Hall-Littlewood Symmetric Functions Research Articles

Quiver Hall–Littlewood functions and Kostka–Shoji polynomials

For any triple $(i,a,\mu)$ consisting of a vertex $i$ in a quiver $Q$, a positive integer $a$, and a dominant $GL_a$-weight $\mu$, we define a quiver current $H^{(i,a)}_\mu$ acting on the tensor power $\Lambda^Q$ of symmetric functions over the vertices of $Q$. These provide a quiver generalization of parabolic Garsia-Jing creation operators in the theory of Hall-Littlewood symmetric functions. For a triple $(\mathbf{i},\mathbf{a},\mu(\bullet))$ of sequences of such data, we define the quiver Hall-Littlewood function $H^{\mathbf{i},\mathbf{a}}_{\mu(\bullet)}$ as the result of acting on $1\in\Lambda^Q$ by the corresponding sequence of quiver currents. The quiver Kostka-Shoji polynomials are the expansion coefficients of $H^{\mathbf{i},\mathbf{a}}_{\mu(\bullet)}$ in the tensor Schur basis. These polynomials include the Kostka-Foulkes polynomials and parabolic Kostka polynomials (Jordan quiver) and the Kostka-Shoji polynomials (cyclic quiver) as special cases. We show that the quiver Kostka-Shoji polynomials are graded multiplicities in the equivariant Euler characteristic of a vector bundle on Lusztig's convolution diagram determined by the sequences $\mathbf{i},\mathbf{a}$. For certain compositions of currents we conjecture higher cohomology vanishing of the associated vector bundle on Lusztig's convolution diagram. For quivers with no branching we propose an explicit positive formula for the quiver Kostka-Shoji polynomials in terms of catabolizable multitableaux. We also relate our constructions to $K$-theoretic Hall algebras, by realizing the quiver Kostka-Shoji polynomials as natural structure constants and showing that the quiver currents provide a symmetric function lifting of the corresponding shuffle product. In the case of a cyclic quiver, we explain how the quiver currents arise in Saito's vertex representation of the quantum toroidal algebra of type $\mathfrak{sl}_r$.

Yang-Baxter random fields and stochastic vertex models

Bijectivization refines the Yang-Baxter equation into a pair of local Markov moves which randomly update the configuration of the vertex model. Employing this approach, we introduce new Yang-Baxter random fields of Young diagrams based on spin q-Whittaker and spin Hall-Littlewood symmetric functions. We match certain scalar Markovian marginals of these fields with (1) the stochastic six vertex model; (2) the stochastic higher spin six vertex model; and (3) a new vertex model with pushing which generalizes the q-Hahn PushTASEP introduced recently in [27]. Our matchings include models with two-sided stationary initial data, and we obtain Fredholm determinantal expressions for the q-Laplace transforms of the height functions of all these models. Moreover, we also discover difference operators acting diagonally on spin q-Whittaker or (stable) spin Hall-Littlewood symmetric functions.

A study of symmetric functions via derived Hall algebra

We construct an infinite-dimensional Heisenberg algebra in the derived Hall algebra for the category of nilpotent representations of Jordan quiver. Using this Heisenberg algebra, we reconstruct the theory of symmetric functions, focusing on Hall-Littlewood symmetric functions and various operators acting on them.

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.

Lax Operator for Macdonald Symmetric Functions

Using the Lax operator formalism, we construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables and of two parameters $q,t$ are their eigenfunctions. We express our operators in terms of the Hall-Littlewood symmetric functions of the same variables and of the parameter $t$ corresponding to the partitions with one part only. Our expression is based on the notion of Baker-Akhiezer function.

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).

0-Hecke Algebra Action on the Stanley-Reisner Ring of the Boolean Algebra

We define an action of the 0-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their (q,t)-analogues introduced by Bergeron and Zabrocki, and to a more general family of noncommutative symmetric functions having parameters associated with paths in binary trees introduced recently by Lascoux, Novelli, and Thibon. We also obtain multivariate quasisymmetric function identities, which specialize to results of Garsia and Gessel on generating functions of multivariate distributions of permutation statistics.

$0$-Hecke algebra action on the Stanley-Reisner ring of the Boolean algebra

We define an action of the $0$-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their $(q,t)$-analogues introduced by Bergeron and Zabrocki. We also obtain multivariate quasisymmetric function identities, which specialize to a result of Garsia and Gessel on the generating function of the joint distribution of five permutation statistics. Nous définissons une action de l’algèbre de Hecke-$0$ de type A sur l’anneau Stanley-Reisner de l’algèbre de Boole. En étudiant cette action, on obtient une famille de fonctions symétriques non commutatives multivariées, qui se spécialisent pour les non commutatives fonctions de Hall-Littlewood symétriques et leur $(q,t)$-analogues introduits par Bergeron et Zabrocki. Nous obtenons également des identités de fonction quasisymmetrique multivariées, qui se spécialisent à la suite de Garsia et Gessel sur la fonction génératrice de la distribution conjointe de cinq statistiques de permutation.

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.

A Plethysm Formula on the Characteristic Map of Induced Linear Characters from $U_n(\mathbb F_q)$ to $GL(\mathbb F_q)$

This paper gives a plethysm formula on the characteristic map of the induced linear characters from the unipotent upper-triangular matrices $U_n(\mathbb F_q)$ to $GL_n(\mathbb F_q)$, the general linear group over finite field $\mathbb F_q$. The result turns out to be a multiple of a twisted version of the Hall-Littlewood symmetric functions $\tilde{P}_n[Y;q]$. A recurrence relation is also given which makes it easy to carry out the computation.

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.

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.

Deformed boson-fermion correspondence, Q-bosons, and topological strings on the conifold

We consider two different physical systems for which the basis of the Hilbert space can be parametrized by Young diagrams: free complex fermions and the phase model of strongly correlated bosons. Both systems have natural, well-known deformations parametrized by a parameter Q: the former one is related to the deformed boson-fermion correspondence introduced by N. Jing, while the latter is the so-called Q-boson, arising also in the context of quantum groups. These deformations are equivalent and can be realized in the same way in the algebra of Hall-Littlewood symmetric functions. Without a deformation, these reduce to Schur functions, which can be used to construct a generating function of plane partitions, reproducing a topological string partition function on $C^3$. We show that a deformation of both systems leads then to a deformed generating function, which reproduces topological string partition function of the conifold, with the deformation parameter Q identified with the size of $P^1$. Similarly, a deformation of the fermion one-point function results in the A-brane partition function on the conifold.

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].

Quantum inverse scattering method for the q-boson model and symmetric functions

The purpose of this paper is to show that the quantum inverse scattering method for the so-called q-boson model has a nice interpretation in terms of the algebra of symmetric functions. In particular, in the case of the phase model (corresponding to q = 0) the creation operator coincides (modulo a scalar factor) with the operator of multiplication by the generating function of complete homogeneous symmetric functions, and the wave functions are expressed via the Schur functions sλ(x). The general case of the q-boson model is related in a similar way to the Hall-Littlewood symmetric functions Pλ(x;q2).

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.