Hall-Littlewood Polynomials Research Articles

Cocharge and Skewing Formulas for Δ-Springer Modules and the Delta Conjecture

Abstract We prove that $\omega \Delta ^{\prime}_{e_{k}}e_{n}|_{t=0}$, the symmetric function in the Delta Conjecture at $t=0$, is a skewing operator applied to a Hall-Littlewood polynomial, and generalize this formula to the Frobenius series of all $\Delta $-Springer modules. We use this to give an explicit Schur expansion in terms of the Lascoux-Schützenberger cocharge statistic on a new combinatorial object that we call a battery-powered tableau. Our proof is geometric, and shows that the $\Delta $-Springer varieties of Levinson, Woo, and the second author are generalized Springer fibers coming from the partial resolutions of the nilpotent cone due to Borho and MacPherson. We also give alternative combinatorial proofs of our Schur expansion for several special cases, and give conjectural skewing formulas for the $t$ and $t^{2}$ coefficients of $\omega \Delta ^{\prime}_{e_{k}}e_{n}$.

Characters of [formula omitted] and vertex operators

In this paper, we present a vertex operator approach to construct and compute all complex irreducible characters of the general linear group GLn(Fq). Green's theory of GLn(Fq) is recovered and enhanced under the realization of the Grothendieck ring of representations RG=⨁n≥0R(GLn(Fq)) as two isomorphic Fock spaces associated to two infinite-dimensional F-equivariant Heisenberg Lie algebras hˆF‾ˆq and hˆF‾q, where F is the Frobenius automorphism of the algebraically closed field F‾q. Under this picture, the irreducible characters are realized by the Bernstein vertex operators for Schur functions, the characteristic functions of the conjugacy classes are realized by the vertex operators for the Hall-Littlewood functions, and the character table is completely given by matrix coefficients of vertex operators of these two types. One of the features of the current approach is a simpler identification of the Fock space RG as the Hall algebra of symmetric functions via vertex operator calculus, and another is that we are able to compute in general the character table, where Green's degree formula is demonstrated as an example.

Bijections between different combinatorial models for q-Whittaker and modified Hall-Littlewood polynomials

Bijections between different combinatorial models for q-Whittaker and modified Hall-Littlewood polynomials

Demazure crystals and the Schur positivity of Catalan functions

Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k$\\end{document}-Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of Uq(slˆℓ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$U_{q}(\\widehat{\\mathfrak{sl}}_{\\ell })$\\end{document}-generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals.

Δ–Springer varieties and Hall–Littlewood polynomials

Abstract The $\Delta $ -Springer varieties are a generalization of Springer fibers introduced by Levinson, Woo and the author that have connections to the Delta Conjecture from algebraic combinatorics. We prove a positive Hall–Littlewood expansion formula for the graded Frobenius characteristic of the cohomology ring of a $\Delta $ -Springer variety. We do this by interpreting the Frobenius characteristic in terms of counting points over a finite field $\mathbb {F}_q$ and partitioning the $\Delta $ -Springer variety into copies of Springer fibers crossed with affine spaces. As a special case, our proof method gives a geometric meaning to a formula of Haglund, Rhoades and Shimozono for the Hall–Littlewood expansion of the symmetric function in the Delta Conjecture at $t=0$ .

Universality for cokernels of random matrix products

For random integer matrices M1,…,Mk∈Matn(Z) with independent entries, we study the distribution of the cokernel Cok(M1⋯Mk) of their product. We show that this distribution converges to a universal one as n→∞ for a general class of matrix entry distributions, and more generally show universal limits for the joint distribution of Cok(M1),Cok(M1M2),…,Cok(M1⋯Mk). Furthermore, we characterize the universal distributions arising as marginals of a natural generalization of the Cohen-Lenstra measure to sequences of abelian groups with maps between them, which weights sequences inversely proportionally to their number of automorphisms. The proofs develop an extension of the moment method of Wood to joint moments of multiple groups, and rely also on the connection to Hall-Littlewood polynomials and symmetric function identities. As a corollary we obtain an explicit universal distribution for coranks of random matrix products over Fp as the matrix size tends to infinity.

Spectral analysis of an open $q$-difference Toda chain with two-sided boundary interactions on the finite integer lattice

A quantum n -particle model consisting of an open q -difference Toda chain with two-sided boundary interactions is placed on a finite integer lattice. The spectrum and eigenbasis are computed by establishing the equivalence with a previously studied q -boson model from which the quantum integrability is inherited. Specifically, the q -boson-Toda correspondence in question yields Bethe Ansatz eigenfunctions in terms of hyperoctahedral Hall–Littlewood polynomials and provides the pertinent solutions of the Bethe Ansatz equations via the global minima of corresponding Yang–Yang-type Morse functions.

