Littlewood Polynomials Research Articles

On theLqnorm of cyclotomic Littlewood polynomials on the unit circle

Letnbe the collection of all (Littlewood) polynomials of degreenwith coefficients in {−1, 1}. In this paper we prove that if (P2ν) is a sequence of cyclotomic polynomialsP2ν∈2ν, thenfor everyq> 2 with somea=a(q) > 1/2 depending only onq, whereThe caseq= 4 of the above result is due to P. Borwein, Choi and Ferguson. We also prove that if (P2ν) is a sequence of cyclotomic polynomialsP2ν∈2ν, thenfor every 0 <q< 2 with some 0 <b=b(q) < 1/2 depending only onq. Similar results are conjectured for Littlewood polynomials of odd degree. Our main tool here is the Borwein–Choi Factorization Theorem.

Haglund–Haiman–Loehr type formulas for Hall–Littlewood polynomials of typeBandC

In previous work we showed that two apparently unrelated formu- las for the Hall-Littlewood polynomials of type A are, in fact, closely related. The first is the tableau formula obtained by specializing q = 0 in the Haglund- Haiman-Loehr formula for Macdonald polynomials. The second is the type A instance of Schwer's formula (rephrased and rederived by Ram) for Hall- Littlewood polynomials of arbitrary finite type; Schwer's formula is in terms of so-called alcove walks, which originate in the work of Gaussent-Littelmann and of the author with Postnikov on discrete counterparts to the Littelmann path model. We showed that the tableau formula follows by Ram's version of Schwer's formula. In this paper, we derive tableau formu- las for the Hall-Littlewood polynomials of type B and C by compressing the corresponding instances of Schwer's formula.

Hall–Littlewood polynomials, alcove walks, and fillings of Young diagrams

A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. The inversion statistic, which is the more intricate one, suffices for specializing a closely related formula to one for the type A Hall–Littlewood Q-polynomials (spherical functions on p-adic groups). An apparently unrelated development, at the level of arbitrary finite root systems, led to Schwer’s formula (rephrased and rederived by Ram) for the Hall–Littlewood P-polynomials of arbitrary type. The latter formula is in terms of so-called alcove walks, which originate in the work of Gaussent–Littelmann and of the author with Postnikov on discrete counterparts to the Littelmann path model. In this paper, we relate the above developments, by deriving a Haglund–Haiman–Loehr type formula for the Hall–Littlewood P-polynomials of type A from Ram’s version of Schwer’s formula via a “compression” procedure.

On some special solutions to periodic Benjamin-Ono equation with discrete Laplacian

We investigate a periodic version of the Benjamin-Ono (BO) equation associated with a discrete Laplacian. We find some special solutions to this equation, and calculate the values of the first two integrals of motion I1 and I2 corresponding to these solutions. It is found that there exists a strong resemblance between them and the spectra for the Macdonald q-difference operators. To better understand the connection between these classical and quantum integrable systems, we consider the special degenerate case corresponding to q=0 in more detail. Namely, we give general solutions to this degenerate periodic BO, obtain explicit formulas representing all the integrals of motions In (n=1,2,…), and successfully identify it with the eigenvalues of Macdonald operators in the limit q→0, i.e. the limit where Macdonald polynomials tend to the Hall–Littlewood polynomials.

HD-QUANTUM VERTEX ALGEBRAS AND BICHARACTERS

We define a new class of quantum vertex algebras, based on the Hopf algebra HD= ℂ[D] of "infinitesimal translations" generated by D. Besides the braiding map describing the obstruction to commutativity of products of vertex operators, HD-quantum vertex algebras have as a main new ingredient a "translation map" that describes the obstruction of vertex operators to satisfying translation covariance. The translation map also appears as obstruction to the state-field correspondence being a homomorphism.We use a bicharacter construction of Borcherds to construct a large class of HD-quantum vertex algebras. One particular example of this construction yields a quantum vertex algebra that contains the quantum vertex operators introduced by Jing in the theory of Hall–Littlewood polynomials.

Hall–Littlewood polynomials and fixed point enumeration

We resolve affirmatively some conjectures of Reiner, Stanton, and White (2004) [12] regarding enumeration of transportation matrices which are invariant under certain cyclic row and column rotations. Our results are phrased in terms of the bicyclic sieving phenomenon introduced by Barcelo, Reiner, and Stanton (2009) [1]. The proofs of our results use various tools from symmetric function theory such as the Stanton–White rim hook correspondence (Stanton and White (1985) [18]) and results concerning the specialization of Hall–Littlewood polynomials due to Lascoux, Leclerc, and Thibon (1994, 1997) [5,6].

