Littlewood Polynomials Research Articles

Square values of Littlewood polynomials

We study the square values of Littlewood polynomials. Using various methods we give all these values for the degrees n=3,5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=3, 5$$\\end{document} and n≤24\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\le 24$$\\end{document} even. Beside this, we gather computational data (by providing all solutions in a certain range) for n odd with n≤17\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\le 17$$\\end{document}. We propose some striking problems for further research, as well.

In Memoriam: Peter Benjamin Borwein (1953–2020)

Peter Borwein had wide-ranging mathematical interests, and was a pioneer in computational and experimental investigations in many areas. We fondly recall some of his interests and contributions in a number of topics in analysis, number theory, and other areas, especially work where computation and experimentation played an important role. This includes his work on a number of extremal problems regarding Littlewood polynomials and other families of integer polynomials with restricted coefficients, questions regarding the Mahler measure of an integer polynomial, problems on merit factors and Barker sequences, the Prouhet–Tarry–Escott problem, problems of Pólya and Turán regarding sums of the Liouville function, and questions in combinatorial geometry regarding arrangements of points and lines.

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 $t=0$ .

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.

On products of polynomials II

We show that if two complex quadratic polynomials in a single variable each have a coefficient 1, then their product must have a coefficient with absolute value at least ( 13 − 3 ) / 2 (\sqrt {13}-3)/2 . This is best possible. There is a more natural and classical formulation using heights. We also present some speculations about higher degree involving Littlewood polynomials.

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.

A Note on Barker Sequences and the L 1 -norm of Littlewood Polynomials

In this note, we investigate the L 1 -norms of Barker polynomials and, more generally, Littlewood polynomials over the unit circle, and give improvements to some existing results.

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.

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.

Diophantine equations for Littlewood polynomials

Diophantine equations for 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.

Do Flat Skew-Reciprocal Littlewood Polynomials Exist?

Polynomials with coefficients in \(\{-1,1\}\) are called Littlewood polynomials. Using special properties of the Rudin–Shapiro polynomials and classical results in approximation theory such as Jackson’s Theorem, de la Vallée Poussin sums, Bernstein’s inequality, Riesz’s Lemma, and divided differences, we give a significantly simplified proof of a recent breakthrough result by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba stating that there exist absolute constants \(\eta _2> \eta _1 > 0\) and a sequence \((P_n)\) of Littlewood polynomials \(P_n\) of degree n such that $$\begin{aligned} \eta _1 \sqrt{n} \le |P_n(z)| \le \eta _2 \sqrt{n} , \qquad z \in {{\mathbb {C}}}, |z| = 1 , \end{aligned}$$confirming a conjecture of Littlewood from 1966. Moreover, the existence of a sequence \((P_n)\) of Littlewood polynomials \(P_n\) is shown in a way that in addition to the above flatness properties a certain symmetry is satisfied by the coefficients of \(P_n\) making the Littlewood polynomials \(P_n\) close to skew-reciprocal.

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.

DIOPHANTINE EQUATIONS FOR POLYNOMIALS WITH RESTRICTED COEFFICIENTS, I (POWER VALUES)

AbstractWe give effective finiteness results for the power values of polynomials with coefficients composed of a fixed finite set of primes; in particular, of Littlewood polynomials.

Step roots of Littlewood polynomials and the extrema of functions in the Takagi class

AbstractWe give a new approach to characterising and computing the set of global maximisers and minimisers of the functions in the Takagi class and, in particular, of the Takagi–Landsberg functions. The latter form a family of fractal functions $f_\alpha:[0,1]\to{\mathbb R}$ parameterised by $\alpha\in(-2,2)$ . We show that $f_\alpha$ has a unique maximiser in $[0,1/2]$ if and only if there does not exist a Littlewood polynomial that has $\alpha$ as a certain type of root, called step root. Our general results lead to explicit and closed-form expressions for the maxima of the Takagi–Landsberg functions with $\alpha\in(-2,1/2]\cup(1,2)$ . For $(1/2,1]$ , we show that the step roots are dense in that interval. If $\alpha\in (1/2,1]$ is a step root, then the set of maximisers of $f_\alpha$ is an explicitly given perfect set with Hausdorff dimension $1/(n+1)$ , where n is the degree of the minimal Littlewood polynomial that has $\alpha$ as its step root. In the same way, we determine explicitly the minima of all Takagi–Landsberg functions. As a corollary, we show that the closure of the set of all real roots of all Littlewood polynomials is equal to $[-2,-1/2]\cup[1/2,2]$ .

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.

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.

Spin q–Whittaker polynomials

We introduce and study a one-parameter generalization of the q–Whittaker symmetric functions. This is a family of multivariate symmetric polynomials, whose construction may be viewed as an application of the procedure of fusion from integrable lattice models to a vertex model interpretation of a one-parameter generalization of Hall–Littlewood polynomials from [3,6,7].We prove branching and Pieri rules, standard and dual (skew) Cauchy summation identities, and an integral representation for the new polynomials.

On Newman and Littlewood polynomials with a prescribed number of zeros inside the unit disk

On Newman and Littlewood polynomials with a prescribed number of zeros inside the unit disk

Abacus-histories and the combinatorics of creation operators

Creation operators act on symmetric functions to build Schur functions, Hall–Littlewood polynomials, and related symmetric functions one row at a time. Haglund, Morse, Zabrocki, and others have studied more general symmetric functions Hα, Cα, and Bα obtained by applying any sequence of creation operators to 1. We develop new combinatorial models for the Schur expansions of these and related symmetric functions using objects called abacus-histories. These formulas arise by chaining together smaller abacus-histories that encode the effect of an individual creation operator on a given Schur function. We give a similar treatment for operators such as multiplication by hm, hm⊥, ω, etc., which serve as building blocks to construct the creation operators. We use involutions on abacus-histories to give bijective proofs of properties of the Bernstein creation operator and Hall–Littlewood polynomials indexed by three-row partitions.