Littlewood Symmetric Functions Research Articles

Refined Littlewood identity for spin Hall–Littlewood symmetric rational functions

Fully inhomogeneous spin Hall–Littlewood symmetric rational functions F λ are multiparameter deformations of the classical Hall–Littlewood symmetric polynomials and can be viewed as partition functions in 𝔰𝔩(2) higher spin six vertex models.

Refined Cauchy identity for spin Hall–Littlewood symmetric rational functions

Fully inhomogeneous spin Hall–Littlewood symmetric rational functions Fλ arise in the context of 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 Fλ'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 equation.We rewrite our Izergin–Korepin type determinant in a different form which includes one of the sets of variables in a completely symmetric way. This determinantal identity might be of independent interest, and also allows to directly link the spin Hall–Littlewood rational functions with (the Hall–Littlewood particular case of) the interpolation Macdonald polynomials. In a different direction, a Schur expansion of our Izergin–Korepin type determinant yields a deformation of Schur symmetric polynomials.In the spin-12 specialization, our refined Cauchy identity leads to a summation identity for eigenfunctions of the ASEP (Asymmetric Simple Exclusion Process), a celebrated stochastic interacting particle system in the Kardar–Parisi–Zhang universality class. This produces explicit integral formulas for certain multitime probabilities in ASEP.

Elementary proof and application of the generating functions for generalized Hall–Littlewood functions

In this note we define a generalization of Hall–Littlewood symmetric functions using a formal group law and we give an elementary proof of the generating function formula for these generalized functions. We also present some applications of this formula.

Hall–Littlewood symmetric functions via the chip-firing game

Via the chip-firing game, a class of Schur positive symmetric functions depending on four parameters is introduced for any labeled connected simple graph. Tableaux formulae are stated to expand such symmetric functions in terms of the complete symmetric functions. In the case of a simple path, the resulting symmetric functions reduce to the transformed Hall–Littlewood symmetric functions when the parameters are suitable specialized.

Law of Large Numbers for infinite random matrices over a finite field

Asymptotic representation theory of general linear groups $$\hbox {GL}(n,F_\mathfrak {q})$$ over a finite field leads to studying probability measures $$\rho $$ on the group $$\mathbb {U}$$ of all infinite uni-uppertriangular matrices over $$F_\mathfrak {q}$$ , with the condition that $$\rho $$ is invariant under conjugations by arbitrary infinite matrices. Such probability measures form an infinite-dimensional simplex, and the description of its extreme points (in other words, ergodic measures $$\rho $$ ) was conjectured by Kerov in connection with nonnegative specializations of Hall–Littlewood symmetric functions. Vershik and Kerov also conjectured the following Law of Large Numbers. Consider an infinite random matrix drawn from an ergodic measure coming from the Kerov’s conjectural classification and its $$n \times n$$ submatrix formed by the first rows and columns. The sizes of Jordan blocks of the submatrix can be interpreted as a (random) partition of $$n$$ , or, equivalently, as a (random) Young diagram $$\lambda (n)$$ with $$n$$ boxes. Then, as $$n\rightarrow \infty $$ , the rows and columns of $$\lambda (n)$$ have almost sure limiting frequencies corresponding to parameters of this ergodic measure. Our main result is the proof of this Law of Large Numbers. We achieve it by analyzing a new randomized Robinson–Schensted–Knuth (RSK) insertion algorithm which samples random Young diagrams $$\lambda (n)$$ coming from ergodic measures. The probability weights of these Young diagrams are expressed in terms of Hall–Littlewood symmetric functions. Our insertion algorithm is a modified and extended version of a recent construction by Borodin and Petrov (2013). On the other hand, our randomized RSK insertion generalizes a version of the RSK insertion introduced by Vershik and Kerov (SIAM J. Algebr. Discret. Math. 7(1):116–124, 1986) in connection with asymptotic representation theory of symmetric groups (which is governed by nonnegative specializations of Schur symmetric functions).

