Symmetric Laurent Polynomials Research Articles

The ∗-Markov Equation for Laurent Polynomials

We consider the $*$-Markov equation for the symmetric Laurent polynomials in three variables with integer coefficients, which is an equivariant analog of the classical Markov equation for integers. We study how the properties of the Markov equation and its solutions are reflected in the properties of the $*$-Markov equation and its solutions.

Spherical functions and local densities on the space of p-adic quaternion Hermitian forms

We introduce the space [Formula: see text] of quaternion Hermitian forms of size [Formula: see text] on a [Formula: see text]-adic field with odd residual characteristic, and define typical spherical functions [Formula: see text] on [Formula: see text] and give their induction formula on sizes by using local densities of quaternion Hermitian forms. Then, we give functional equation of spherical functions with respect to [Formula: see text], and define a spherical Fourier transform on the Schwartz space [Formula: see text] which is Hecke algebra [Formula: see text]-injective map into the symmetric Laurent polynomial ring of size [Formula: see text]. Then, we determine the explicit formulas of [Formula: see text] by a method of the author’s former result. In the last section, we give precise generators of [Formula: see text] and determine all the spherical functions for [Formula: see text], and give the Plancherel formula for [Formula: see text].

Causal FIR symmetric paraunitary matrix extension and construction of symmetric tight M-dilated framelets

We consider a causal FIR symmetric paraunitary (PU) matrix extension with parameters, and the constructions of symmetric orthogonal wavelets and symmetric tight framelets with an integer dilation factor M ⩾ 2 . Firstly, we propose an algorithm for factorizing a causal FIR symmetric Laurent polynomial vector into the product of order-one paraunitary matrices and constant vector. Secondly, based on the factorization algorithm, we propose a method for a causal symmetric PU extension with parameters of a Laurent polynomial vector. This method enables us to parameterize polyphase matrix whose first row is equal to the Laurent polynomial vector composed of polyphase components of given lowpass filter. And this symmetric PU extension provides a minimal factorization structure. Thirdly, we consider the constructions of symmetric orthogonal wavelets and symmetric tight framelets with integer dilation factor by the causal symmetric PU extension. Finally, several examples are provided to illustrate the construction methods proposed in this paper.

Dualities in the q‐Askey Scheme and Degenerate DAHA

AbstractThe Askey–Wilson polynomials are a four‐parameter family of orthogonal symmetric Laurent polynomials that are eigenfunctions of a second‐order q‐difference operator L, and of a second‐order difference operator in the variable n with eigenvalue . Then, L and multiplication by generate the Askey–Wilson (Zhedanov) algebra. A nice property of the Askey–Wilson polynomials is that the variables z and n occur in the explicit expression in a similar and to some extent exchangeable way. This property is called duality. It returns in the nonsymmetric case and in the underlying algebraic structures: the Askey–Wilson algebra and the double affine Hecke algebra (DAHA). In this paper, we follow the degeneration of the Askey–Wilson polynomials until two arrows down and in four different situations: for the orthogonal polynomials themselves, for the degenerate Askey–Wilson algebras, for the nonsymmetric polynomials, and for the (degenerate) DAHA and its representations.

Euclidean Algorithm for Extension of Symmetric Laurent Polynomial Matrix and Its Application in Construction of Multiband Symmetric Perfect Reconstruction Filter Bank

For a given pair of s-dimensional real Laurent polynomials ( a → ( z ) , b → ( z ) ) , which has a certain type of symmetry and satisfies the dual condition b → ( z ) T a → ( z ) = 1 , an s × s Laurent polynomial matrix A ( z ) (together with its inverse A - 1 ( z ) ) is called a symmetric Laurent polynomial matrix extension of the dual pair ( a → ( z ) , b → ( z ) ) if A ( z ) has similar symmetry, the inverse A - 1 ( Z ) also is a Laurent polynomial matrix, the first column of A ( z ) is a → ( z ) and the first row of A - 1 ( z ) is ( b → ( z ) ) T . In this paper, we introduce the Euclidean symmetric division and the symmetric elementary matrices in the Laurent polynomial ring and reveal their relation. Based on the Euclidean symmetric division algorithm in the Laurent polynomial ring, we develop a novel and effective algorithm for symmetric Laurent polynomial matrix extension. We also apply the algorithm in the construction of multi-band symmetric perfect reconstruction filter banks.

Twisted Alexander Polynomials of Hyperbolic Knots

We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to . It is an unambiguous symmetric Laurent polynomial whose coefficients lie in the trace field of the knot. It contains information about genus, fibering, and chirality, and moreover, is powerful enough to sometimes detect mutation. We calculated this invariant numerically for all 313 209 hyperbolic knots in S 3 with at most 15 crossings, and found that in all cases it gave a sharp bound on the genus of the knot and determined both fibering and chirality. We also study how such twisted Alexander polynomials vary as one moves around in an irreducible component X 0 of the -character variety of the knot group. We show how to understand all of these polynomials at once in terms of a polynomial whose coefficients lie in the function field of X 0. We use this to help explain some of the patterns observed for knots in S 3, and explore a potential relationship between this universal polynomial and the Culler–Shalen theory of surfaces associated to ideal points.

