Hall-Littlewood Polynomials Research Articles

Combinatorial Properties of Rogers-Ramanujan Type Identities Arising from Hall-Littlewood Polynomials

Here we consider the q-series coming from the Hall-Littlewood polynomials, $$\begin{array}{l}{R_{v} (a, b; q) = {\sum_{\mathop {\lambda}\limits_{\lambda_1 \leq a}}} q^{c | \lambda |} P_{2\lambda}(1, q, q^{2}, \ldots ; q^{2b+d}).}\end{array}$$ These series were defined by Griffin, Ono, and Warnaar in their work on the framework of the Rogers-Ramanujan identities. We devise a recursive method for computing the coefficients of these series when they arise within the Rogers-Ramanujan framework. Furthermore, we study the congruence properties of certain quotients and products of these series, generalizing the famous Ramanujan congruence $$p(5n+4) \equiv 0\quad ({\rm mod}\, 5).$$

Hall-Littlewood Polynomials and Cohen-Lenstra Heuristics for Jacobians of Random Graphs

Cohen-Lenstra heuristics for Jacobians of random graphs give rise to random partitions. We connect these random partitions to the Hall-Littlewood polynomials of symmetric function theory, and use this connection to give combinatorial proofs of properties of these random partitions. In addition, we use Markov chains to give an algorithm for generating these partitions.

Construction and deconstruction of single instanton Hilbert series

Many methods exist for the construction of the Hilbert series describing the moduli spaces of instantons. We explore some of the underlying group theoretic relationships between these various constructions, including those based on the Coulomb branches and Higgs branches of SUSY quiver gauge theories, as well as those based on generating functions derivable from the Weyl Character Formula. We show how the character description of the reduced single instanton moduli space of any Classical or Exceptional group can be deconstructed faithfully in terms of characters or modified Hall-Littlewood polynomials of its regular semi-simple subgroups. We derive and utilise Highest Weight Generating functions, both for the characters of Classical or Exceptional groups and for the Hall-Littlewood polynomials of unitary groups. We illustrate how the root space data encoded in extended Dynkin diagrams corresponds to relationships between the Coulomb branches of quiver gauge theories for instanton moduli spaces and those for T(SU(N)) moduli spaces.

Hall–Littlewood polynomials and characters of affine Lie algebras

The Weyl–Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of Kac–Moody algebras. It is not, however, a formula that is combinatorial in nature, obscuring positivity. In this paper we show that the theory of Hall–Littlewood polynomials may be employed to prove Littlewood-type combinatorial formulas for the characters of certain highest weight modules of the affine Lie algebras Cn(1), A2n(2) and Dn+1(2). Through specialisation this yields generalisations for Bn(1), Cn(1), A2n−1(2), A2n(2) and Dn+1(2) of Macdonald's identities for powers of the Dedekind eta-function. These generalised eta-function identities include the Rogers–Ramanujan, Andrews–Gordon and Göllnitz–Gordon q-series as special, low-rank cases.

A Generalization of Tokuyama's Formula to the Hall-Littlewood Polynomials

A theorem due to Tokuyama expresses Schur polynomials in terms of Gelfand-Tsetlin patterns, providing a deformation of the Weyl character formula and two other classical results, Stanley's formula for the Schur $q$-polynomials and Gelfand's parametrization for the Schur polynomials. We generalize Tokuyama's formula to the Hall-Littlewood polynomials by extending Tokuyama's statistics. Our result, in addition to specializing to Tokuyama's result and the aforementioned classical results, also yields connections to the monomial symmetric function and a new deformation of Stanley's formula.

Hall–Littlewood polynomials and vector bundles on the Hilbert scheme

Let E be the bundle defined by applying a polynomial functor to the tautological bundle on the Hilbert scheme of n points in the complex plane. By a result of Haiman [5], the Čech cohomology groups Hi(E) vanish for all i>0. It follows that the equivariant Euler characteristic with respect to the standard two-dimensional torus action has nonnegative integer coefficients in the torus variables z1,z2, because they count the dimensions of the weight spaces of H0(E). We derive a formula for this Euler characteristic using residue formulas for the Euler characteristic coming from the description of the Hilbert scheme as a quiver variety [13,14]. We evaluate this expression using Jing's Hall–Littlewood vertex operator with parameter z1 [7], and a new vertex operator formula given in Proposition 1 below. We conjecture that the summand in this formula is a polynomial in z1 with nonnegative integer coefficients, a special case of which was known to Lascoux and Schützenberger [8].

