involution defined abstractly on the representations of quantum groups. We also prove that the Macdonald scalar product coincides with the abstract Kashiwara form. This implies, in particular, that the Macdonald polynomials form an intermediate basis between the canonical basis and the dual canonical basis, and the coefficients of the transition matrix are necessarily bar invariant. We also verify that the Macdonald polynomials (after a natural rescaling) form a sublattice in the canonical basis lattice which is invariant under the divided powers action. The transition matrix with respect to this rescaling is integral and we conjecture its positivity. For levelk, we expect a similar relation between the canonical basis and Macdonald polynomials withq2=tk.

The cooperations of MU ∗(CP ∞) and MU ∗(HP ∞) and the primitive generators

We review the Adams-Quillen description of the MU ∗(MU) comodule structure on MU ∗(CP ∞) (Section 1) and obtain a workable formula for the comodule structure on MU ∗(HP ∞) (Section 3). We use this to obtain a complete picture of the bottom rows of the Adams–Novikov spectral sequences for CP ∞ (Section 2) and HP ∞ (Section 4). In the process we make note of a very convenient form of the Hattori–Stong theorem due to Liulevicius (Section 2) and describe the standard map H ∗( MSp;Z)→H ∗( MU;Z) (Section 3) by arguments on algebras of symmetric functions.

IV. Seventh memoir on the partition of numbers. A detailed study of the enumeration of the partitions of multipartite numbers

In this paper I give the complete solution of the problem of the partition of multipartite numbers. This is the same subject as that named by Sylvester, "Compound Denumeration." Twenty-nine years have elapsed since I announced that the algebra of symmetric functions is co-extensive with the grand problems of the combinatory analysis. The theory of symmetric function supplies generating functions which enumerate all the combinations, while the operators of Hammond are the instruments which are effective in actually evaluating the coefficients of the terms of the expanded generating functions. When these operators fell from the hands of Hammond, they were already of much service as mining tools in extracting the ore from the mine :field of symmetric functions; but they were only partially adequate. They required sharpening and general adaptation to the work in hand. The first step was to decompose an operator of given order into the sum of a number of operators in correspondence with every partition of the number which defines the order. Since there is a Hammond operator corresponding to every positive integer, this process resulted in there being an operator in correspondence with every partition of every integer. The outcome of this decomposition was that the operators were able to deal with the symmetric operands in a much more effective manner. The surface material of the mine could not only be removed, but the strata to a considerable depth could be dealt with. But this was not sufficient. It became necessary to effect a further decomposition by showing that ·every partition operator could be represented by a sum of composition operators. There emerged a composition operator in correspondence with every permutation of the parts of the partition of the operator. The operators at once became effective in dealing with the material in the lower strata of the mine field. The operators had, in fact, been handled with particular reference to the operands with which they were to be associated. It was now necessary to deal with the material of the mine with particular reference to the tools which had been forged. To evaluate the coefficients we have to operate repeatedly with the appropriate operators until a numerical result is reached. In order to accomplish with facility and to establish laws we have to put the generating functions form that these operations are carried out in a regular and simple manner. To make my meaning clear, I will instance the case of the simple operation of differentiation and the exponential function e αx. We have ∂ x e αx =ae αx , the effect of the operation being, to merely the operand by magnitude α.