On the Hall algebra of an elliptic curve, I

In this article we describe the Hall algebra H_X of an elliptic curve X defined over a finite field and show that the group SL(2,Z) of exact auto-equivalences of the derived category D^b(Coh(X)) acts on the Drinfeld double DH_X of H_X by algebra automorphisms. Next, we study a certain natural subalgebra U_X of DH_X for which we give a presentation by generators and relations. This algebra turns out to be a flat two-parameter deformation of the ring of diagonal invariants C[x_1^{\pm 1}, ..., y_1^{\pm 1},...]^{S_{\infty}}, i.e. the ring of symmetric Laurent polynomials in two sets of countably many variables under the simultaneous symmetric group action.

Nonsymmetric Askey–Wilson polynomials as vector-valued polynomials

Nonsymmetric Askey–Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl–Cherednik operator of which they are eigenfunctions, is represented as a 2 × 2 matrix-valued operator. As a new result made possible by this approach we obtain positive definiteness of the inner product in the orthogonality relations, under certain constraints on the parameters. A limit transition to nonsymmetric little q-Jacobi polynomials also becomes possible in this way. Nonsymmetric Jacobi polynomials are considered as limits both of the Askey–Wilson and of the little q-Jacobi case.

Three-term relations between interpolation polynomials for a [formula omitted]-type basic hypergeometric series

We present an explicit expression for the q-difference system of rank n+1 satisfied by a BCn-type Jackson integral (q-series) as first order simultaneous q-difference equations with a concrete basis. For this purpose we introduce two types of symmetric Laurent polynomials called the interpolation polynomials. Three-term relations between the interpolation polynomials play an essential role in deriving the q-difference system. As an application, we give another proof for the product formula of the q-integral introduced by Gustafson.

Explicit construction of symmetric orthogonal wavelet frames in [formula omitted

Recently, some researchers propose the concept of orthogonal wavelet frames, which are useful for multiple access communication systems. In this article, we first give two explicit algorithms for constructing paraunitary symmetric matrices ( p.s.m. for short), whose entries are symmetric or antisymmetric Laurent polynomials. We also give two algorithms for constructing orthogonal wavelet frames from existing tight or dual wavelet frames in L 2 ( R s ) . The constructed orthogonal wavelet frames are also tight or dual ones. Furthermore, based on the constructed p.s.m. and the existing symmetric tight (dual) wavelet frames, we can obtain symmetric orthogonal ( s.o. for short) tight (dual) wavelet frames in L 2 ( R s ) . From the constructed s.o. wavelet frames in L 2 ( R s ) , we can obtain s.o. wavelet frames in L 2 ( R m ) by the projection method, where m ≤ s . To illustrate our results, we construct s.o. wavelet frames in L 2 ( R ) and L 2 ( R 2 ) from the quadratic B-spline B 3 ( x ) . Especially, in Example 2, we obtain nonseparable tight 2 I 2 -wavelet frames in L 2 ( R 2 ) from a separable tight 2 I 2 -wavelet frame constructed by tensor product.

Double Affine Hecke Algebras and Bispectral Quantum Knizhnik-Zamolodchikov Equations

We use the double affine Hecke algebra of type GL_N to construct an explicit consistent system of q-difference equations, which we call the bispectral quantum Knizhnik-Zamolodchikov (BqKZ) equations. BqKZ includes, besides Cherednik's quantum affine KZ equations associated to principal series representations of the underlying affine Hecke algebra, a compatible system of q-difference equations acting on the central character of the principal series representations. We construct a meromorphic self-dual solution \Phi of BqKZ which, upon suitable specializations of the central character, reduces to symmetric self-dual Laurent polynomial solutions of quantum KZ equations. We give an explicit correspondence between solutions of BqKZ and solutions of a particular bispectral problem for the Ruijsenaars' commuting trigonometric q-difference operators. Under this correspondence \Phi becomes a self-dual Harish-Chandra series solution \Phi^+ of the bispectral problem. Specializing the central character as above, we recover from \Phi^+ the symmetric self-dual Macdonald polynomials.

Some quotient algebras arising from the quantum toroidal algebra Uq(sl₂(C_γ))

Some quotient algebras arising from the quantum toroidal algebra $U_{q}(\mathit{sl}_{n+1}(\mathcal{C}_{\gamma}))$ ($n\ge 2$) are considered. They are related to integrable highest weight representations of the algebra and are shown to be isomorphic to tensor products of two algebras of symmetric Laurent polynomials and Macdonald's difference operators.

Zeros of symmetric Laurent polynomials of type (BC)n and Koornwinder–Macdonald polynomials specialized at $t^{k+1}q^{r-1}=1$

A characterization of the space of symmetric Laurent polynomials of type ( BC ) n , which vanish on a certain set of submanifolds, is given by using the Koornwinder–Macdonald polynomials. A similar characterization was given previously for symmetric polynomials of type A n by using the Macdonald polynomials. We use a new method which exploits the duality relation. The method simplifies a part of the proof in the A n case.