Shifted Quantum Groups and Matter Multiplets in Supersymmetric Gauge Theories

The notion of shifted quantum groups has recently played an important role in algebraic geometry. This subtle modification of the original definition brings more flexibility in the representation theory of quantum groups. The first part of this paper presents new mathematical results for the shifted quantum toroidal $$\mathfrak {gl}(1)$$ and quantum affine $$\mathfrak {sl}(2)$$ algebras (resp. denoted $${\ddot{U}}_{q_1,q_2}^{\varvec{\mu }}(\mathfrak {gl}(1))$$ and $${\dot{U}}_q^{\varvec{\mu }}(\mathfrak {sl}(2))$$ ). It defines several new representations, including finite dimensional highest $$\ell $$ -weight representations for the toroidal algebra, and a vertex representation of $${\dot{U}}_q^{\varvec{\mu }}(\mathfrak {sl}(2))$$ acting on Hall–Littlewood polynomials. It also explores the relations between representations of $${\dot{U}}_q^{\varvec{\mu }}(\mathfrak {sl}(2))$$ and $${\ddot{U}}_{q_1,q_2}^{\varvec{\mu }}(\mathfrak {gl}(1))$$ in the limit $$q_1\rightarrow \infty $$ ( $$q_2$$ fixed), and present the construction of several new intertwiners. These results are used in the second part to construct BPS observables for 5d $${{\mathcal {N}}}=1$$ and 3d $${{\mathcal {N}}}=2$$ gauge theories. In particular, it is shown that 5d hypermultiplets and 3d chiral multiplets can be introduced in the algebraic engineering framework using shifted representations, and the Higgsing procedure is revisited from this perspective.

Linear transformations of vertex operators presentations of coupled Hall–Littlewood functions

Linear transformations of vertex operators presentations of coupled Hall–Littlewood functions

Linear Transformations of Vertex Operators of Hall–Littlewood Polynomials

We study the effect of linear transformations on quantum fields with applications to vertex operator presentations of symmetric functions. Properties of linearly transformed quantum fields and corresponding transformations of Hall–Littlewood polynomials are described, including preservation of commutation relations, stability, explicit combinatorial formulas, and generating functions. We prove that specializations of linearly transformed Hall–Littlewood polynomials describe all polynomial tau-functions of the KP and BKP hierarchies. Examples of linear transformations are related to multiparameter symmetric functions, Grothendieck polynomials, deformations by cyclotomic polynomials, and some other variations of Schur symmetric functions that exist in the literature.

Quantum field presentation for generalized Hall–Littlewood functions

In this paper, we give the definition and quantum field presentation for a generalized Hall–Littlewood function using vertex operators, which is an extension of the Hall–Littlewood function. Furthermore, these generalized Hall–Littlewood functions can be deduced to universal characters at t = 0 and generalized Schur Q-functions at t = −1.

A Shuffle Theorem for Paths Under Any Line

AbstractWe generalize the shuffle theorem and its$(km,kn)$version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the$(km,kn)$Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whosexandyintercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of$\operatorname {\mathrm {GL}}_{l}$characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials.

Action of Virasoro operators on Hall–Littlewood polynomials

In this paper, we prove formulas for the action of Virasoro operators on Hall–Littlewood polynomials at roots of unity.

Hall–Littlewood Polynomials, Boundaries, and p-Adic Random Matrices

Abstract We prove that the boundary of the Hall–Littlewood $t$-deformation of the Gelfand–Tsetlin graph is parametrized by infinite integer signatures, extending results of Gorin [23] and Cuenca [15] on boundaries of related deformed Gelfand–Tsetlin graphs. In the special case when $1/t$ is a prime $p$, we use this to recover results of Bufetov and Qiu [12] and Assiotis [1] on infinite $p$-adic random matrices, placing them in the general context of branching graphs derived from symmetric functions. Our methods rely on explicit formulas for certain skew Hall–Littlewood polynomials. As a separate corollary to these, we obtain a simple expression for the joint distribution of the cokernels of products $A_1, A_2A_1, A_3A_2A_1,\ldots $ of independent Haar-distributed matrices $A_i$ over ${\mathbb {Z}}_p$, generalizing the explicit formula for the classical Cohen–Lenstra measure.

Limits and fluctuations of p-adic random matrix products

We show that singular numbers (also known as elementary divisors, invariant factors or Smith normal forms) of products and corners of random matrices over $${\mathbb {Q}}_p$$ are governed by the Hall–Littlewood polynomials, in a structurally identical manner to the known relations between singular values of complex random matrices and Heckman–Opdam hypergeometric functions. This implies that the singular numbers of a product of corners of Haar-distributed elements of $$\mathrm {GL}_N({\mathbb {Z}}_p)$$ form a discrete-time Markov chain distributed as a Hall–Littlewood process, with the number of matrices in the product playing the role of time. We give an exact sampling algorithm for the Hall–Littlewood processes which arise by relating them to an interacting particle system similar to PushTASEP. By analyzing the asymptotic behavior of this particle system, we show that the singular numbers of such products obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions. In the limit of large matrix size, we also show that the analogues of the Lyapunov exponents for matrix products have universal limits within this class of $$\mathrm {GL}_N({\mathbb {Z}}_p)$$ corners.