A Gysin formula for Hall-Littlewood polynomials

We give a formula for pushing forward the classes of Hall-Littlewood polynomials in Grassmann bundles, generalizing Gysin formulas for Schur S S - and P P -functions.

T ρ σ (G) theories and their Hilbert series

We give an explicit formula for the Higgs and Coulomb branch Hilbert series for the class of 3d $$ \mathcal{N}=4 $$ superconformal gauge theories T (G) corresponding to a set of D3 branes ending on NS5 and D5-branes, with or without O3 planes. Here G is a classical group, σ is a partition of G and ρ a partition of the dual group G ∨. In deriving such a formula we make use of the recently discovered formula for the Hilbert series of the quantum Coulomb branch of $$ \mathcal{N}=4 $$ superconformal theories. The result can be expressed in terms of a generalization of a class of symmetric functions, the Hall-Littlewood polynomials, and can be interpreted in mathematical language in terms of localization. We mainly consider the case G = SU(N) but some interesting results are also given for orthogonal and symplectic groups.

Modular Macdonald functions and generalized Newton's identity

Based on a generalized Newton's identity, we construct a family of symmetric functions which deform the modular Hall–Littlewood functions.

Coulomb branch Hilbert series and Hall-Littlewood polynomials

There has been a recent progress in understanding the chiral ring of 3d $\mathcal{N}=4$ superconformal gauge theories by explicitly constructing an exact generating function (Hilbert series) counting BPS operators on the Coulomb branch. In this paper we introduce Coulomb branch Hilbert series in the presence of background magnetic charges for flavor symmetries, which are useful for computing the Hilbert series of more general theories through gluing techniques. We find a simple formula of the Hilbert series with background magnetic charges for $T_\rho(G)$ theories in terms of Hall-Littlewood polynomials. Here $G$ is a classical group and $\rho$ is a certain partition related to the dual group of $G$. The Hilbert series for vanishing background magnetic charges show that Coulomb branches of $T_\rho(G)$ theories are complete intersections. We also demonstrate that mirror symmetry maps background magnetic charges to baryonic charges.

Coulomb branch Hilbert series and three dimensional Sicilian theories

We evaluate the Coulomb branch Hilbert series of mirrors of three dimensional Sicilian theories, which arise from compactifying the $6d$ $(2,0)$ theory with symmetry $G$ on a circle times a Riemann surface with punctures. We obtain our result by gluing together the Hilbert series for building blocks $T_{\mathbf{\rho}}(G)$, where $\mathbf{\rho}$ is a certain partition related to the dual group of $G$, which we evaluated in a previous paper. The result is expressed in terms of a class of symmetric functions, the Hall-Littlewood polynomials. As expected from mirror symmetry, our results agree at genus zero with the superconformal index prediction for the Higgs branch Hilbert series of the Sicilian theories and extend it to higher genus. In the $A_1$ case at genus zero, we also evaluate the Coulomb branch Hilbert series of the Sicilian theory itself, showing that it only depends on the number of external legs.

Diagonalization of the infinite q-boson system

We present a hierarchy of commuting operators in Fock space containing the q-boson Hamiltonian on Z and show that the operators in question are simultaneously diagonalized by Hall–Littlewood functions and have absolutely continuous spectrum. As an application, the n-particle scattering operator is computed explicitly.

A three shuffle case of the compositional parking function conjecture

We prove here that the polynomial 〈∇Cp1,eahbhc〉q,t-enumerates, by the statistics dinv and area, the parking functions whose supporting Dyck path touches the main diagonal according to the composition p⊨a+b+c and has a reading word which is a shuffle of one decreasing word and two increasing words of respective sizes a, b, c. Here Cp1 is a rescaled Hall–Littlewood polynomial and ∇ is the Macdonald eigen-operator introduced by Bergeron and Garsia. This is our latest progress in a continued effort to settle the decade old shuffle conjecture of Haglund et al. This result includes as special cases all previous results connected with the shuffle conjecture such as the q,t-Catalan, Schröder and two shuffle results of Haglund as well as their compositional refinements recently obtained by the authors. It also confirms the possibility that the approach adopted by the authors has the potential to yield a resolution of the shuffle parking function conjecture as well as its compositional refinement more recently proposed by Haglund, Morse and Zabrocki.

