Theory Of Symmetric Functions Research Articles

Conjectured Statistics for the Higher $q,t$-Catalan Sequences

This article describes conjectured combinatorial interpretations for the higher $q,t$-Catalan sequences introduced by Garsia and Haiman, which arise in the theory of symmetric functions and Macdonald polynomials. We define new combinatorial statistics generalizing those proposed by Haglund and Haiman for the original $q,t$-Catalan sequence. We prove explicit summation formulas, bijections, and recursions involving the new statistics. We show that specializations of the combinatorial sequences obtained by setting $t=1$ or $q=1$ or $t=1/q$ agree with the corresponding specializations of the Garsia-Haiman sequences. A third statistic occurs naturally in the combinatorial setting, leading to the introduction of $q,t,r$-Catalan sequences. Similar combinatorial results are proved for these trivariate sequences.

The matrix Euler-Fermat theorem

We prove many congruences for binomial and multinomial coefficients as well as for the coefficients of the Girard-Newton formula in the theory of symmetric functions. These congruences also imply congruences (modulo powers of primes) for the traces of various powers of matrices with integer elements. We thus have an extension of the matrix Fermat theorem similar to Euler's extension of the numerical little Fermat theorem.

Symmetric functions in noncommuting variables

Consider the algebra Q >of formal power series in countably many noncommuting variables over the rationals. The subalgebra Π(x 1 , x 2 ,...) of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, complete homogeneous, and Schur symmetric functions as well as investigating their properties.

A generalization of Weyl's identity for Dn

The expansion in Schur functions of the product ∏ i< j (1− x i x j ) is well known. It is more or less equivalent to Weyl's identity for root-systems of type D n . In this paper we obtain the expansion in Schur functions of the product ∏ i< j (1−∑ ℓ>0 a ℓ x i x j ( x i ℓ− x j ℓ)/( x i − x j )), thus generalizing Weyl's identity. We obtain this result by systematic calculation in fermionic Fock space. But familiarity with the latter is not a prerequisite for reading this paper. We develop from scratch the modest amount of theory that we need in elementary and purely algebraic fashion, taking pains to integrate the theory with classical symmetric function theory. The tools developed in this paper ought to have many further applications, e.g., to random matrix theory and to computational abelian function theory.

Hypersymmetric functions and Pochhammers of 2 × 2 nonautonomous matrices

We introduce the hypersymmetric functions of 2 × 2 nonautonomous matrices and show that they are related, by simple expressions, to the Pochhammers (factorial polynomials) of these matrices. The hypersymmetric functions are generalizations of the associated elementary symmetric functions, and for a specific class of 2 × 2 matrices, having a high degree of symmetry, they reduce to these latter functions. This class of matrices includes rotations, Lorentz boosts, and discrete time generators for the harmonic oscillators. The hypersymmetric functions are defined over four sets of independent indeterminates using a triplet of interrelated binary partitions. We work out the algebra of this triplet of partitions and then make use of the results in order to simplify the expressions for the hypersymmetric functions for a special class of matrices. In addition to their obvious applications in matrix theory, in coupled difference equations, and in the theory of symmetric functions, the results obtained here also have useful applications in problems involving successive rotations, successive Lorentz transformations, discrete harmonic oscillators, and linear two‐state systems.

Conjugacy class properties of the extension of GL( n, q) generated by the inverse transpose involution

Letting τ denote the inverse transpose automorphism of GL( n, q), a formula is obtained for the number of g in GL( n, q) so that gg τ is equal to a given element h. This generalizes a result of Gow and Macdonald for the special case that h is the identity. We conclude that for g random, gg τ behaves like a hybrid of symplectic and orthogonal groups. It is shown that our formula works well with both cycle index generating functions and asymptotics, and is related to the theory of random partitions. The derivation makes use of models of representation theory of GL( n, q) and of symmetric function theory, including a new identity for Hall–Littlewood polynomials. We obtain information about random elements of finite symplectic groups in even characteristic, and explicit bounds for the number of conjugacy classes and centralizer sizes in the extension of GL( n, q) generated by the inverse transpose automorphism. We give a second approach to these results using the theory of bilinear forms over a field. The results in this paper are key tools in forthcoming work of the authors on derangements in actions of almost simple groups, and we give a few examples in this direction.