On Newman polynomials which divide no Littlewood polynomial

Recall that a polynomial P(x) G Z[x] with coefficients 0, 1 and constant term 1 is called a Newman polynomial, whereas a polynomial with coefficients -1, 1 is called a Littlewood polynomial. Is there an algebraic number ol which is a root of some Newman polynomial but is not a root of any Littlewood polynomial? In other words (but not equivalently), is there a Newman polynomial which divides no Littlewood polynomial? In this paper, for each Newman polynomial P of degree at most 8, we find a Littlewood polynomial divisible by P. Moreover, it is shown that every trinomial l + uxa +vxb, where a < b are positive integers and u, v € {1, 1}, so, in particular, every Newman trinomial 1 + xa + xb, divides some Littlewood polynomial. Nevertheless, we prove that there exist Newman polynomials which divide no Littlewood polynomial, e.g., x9+x6+x2+x+l. This example settles the problem 006:07 posed by the first named author at the 2006 West Coast Number Theory conference. It also shows that the sets of roots of Newman polynomials Vjv» Littlewood polynomials Vc and {-1,0,1} polynomials V are distinct in the sense that between them there are only trivial relations V/v C V and Vc C V. Moreover, V jz Vc U Vjsf. The proofs of several main results (after some preparation) are computational.

Heights of powers of Newman and Littlewood polynomials

Heights of powers of Newman and Littlewood polynomials

Unimodular Roots of Special Littlewood Polynomials

AbstractWe call α(z) = a0 + a1z + · · · + an–1zn–1 a Littlewood polynomial if aj = ±1 for all j. We call α(z) self-reciprocal if α(z) = zn–1α(1/z), and call α(z) skewsymmetric if n = 2m + 1 and am+j = (–1)jam–j for all j. It has been observed that Littlewood polynomials with particularly high minimum modulus on the unit circle in ℂ tend to be skewsymmetric. In this paper, we prove that a skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle, as well as providing a new proof of the known result that a self-reciprocal Littlewood polynomial must have a zero on the unit circle.

Littlewood Pisot numbers

A Pisot number is a real algebraic integer, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is a Littlewood polynomial, one with { + 1 ,- 1 } -coefficients, and shows that they form an increasing sequence with limit 2. It is known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Littlewood polynomials. Finally, we prove that every reciprocal Littlewood polynomial of odd degree n ⩾ 3 has at least three unimodular roots.

Norms of cyclotomic Littlewood polynomials

The main result of this paper gives an explicit computation of the are primes that are not necessarily distinct.A corollary of this is the following theorem.THEOREM. If\[ f(x)=\Phi_{p_1}(\e_1 x)\Phi_{p_2}(\e_2 x^{p_1})\cdots \Phi_{p_r}(\e_r x^{p_1 p_2\cdots p_{r-1}}), \]where N = p1p2···prand εj = ±, then \[ \frac{\| f\|_4^4}{N^2} &amp; \ge &amp; \frac{\big\|\Phi_2(-x)\Phi_2(-x^2)\cdots \Phi_2\big({-}x^{2^{r-1}}\big) \big\|_4^4}{4^r}\\&amp; = &amp; \frac{\big(\frac12 +\frac{5}{34}\sqrt{17}\big)(1+\sqrt{17})^r - \big({-}\frac12 +\frac{5}{34}\sqrt{17}\big)(1-\sqrt{17})^r}{4^r}. \] In particular the minimum possible normalized L4 norm of any polynomial of the above form is attained by \[ \Phi_2(-x)\Phi_2(-x^2)\cdots \Phi_2\big({-}x^{2^{r-1}}\big). \]

Combinatorial theory of Macdonald polynomials I: Proof of Haglund's formula

Haglund recently proposed a combinatorial interpretation of the modified Macdonald polynomials H(mu). We give a combinatorial proof of this conjecture, which establishes the existence and integrality of H(mu). As corollaries, we obtain the cocharge formula of Lascoux and Schutzenberger for Hall-Littlewood polynomials, a formula of Sahi and Knop for Jack's symmetric functions, a generalization of this result to the integral Macdonald polynomials J(mu), a formula for H(mu) in terms of Lascoux-Leclerc-Thibon polynomials, and combinatorial expressions for the Kostka-Macdonald coefficients K(lambda,mu) when mu is a two-column shape.

Conjugacy class properties of the extension of GL( n, q) generated by the inverse transpose involution

Letting τ denote the inverse transpose automorphism of GL( n, q), a formula is obtained for the number of g in GL( n, q) so that gg τ is equal to a given element h. This generalizes a result of Gow and Macdonald for the special case that h is the identity. We conclude that for g random, gg τ behaves like a hybrid of symplectic and orthogonal groups. It is shown that our formula works well with both cycle index generating functions and asymptotics, and is related to the theory of random partitions. The derivation makes use of models of representation theory of GL( n, q) and of symmetric function theory, including a new identity for Hall–Littlewood polynomials. We obtain information about random elements of finite symplectic groups in even characteristic, and explicit bounds for the number of conjugacy classes and centralizer sizes in the extension of GL( n, q) generated by the inverse transpose automorphism. We give a second approach to these results using the theory of bilinear forms over a field. The results in this paper are key tools in forthcoming work of the authors on derangements in actions of almost simple groups, and we give a few examples in this direction.