Hall-Littlewood Polynomials in terms of Yamanouchi words

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions to several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram $δ ⊂ \mathbb{Z} \times \mathbb{Z}$, written as $\widetilde H_δ (X;q,t)$ and $\widetilde P_δ (X;t)$, respectively. We then give an explicit Schur expansion of $\widetilde P_δ (X;t)$ as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function $R_γ ,δ (X)$ as a refinement of $\widetilde P_δ$ and similarly describe its Schur expansion. We then analysize $R_γ ,δ (X)$ to determine the leading term of its Schur expansion. To gain these results, we associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_δ$ . In the case where a subgraph of $\mathcal{H}_δ$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.

Boundary Interactions for the Semi-Infinite q-Boson System and Hyperoctahedral Hall-Littlewood Polynomials

We present a semi-infinite q-boson system endowed with a four-parameter boundary interaction. The n-particle Hamiltonian is diagonalized by generalized Hall-Littlewood polynomials with hyperoctahedral symmetry that arise as a degeneration of the Macdonald-Koornwinder polynomials and were recently studied in detail by Venkateswaran.

The Semi-Infinite q-Boson System with Boundary Interaction

Upon introducing a one-parameter quadratic deformation of the q-boson algebra and a diagonal perturbation at the end point, we arrive at a semi-infinite q-boson system with a two-parameter boundary interaction. The eigenfunctions are shown to be given by Macdonald's hyperoctahedral Hall-Littlewood functions of type BC. It follows that the n-particle spectrum is bounded and absolutely continuous and that the corresponding scattering matrix factorizes as a product of two-particle bulk and one-particle boundary scattering matrices.

Generalization of the Macdonald formula for Hall–Littlewood polynomials

We study the Gaussent–Littelmann formula for Hall–Littlewood polynomials and we develop combinatorial tools to describe the formula in a purely combinatorial way for type An, Bn and Cn. This description is in terms of Young tableaux and arises from identifying one-skeleton galleries that appear in the Gaussent–Littelmann formula with Young tableaux. Furthermore, we show by using these tools that the Gaussent–Littelmann formula and the well-known Macdonald formula for Hall–Littlewood polynomials for type An are the same up to an additional factorization.

Charge on tableaux and the poset of k-shapes

A poset on a certain class of partitions known as k-shapes was introduced in [7] to provide a combinatorial rule for the expansion of a k−1-Schur function into k-Schur functions at t=1. The main ingredient in this construction was a bijection, which we call the weak bijection, that associates to a k-tableau a pair made out of a k−1-tableau and a path in the poset of k-shapes. We define here a concept of charge on k-tableaux (which conjecturally gives a combinatorial interpretation for the expansion coefficients of Hall–Littlewood polynomials into k-Schur functions), and show that it is compatible in the standard case with the weak bijection. In particular, we obtain that the usual charge of a standard tableau of size n is equal to the sum of the charges of its corresponding paths in the poset of k-shapes, for k=2,3,…,n.

Transition matrices for symmetric and quasisymmetric Hall–Littlewood polynomials

We introduce explicit combinatorial interpretations for the coefficients in some of the transition matrices relating to skew Hall–Littlewood polynomials Pλ/μ(t) and Hivertʼs quasisymmetric Hall–Littlewood polynomials Gγ(t). More specifically, we provide:1.the G-expansions of the Hall–Littlewood polynomials Pλ(t), the monomial quasisymmetric polynomials Mα, the quasisymmetric Schur polynomials Sα, and the peak quasisymmetric functions Kα;2.an expansion of Pλ/μ(t) in terms of the Fαʼs. The F-expansion of Pλ/μ(t) is facilitated by introducing starred tableaux.

A metaplectic Casselman-Shalika formula for GL r

We provide formulas for various bases of spherical Whittaker functions on the $n$-fold metaplectic cover of ${\rm GL}_r$ over a $p$-adic field and show that there is a basis of symmetric functions in the complex parameter. In addition we relate a specific spherical Whittaker function to the $p$-power part of the Weyl group multiple Dirichlet series for the root system of type $A_{r-1}$ constructed from $n$th order Gauss sums. We also show that the zonal spherical functions can be computed explicitly in terms of Hall-Littlewood polynomials as in Macdonald's formula for ${\rm GL}_r$.