A Lift of the Schur and Hall–Littlewood Bases to Non-commutative Symmetric Functions

AbstractWe introduce a new basis of the algebra of non-commutative symmetric functions whose images under the forgetful map are Schur functions when indexed by a partition. Dually, we build a basis of the quasi-symmetric functions that expand positively in the fundamental quasi-symmetric functions. We then use the basis to construct a non-commutative lift of theHall–Littlewood symmetric functions with similar properties to their commutative counterparts.

0-Hecke algebra actions on coinvariants and flags

The 0-Hecke algebra H n (0) is a deformation of the group algebra of the symmetric group $\mathfrak{S}_{n}$ . We show that its coinvariant algebra naturally carries the regular representation of H n (0), giving an analogue of the well-known result for $\mathfrak{S}_{n}$ by Chevalley–Shephard–Todd. By investigating the action of H n (0) on coinvariants and flag varieties, we interpret the generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. We also study the action of H n (0) on the cohomology rings of the Springer fibers, and similarly interpret the (non-commutative) Hall–Littlewood symmetric functions indexed by hook shapes.

Macdonald operators at infinity

We construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables x 1,x 2,? and of two parameters q,t are their eigenfunctions. These operators are defined as limits at N?? of renormalized Macdonald operators acting on symmetric polynomials in the variables x 1,?,x N . They are differential operators in terms of the power sum variables $p_{n}=x_{1}^{n}+x_{2}^{n}+\cdots$ and we compute their symbols by using the Macdonald reproducing kernel. We express these symbols in terms of the Hall---Littlewood symmetric functions of the variables x 1,x 2,?. Our result also yields elementary step operators for the Macdonald symmetric functions.

Class expansion of some symmetric functions in Jucys–Murphy elements

We present a method to compute the class expansion of a symmetric function in the Jucys–Murphy elements of the symmetric group. We apply this method to one-row Hall–Littlewood symmetric functions, which interpolate between power sums and complete symmetric functions.

Positivity for Kac polynomials and DT-invariants of quivers

We give a cohomological interpretation of both the Kac polynomial and the rened Donaldson-Thomas-invariants of quivers. This interpretation yields a proof of a conjecture of Kac from 1982 and gives a new perspective on recent work of Kontsevich{Soibelman. This is achieved by computing, via an arithmetic Fourier transform, the dimensions of the isotypical components of the cohomology of associated Nakajima quiver varieties under the action of a Weyl group. The generating function of the corresponding Poincar e polynomials is an extension of Hua’s formula for Kac polynomials of quivers involving Hall{Littlewood symmetric functions. The resulting formulae contain a wide range of information on the geometry of the quiver varieties.

Narayana polynomials and Hall–Littlewood symmetric functions

We show that Narayana polynomials are a specialization of row Hall–Littlewood symmetric functions. Using λ-ring calculus, we generalize to Narayana polynomials the formulas of Koshy and Jonah for Catalan numbers.

Hall–Littlewood Operators in the Theory of Parking Functions and Diagonal Harmonics

In a recent work, Jim Haglund, Jennifer Morse, and Mike Zabrocki proved a variety of identities involving Hall–Littlewood symmetric functions indexed by compositions. When they applied ∇ to these symmetric functions, the resulting identities and computer data led them to some truly remarkable refinements of the shuffle conjecture. We prove here the symmetric function side of a recursion which when combined with a recent parking function recursion of Angela Hicks [18] settles some special cases of the Haglund–Morse–Zabrocki conjectures. Our main result of a compositional q,t-Catalan and Schröder theorem yields, as a consequence, surprisingly simple new proofs of the original q,t-Catalan and Schröder results.

Combinatorial Hopf algebras, noncommutative Hall–Littlewood functions, and permutation tableaux

We introduce a new family of noncommutative analogues of the Hall–Littlewood symmetric functions. Our construction relies upon Tevlin's bases and simple q-deformations of the classical combinatorial Hopf algebras. We connect our new Hall–Littlewood functions to permutation tableaux, and also give an exact formula for the q-enumeration of permutation tableaux of a fixed shape. This gives an explicit formula for: the steady state probability of each state in the partially asymmetric exclusion process (PASEP); the polynomial enumerating permutations with a fixed set of weak excedances according to crossings; the polynomial enumerating permutations with a fixed set of descent bottoms according to occurrences of the generalized pattern 2–31.