Recurrent sequences and Schur functions

We show that some classical determinants in the theory of symmetric functions can be interpreted in terms of recurrent sequences. Conversely we generalize determinantal expressions of Schur functions, by taking several recurrent sequences having same characteristic polynomial, or by prolonging sequences to negative indices. Finally, we give some recurrent sequences associated to plethysm of symmetric functions, for example with characteristic polynomial having roots the same powers of the roots of the original characteristic polynomial.

Vicious walkers, friendly walkers, and young tableaux. III. Between two walls

We derive exact and asymptotic results for the number of star and watermelon configurations of vicious walkers confined to lie between two impenetrable walls, as well as corresponding results for the analogous problem of ∞-friendly walkers. Our proofs make use of results from symmetric function theory and the theory of basic hypergeometric series.

The plethysm technique applied to the classification of nuclear states

A short summary of the theory of symmetric group and symmetric functions needed to follow the theory of Schur functions and plethysms is presented. One then defines plethysm, gives its properties and presents a procedure for its calculation. Finally, some aplications in atomic physics and nuclear structure are given.

Q-Analogs of symmetric function operators

For any homomorphism V on the space of symmetric functions, we introduce an operation that creates a q-analog of V. By giving several examples we demonstrate that this quantization occurs naturally within the theory of symmetric functions. In particular, we show that the Hall–Littlewood symmetric functions are formed by taking this q-analog of the Schur symmetric functions and the Macdonald symmetric functions appear by taking the q-analog of the Hall–Littlewood symmetric functions in the parameter t. This relation is then used to derive recurrences on the Macdonald q, t-Kostka coefficients. Résumé Pour un homomorphisme V sur l'espace des fonctions symétriques, nous présentons une opération qui crée un q-analogue de V. En donnant plusieurs exemples nous démontrons que cette quantization se produit naturellement dans la théorie de fonctions symétriques. En particulier, nous prouvons que les fonctions symétriques de Hall–Littlewood sont constituées en prenant ce q-analogue des fonctions symétriques de Schur et les fonctions symétriques de Macdonald apparaissent en prenant le q-analogue des fonctions symétriques de Hall–Littlewood dans le paramètre t. Cette relation est alors employée pour dériver des récurrence sur les coefficients Macdonald q, t-Kostka.

Finite Affine Groups: Cycle Indices, Hall–Littlewood Polynomials, and Probabilistic Algorithms

The study of asymptotic properties of the conjugacy class of a random element of the finite affine group leads one to define a probability measure on the set of all partitions of all positive integers. Four different probabilistic understandings of this measure are given—three using symmetric function theory and one using Markov chains. This leads to non-trivial enumerative results. Cycle index generating functions are derived and are used to compute the large dimension limiting probabilities that an element of the affine group is separable, cyclic, or semisimple and to study the convergence to these limits. The semisimple limit involves both Rogers–Ramanujan identities. This yields the first examples of such computations for a maximal parabolic subgroup of a finite classical group.

Toeplitz Minors

We give a new proof of the strong Szegö limit theorem estimating the determinants of Toeplitz matrices using symmetric function theory. We also obtain asymptotics for Toeplitz minors.

Applications of Symmetric Functions to Cycle and Increasing Subsequence Structure after Shuffles

Using symmetric function theory, we study the cycle structure and increasing subsequence structure of permutations after iterations of various shuffling methods. We emphasize the role of Cauchy type identities and variations of the Robinson-Schensted-Knuth correspondence.

Generalized Riffle Shuffles and Quasisymmetric Functions

Given a probability distribution on a totally ordered set, we define for each $ n \geq 1 $ a related distribution on the symmetric group $ \frak S_n $ , called the QS-distribution. It is a generalization of the q-shuffle distribution considered by Bayer, Diaconis, and Fulman. The QS-distribution is closely related to the theory of quasisymmetric functions and symmetric functions. We obtain explicit formulas in terms of quasisymmetric and symmetric functions for the probability that a random permutation from the QS-distribution satisfies various properties, such as having a given descent set, cycle structure, or shape under the Robinson-Schensted-Knuth algorithm.