A functional model for the tensor product of level 1 highest and level −1 lowest modules for the quantum affine algebra [formula omitted

Let V( Λ i ) (resp., V(− Λ j )) be a fundamental integrable highest (resp., lowest) weight module of U q( sl 2) . The tensor product V( Λ i )⊗ V(− Λ j ) is filtered by submodules F n=U q( sl 2)(v i⊗ v n−i) , n≥0, n≡i−j mod 2 , where v i ∈ V( Λ i ) is the highest vector and v n−i∈V(−Λ j) is an extremal vector. We show that F n / F n+2 is isomorphic to the level 0 extremal weight module V( n( Λ 1− Λ 0)). Using this we give a functional realization of the completion of V( Λ i )⊗ V(− Λ j ) by the filtration ( F n ) n≥0 . The subspace of V( Λ i )⊗ V(− Λ j ) of sl 2 -weight m is mapped to a certain space of sequences (P n,l) n≥0,n≡i−j mod 2,n−2l=m , whose members P n, l = P n, l ( X 1,…, X l ∣ z 1,…, z n ) are symmetric polynomials in X a and symmetric Laurent polynomials in z k , with additional constraints. When the parameter q is specialized to −1 , this construction settles a conjecture which arose in the study of form factors in integrable field theory.

Form Factors and Action of on 8-Cycles

Let be a sequence of skew-symmetric polynomials in X 1,⋯,X l satisfying deg X j P n,l ≤n−1, whose coefficients are symmetric Laurent polynomials in z 1 ,⋯,z n . We call p an ∞-cycle if holds for all n,l. These objects arise in integral representations for form factors of massive integrable field theory, i.e., the SU(2)-invariant Thirring model and the sine-Gordon model. The variables α a =−logX a are the integration variables and β j =logz j are the rapidity variables. To each ∞-cycle there corresponds a form factor of the above models. Conjecturally all form-factors are obtained from the ∞-cycles. In this paper, we define an action of on the space of ∞-cycles. There are two sectors of ∞-cycles depending on whether n is even or odd. Using this action, we show that the character of the space of even (resp. odd) ∞-cycles which are polynomials in z 1 ,⋯,z n is equal to the level (−1) irreducible character of with lowest weight −Λ0 (resp. −Λ1). We also suggest a possible tensor product structure of the full space of ∞-cycles.

SYMMETRIC LIFTING FACTORIZATION AND MATRIX REPRESENTATION OF BIORTHOGONAL WAVELET TRANSFORMS

This paper introduces some properties of symmetric Laurent polynomials, and then extends Euclidean algorithm to symmetric Laurent polynomials. The new results are used to investigate factorization of polyphase matrix for biorthogonal finite filters. It is shown that there exists one and only one symmetric factorization of the polyphase matrix, and the symmetric factorization can be determined directly and efficiently by Euclidean algorithm for symmetric Laurent polynomials. Finally, symmetric implementation and matrix representation of biorthogonal wavelet transforms are introduced, and the study demonstrates that the symmetric implementation has the least multiplication number in all lifting implementations, and it is equivalent to a matrix transform on finite dimensional vector space.

Yangian Gelfand-Zetlin Bases, $\glN$ -Jack Polynomials and Computation of Dynamical Correlation Functions in the Spin Calogero-Sutherland Model

We consider the gl(N)-invariant Calogero-Sutherland Models with N=1,2,3,... in a unified framework, which is the framework of Symmetric Polynomials. By the framework we mean an isomorphism between the space of states of the gl(N)-invariant Calogero-Sutherland Model and the space of Symmetric Laurent Polynomials. In this framework it becomes apparent that all gl(N)-invariant Calogero-Sutherland Models are manifestations of the same entity, which is the commuting family of Macdonald Operators. Macdonald Operators depend on two parameters $q$ and $t$. The Hamiltonian of gl(N)-invariant Calogero-Sutherland Model belongs to a degeneration of this family in the limit when both $q$ and $t$ approach the N-th elementary root of unity. This is a generalization of the well-known situation in the case of Scalar Calogero-Sutherland Model (N=1). In the limit the commuting family of Macdonald Operators is identified with the maximal commutative sub-algebra in the Yangian action on the space of states of the gl(N)-invariant Calogero-Sutherland Model. The limits of Macdonald Polynomials which we call gl(N)-Jack Polynomials are eigenvectors of this sub-algebra and form Yangian Gelfand-Zetlin bases in irreducible components of the Yangian action. The gl(N)-Jack Polynomials describe the orthogonal eigenbasis of gl(N)-invariant Calogero-Sutherland Model in exactly the same way as Jack Polynomials describe the orthogonal eigenbasis of the Scalar Model (N=1). For each known property of Macdonald Polynomials there is a corresponding property of gl(N)-Jack Polynomials. As a simplest application of these properties we compute two-point Dynamical Spin-Density and Density Correlation Functions in the gl(2)-invariant Calogero-Sutherland Model at integer values of the coupling constant.