Ribbon Schur operators

A new combinatorial approach to the ribbon tableaux generating functions and q -Littlewood–Richardson coefficients of Lascoux et al. [A. Lascoux, B. Leclerc, J.-Y. Thibon, Ribbon tableaux, Hall–Littlewood symmetric functions, quantum affine algebras, and unipotent varieties, J. Math. Phys. 38 (3) (1997) 1041–1068] is suggested. We define operators which add ribbons to partitions and following Fomin and Greene [S. Fomin, C. Greene, Noncommutative Schur functions and their applications, Discrete Math. 193 (1998) 179–200] study non-commutative symmetric functions in these operators. This allows us to give combinatorial interpretations for some (skew) q -Littlewood–Richardson coefficients whose non-negativity appears not to be known. Our set-up also leads to a new proof of the action of the Heisenberg algebra on the Fock space of U q ( s l ̂ n ) due to Kashiwara et al. [M. Kashiwara, T. Miwa, E. Stern, Decomposition of q -deformed Fock spaces, Selecta Math. 1 (1996) 787–805].

Rogers–Szegö polynomials and Hall–Littlewood symmetric functions

We use Rogers–Szegö polynomials to unify some well-known identities for Hall–Littlewood symmetric functions due to Macdonald and Kawanaka.

Inversion of the Pieri formula for Macdonald polynomials

We give the explicit analytic development of Macdonald polynomials in terms of “modified complete” and elementary symmetric functions. These expansions are obtained by inverting the Pieri formula. Specialization yields similar developments for monomial, Jack and Hall–Littlewood symmetric functions.

Breakthroughs in the theory of Macdonald polynomials

In 1998, I. G. Macdonald (1) introduced a remarkable new basis for the space of symmetric functions. The elements of this basis are denoted , where λ is a partition and p, q are two free parameters. The 's, which are now called “Macdonald polynomials,” specialize to many of the well known bases for the symmetric functions, by suitable choices of the parameters q and t. In fact, we can obtain in this manner the Schur functions, the Hall–Littlewood symmetric functions, the Jack symmetric functions, the zonal symmetric functions, the zonal spherical functions, and the elementary and monomial symmetric functions.

Hall–Littlewood functions and the [formula omitted] Rogers–Ramanujan identities

We prove an identity for Hall–Littlewood symmetric functions labelled by the Lie algebra A 2 . Through specialization this yields a simple proof of the A 2 Rogers–Ramanujan identities of Andrews, Schilling and the author.

Hall–Littlewood Vertex Operators and Generalized Kostka Polynomials

A family of vertex operators that generalizes those given by Jing for the Hall–Littlewood symmetric functions is presented. These operators produce symmetric functions related to the Poincaré polynomials referred to as generalized Kostka polynomials in the same way that Jing's operator produces symmetric functions related to Kostka–Foulkes polynomials. These operators are then used to derive commutation relations and new relations involving the generalized Kostka coefficients. Such relations may be interpreted as identities in the (GL(n)×C)-equivariant K-theory of the nullcone.

Ribbon Operators and Hall–Littlewood Symmetric Functions

Given a partition λ=(λ1, λ2, …, λk), let λrc=(λ2−1, λ3−1, …, λk−1). It is easily seen that the diagram λ/λrc is connected and has no 2×2 subdiagrams, we shall call it a ribbon. To each ribbon R, we associate a symmetric function operator SR. We may define the major index of a ribbon maj(R) to be the major index of any permutation that fits the ribbon. This paper is concerned with the operator Hq1k=∑Rqmaj(R)SR where the sum is over all 2k−1 ribbons of size k. We show here that Hq1k has truly remarkable properties, in particular, it is a Rodriguez operator that adds a column to the Hall–Littlewood symmetric functions. We believe that some of the tools we introduce here to prove our results should also be of independent interest and may be useful for establishing further symmetric function identities.