Schur function analogs for a filtration of the symmetric function space

We work here with the linear span Λ t ( k) of Hall–Littlewood polynomials indexed by partitions whose first part is no larger than k. The sequence of spaces Λ t ( k) yields a filtration of the space Λ of symmetric functions in an infinite alphabet X. In joint work with Lascoux [4] we gave a combinatorial construction of a family of symmetric polynomials { A λ ( k) [ X; t]} λ 1⩽ k , with N[t] -integral Schur function expansions, which we conjectured to yield a basis for Λ t ( k) . Our primary motivation for this construction is to provide a positive integral factorization of the Macdonald q, t-Kostka matrix. More precisely, we conjecture that the connection coefficients expressing the Hall–Littlewood or Macdonald polynomials belonging to Λ t ( k) in terms of the basis { A λ ( k) [ X; t]} λ 1⩽ k are polynomials in N[q,t] . We give here a purely algebraic construction of a new family { s λ ( k) [ X; t]} λ 1⩽ k of polynomials in Λ t ( k) which we conjecture is identical to { A λ ( k) [ X; t]} λ 1⩽ k . We prove that { s λ ( k) [ X; t]} λ 1⩽ k is in fact a basis of Λ t ( k) and derive several further properties including that s λ ( k) [ X; t] reduces to the Schur function s λ [ X] for sufficiently large k. We also state a number of conjectures which reveal that the polynomials { s λ ( k) [ X; t]} λ 1⩽ k are in fact the natural analogues of Schur functions for the space Λ t ( k) .

Towards a Schubert calculus for complex reflection groups

The cohomology ring of the flag manifold associated to any compact Lie group has a simple description in terms of the Weyl group of . We discuss the relation of our calculation to Hall–Littlewood polynomials at roots of unity.

Finite Affine Groups: Cycle Indices, Hall–Littlewood Polynomials, and Probabilistic Algorithms

The study of asymptotic properties of the conjugacy class of a random element of the finite affine group leads one to define a probability measure on the set of all partitions of all positive integers. Four different probabilistic understandings of this measure are given—three using symmetric function theory and one using Markov chains. This leads to non-trivial enumerative results. Cycle index generating functions are derived and are used to compute the large dimension limiting probabilities that an element of the affine group is separable, cyclic, or semisimple and to study the convergence to these limits. The semisimple limit involves both Rogers–Ramanujan identities. This yields the first examples of such computations for a maximal parabolic subgroup of a finite classical group.

Spherical Harmonic Analysis on Buildings of Type à n

To any locally finite thick building \(\) of type \(\) there is naturally associated a commutative algebra \(\) of operators. When \(\) is constructed from a local field F with local ring \(\), and \(\), then \(\) is isomorphic to the convolution algebra of compactly supported bi-K-invariant functions on PGL(n+1,F). We give a proof, valid for any \(\), that the multiplicative functionals on \(\) may all be expressed in terms of Hall–Littlewood polynomials. Regarding \(\) as a subalgebra of the C*-algebra of bounded operators on the space \(\) of square summable functions on the vertex set \(\) of \(\), we find the spectrum of the C*-algebra \(\), the closure of \(\). This generalizes results obtained in [3] when n = 1 and in [5] when n = 2.

Sequences of Symmetric Polynomials and Combinatorial Properties of Tableaux

In 1977 G. P. Thomas showed that the sequence of Schur polynomials associated to a partitionλcan be comfortably generated from the sequence of variablesx=(x1, x2, x3, …) by the application of mixed Baxter/multiplication operators, which in turn can be easily computed from the set SYT(λ) of standard Young tableaux of shapeλ. We generalize this construction, thereby making possible the explicit and effective computation of the Hall–Littlewood, Jack, and Macdonald polynomials used in representation theory, combinatorics, multivariate statistics, and quantum algebra. These generalized formulas have a pleasing recursive structure with respect to the Young lattice and they can easily be specialized to yield “skew” forms in all cases and “super” forms in the Schur case. We introduce and investigate: (1) the “descent polynomial of a partitionλ,” which arises naturally in the enumeration of semistandard Young tableaux of shapeλ; (2) the Boolean latticeG(ζ) associated to anyζ∈SYT(λ), which is fundamental for the “weighted” generalization of Thomas' approach to Schur polynomials; and (3) an action of the symmetric groups on semistandard Young tableaux, which is connected with Knuth's combinatorial proof of the symmetry of Schur functions. Moreover, we argue that a generalization of Thomas' approach is a natural starting point in search of “universal weighted symmetric functions.”

Quantum Kac– Algebras and Vertex Representations

We introduce an affinization of the quantum Kac–Moody algebra associated to a symmetric generalized Cartan matrix. Based on the affinization, we construct a representation of the quantum Kac–Moody algebra by vertex operators from bosonic fields. We also obtain a combinatorial indentity about Hall–Littlewood polynomials.

Boson–fermion correspondence for Hall–Littlewood polynomials

The boson–fermion correspondence is given for the case of the Hall–Littlewood polynomials, which is a t-deformation of the correspondence realizing the Schur polynomials.