Harmonic analysis of boxed hyperoctahedral Hall-Littlewood polynomials

For positive integers n and c with c≥2n (= the Coxeter number of the hyperoctahedral group of signed permutations of degree n), we present a finite-dimensional discrete orthogonality relation for Macdonald's three-parameter hyperoctahedral Hall-Littlewood polynomials of degree at most c in each of the n variables. These polynomials are labeled by partitions λ that fit inside a rectangular box of shape cn, i.e. the partitions in question are of length ≤n and have parts of size ≤c. We employ coordinate patches around the vertices of the alcove of boxed partitions Λ(n,c)={λ⊆cn} to establish the self-adjointness of a one-parameter family of commuting discrete difference operators ▪ acting on functions f:Λ(n,c)→C. By construction, the basis of hyperoctahedral Hall-Littlewood polynomials constitutes a joint eigenbasis for ▪ with simple spectrum, which gives rise to our discrete orthogonality relation. From the point of view of quantum integrable particle dynamics, the present geometric construction establishes the orthogonality of the Bethe Ansatz eigenfunctions for a recently studied q-boson system on a finite lattice with integrable open-end boundary conditions. The Bethe Ansatz equations enter in the geometric picture as compatibility conditions between coordinate patches stemming from distinct vertices. Two applications of the orthogonality relations are highlighted: (i) a cubature rule for the integration of symmetric functions with respect to the Haar measure on the compact symplectic group Sp(n), and (ii) a Verlinde formula for the structure constants of a deformation of the Wess-Zumino-Witten fusion ring associated with the affine Lie algebra of type Cˆn (= Cn(1)).

Affine Pieri rule for periodic Macdonald spherical functions and fusion rings

Let gˆ be an untwisted affine Lie algebra or the twisted counterpart thereof (which excludes the affine Lie algebras of type BCˆn=A2n(2)). We present an affine Pieri rule for a basis of periodic Macdonald spherical functions associated with gˆ. In type Aˆn−1=An−1(1) the formula in question reproduces an affine Pieri rule for cylindric Hall-Littlewood polynomials due to Korff, which at t=0 specializes in turn to a well-known Pieri formula in the fusion ring of genus zero slˆ(n)c-Wess-Zumino-Witten conformal field theories.

Generalized Q-functions for GKM

Recently we explained that the classical Q Schur functions stand behind various well known properties of the cubic Kontsevich model, and the next step is to ask what happens in this approach to the generalized Kontsevich model (GKM) with monomial potential Xn+1. We suggest to use the Hall-Littlewood polynomials at the parameter equal to the n-th root of unity as a generalization of the Q Schur functions from n=2 to arbitrary n>2. They are associated with n-strict Young diagrams and are independent of time-variables pkn with numbers divisible by n. These are exactly the properties possessed by the generalized Kontsevich model (GKM), thus its partition function can be expanded in such functions Q(n). However, the coefficients of this expansion remain to be properly identified. At this moment, we have not found any “superintegrability” property <character>∼character, which expressed these coefficients through the values of Q at delta-loci in the n=2 case. This is not a big surprise, because for n>2 our suggested Q functions are not looking associated with characters.

Demazure crystals for specialized nonsymmetric Macdonald polynomials

We give an explicit, nonnegative formula for the expansion of nonsymmetric Macdonald polynomials specialized at t=0 in terms of Demazure characters. Our formula results from constructing Demazure crystals whose characters are the nonsymmetric Macdonald polynomials, which also gives a new proof that these specialized nonsymmetric Macdonald polynomials are positive graded sums of Demazure characters. Demazure crystals are certain truncations of classical crystals that give a combinatorial skeleton for Demazure modules. To prove our construction, we develop further properties of Demazure crystals, including an efficient algorithm for computing their characters from highest weight elements. As a corollary, we obtain a new formula for the Schur expansion of Hall–Littlewood polynomials in terms of a simple statistic on highest weight elements of our crystals.

$q$-Deformation of the Kac-Sylvester tridiagonal matrix

Upon replacing integers by $q$-integers, one arrives at a natural $q$-deformation of a remarkably ubiquitous tridiagonal integer matrix whose eigenvalues and eigenvectors were first computed by J.J. Sylvester and M. Kac, respectively. The present note computes the eigenvalues and eigenvectors of this novel $q$-deformed Kac-Sylvester matrix through a connection with Macdonald’s hyperoctahedral Hall-Littlewood polynomials. This yields the spectrum as a minimizer of an associated Morse function reminiscent of Stieltjes’ electrostatic potential for the roots of classical orthogonal polynomials.