Random matrix theory over finite fields

The first part of this paper surveys generating functions methods in the study of random matrices over finite fields, explaining how they arose from theoretical need. Then we describe a probabilistic picture of conjugacy classes of the finite classical groups. Connections are made with symmetric function theory, Markov chains, Rogers-Ramanujan type identities, potential theory, and various measures on partitions.

Applications of the Brauer Complex: Card Shuffling, Permutation Statistics, and Dynamical Systems

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. Using the Brauer complex, it is proved that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of Cellini using the affine Weyl group. Formulas for Cellini's measure in type A are found. This leads to new models of card shuffling and has interesting combinatorial and number-theoretic consequences. An analysis of type C gives another solution to a problem of Rogers in dynamical systems: the enumeration of unimodal permutations by cycle structure. The proof uses the factorization theory of palindromic polynomials over finite fields. Contact is made with symmetric function theory.

The Waring formula and fusion rings

The potential that generates the cohomology ring of the Grassmannian is given in terms of the elementary symmetric functions using the Waring formula that computes the power sum of roots of an algebraic equation in terms of its coefficients. As a consequence, the fusion potential for su( N) K is obtained. This potential is the explicit Chebyshev polynomial in several variables of the first kind. We also derive the fusion potential for sp( N) K from a reciprocal algebraic equation. This potential is identified with another Chebyshev polynomial in several variables. We display a connection between these fusion potentials and generalized Fibonacci and Lucas numbers. In the case of su( N) K the generating function for the generalized Fibonacci numbers obtained are in agreement with Lascoux using the theory of symmetric functions. For sp( N) K , however, the generalized Fibonacci numbers are obtained from new sequences.

Differential Symmetric Functions

A (generalized) Wronskian is a determinant of the form det\( [D^{\alpha_i} x_j]_{1 \leq i, j \leq n} \), where D is the differential operator. Wronskians are differential analogs of alternants. Imitating Jacobi's definition of Schur functions as quotients of two alternants, we define differential Schur functions as quotients of two Wronskians. Differential Schur functions are absolute invariants of the general linear group acting on the vector space spanned by the “functions”x1, x2,...,xn and they generate all such invariants. Thus, they provide the building blocks for a theory of differential symmetric functions. In this paper, we prove analogs of two basic identities: the Nagelsbach identity, which expressed Schur functions as determinants of elementary symmetric functions, and the Jacobi-Trudi identity, which expresses Schur functions as determinants of complete homogeneous symmetric functions. The proofs offer insight into the nature of these identities. In particular, the Jacobi-Trudi identity can be viewed as a change-of-basis formula.

A Combinatorial Proof of a Recursion for the q-Kostka Polynomials

The Kostka numbers Kλμ play an important role in symmetric function theory, representation theory, combinatorics and invariant theory. The q-Kostka polynomials Kλμ(q) are the q-analogues of the Kostka numbers and generalize and extend the mathematical meaning of the Kostka numbers. Lascoux and Schützenberger proved one can attach a non-negative integer statistic called charge to a semistandard tableau of shape λ and content μ such that the Kostka polynomial Kλμ(q) is the generating function for charge on those semistandard tableaux. We will give two new descriptions of charge and prove several new properties of this statistic. These new descriptions of charge will be used to give a combinatorial proof of a content reducing recursion for the q-Kostka polynomials originally proved by A. M. Garsia and C. Procesi (1992, Adv. in Math.94, 82–138).

The Eigenvalue Distribution of a Random Unipotent Matrix in Its Representation on Lines

The eigenvalue distribution of a uniformly chosen random finite unipotent matrix in its permutation action on lines is studied. We obtain bounds for the mean number of eigenvalues lying in a fixed arc of the unit circle and offer an approach to other asymptotics. For the case of all unipotent matrices, the proof gives a probabilistic interpretation to identities of Macdonald from symmetric function theory. For the case of upper triangular matrices over a finite field, connections between symmetric function theory and a probabilistic growth algorithm of Borodin and Kirillov emerge.