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Two linear transformations each tridiagonal with respect to an eigenbasis of the other: comments on the split decomposition
Abstract
Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A*:V→V that satisfy both conditions below:(i)There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A* is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal.We call such a pair a Leonard pair on V. Referring to the above Leonard pair, it is known there exists a decomposition of V into a direct sum of one-dimensional subspaces, on which A acts in a lower bidiagonal fashion and A* acts in an upper bidiagonal fashion. This is called the split decomposition. In this paper, we give two characterizations of a Leonard pair that involve the split decomposition.
... A Leonard pair satisfies two relations called the tridiagonal relations [17]; see Lemma 2.4 below. Notable papers about Leonard pairs are [17][18][19][20][21]. There is a generalization of a Leonard pair called a tridiagonal pair. ...
... (ii): Similar to the proof of (i). (iii): Recall the equation(20). By (i), we have E 0 A * E j = 0 for 1 < j < d. ...
... By (i), we have E 0 A * E j = 0 for 1 < j < d. Applying this to the equation(20), we have the equation ...
A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let $M$ denote a Hessenberg matrix. Then $M$ is called circular whenever the upper-right corner entry of $M$ is nonzero and every other entry above the superdiagonal is zero. A circular Hessenberg pair consists of two diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a circular Hessenberg fashion. Let $A, A^*$ denote a circular Hessenberg pair. We investigate six bases for the underlying vector space that we find attractive. We display the transition matrices between certain pairs of bases among the six. We also display the matrices that represent $A$ and $A^*$ with respect to the six bases. We introduce a special type of circular Hessenberg pair, said to be recurrent. We show that a circular Hessenberg pair $A, A^*$ is recurrent if and only if $A, A^*$ satisfy the tridiagonal relations. For a circular Hessenberg pair, there is a related object called a circular Hessenberg system. We classify up to isomorphism the recurrent circular Hessenberg systems. To this end, we construct four families of recurrent circular Hessenberg systems. We show that every recurrent circular Hessenberg system is isomorphic to a member of one of the four families.
... The q-Onsager algebra (see [16,19,23,151,153]) is the tridiagonal algebra for the case β = q 2 + q −2 , γ = 0, γ * = 0, ̺ = 0, ̺ * = 0. The Askey-Wilson algebra (see [166]) is defined by two generators subject to the Askey-Wilson relations (139), (140). This algebra has a central extension (see [144]) that we now describe. ...
... Additional results concerning the split decomposition and the shape can be found in [74,97,98,106,108,112,113,140,142,157]. Some miscellaneous results about tridiagonal pairs and systems can be found in [1,4,28,85,87]. ...
... Observe that in (139) the right-hand side is a polynomial in A, and therefore commutes with A. Thus (139) implies (135). Similarly (140) implies (136). If A, A * is TB then the scalars γ, γ * , ω, η, η * are all zero, and (139), (140) become (42), (43). ...
There is a concept in linear algebra called a tridiagonal pair. The concept was motivated by the theory of $Q$-polynomial distance-regular graphs. We give a tutorial introduction to tridiagonal pairs, working with a special case as a concrete example. The special case is called totally bipartite, or TB. Starting from first principles, we give an elementary but comprehensive account of TB tridiagonal pairs. The following topics are discussed: (i) the notion of a TB tridiagonal system; (ii) the eigenvalue array; (iii) the standard basis and matrix representations; (iv) the intersection numbers; (v) the Askey-Wilson relations; (vi) a recurrence involving the eigenvalue array; (vii) the classification of TB tridiagonal systems; (viii) self-dual TB tridiagonal pairs and systems; (ix) the $\mathbb{Z}_3$-symmetric Askey-Wilson relations; (x) some automorphisms and antiautomorphisms associated with a TB tridiagonal pair; (xi) an action of the modular group ${\rm PSL}_2(\mathbb{Z})$ associated with a TB tridiagonal pair.
... Hence, all polynomial solutions for (1.3) should be orthogonal AWP. Although for the finite-dimensional case the problem was effectively solved by Terwilliger in [18], we present here the finite-dimensional version of the generalized Bochner theorem as well. The main reason is that our method of proof is essentially different and deals directly with difference equation (1.3) for polynomials, whereas in the Terwilliger paper [18] another (a purely algebraic) approach is presented. ...
... Although for the finite-dimensional case the problem was effectively solved by Terwilliger in [18], we present here the finite-dimensional version of the generalized Bochner theorem as well. The main reason is that our method of proof is essentially different and deals directly with difference equation (1.3) for polynomials, whereas in the Terwilliger paper [18] another (a purely algebraic) approach is presented. ...
... We just related the representation theories of ⊠ q and A q . To illuminate this relationship we bring in the concept of a Leonard pair [25][26][27][28][29][30]33] and tridiagonal pair [5,6,9,15]. Roughly speaking, a Leonard pair consists of two diagonalizable linear transformations of a finitedimensional vector space, each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other one [25, Definition 1.1]. ...
... Comparing (29), (30) we obtain ...
Let $\mathbb F$ denote an algebraically closed field, and fix a nonzero $q
\in \mathbb F$ that is not a root of unity. We consider the $q$-tetrahedron
algebra $\boxtimes_q$ over $\mathbb F$. It is known that each
finite-dimensional irreducible $\boxtimes_q$-module of type 1 is a tensor
product of evaluation modules. This paper contains a comprehensive description
of the evaluation modules for $\boxtimes_q$. This description includes the
following topics. Given an evaluation module $V$ for $\boxtimes_q$, we display
24 bases for $V$ that we find attractive. For each basis we give the matrices
that represent the $\boxtimes_q$-generators. We give the transition matrices
between certain pairs of bases among the 24. It is known that the cyclic group
$\Z_4$ acts on $\boxtimes_q$ as a group of automorphisms. We describe what
happens when $V$ is twisted via an element of $\Z_4$. We discuss how evaluation
modules for $\boxtimes_q$ are related to Leonard pairs of $q$-Racah type.
... When N is finite, the couple of operators H and X form by definition a Leonard pair [19]. One can deduce that the eigenvalues {ω k } of H are pairwise distinct and similarly for the eigenvalues {λ n } of X (see Lemma 1.3. in [20]). Leonard pairs have been classified [19] and shown to be in correspondence with the orthogonal polynomial families of the truncating part of the Askey tableau. ...
Entanglement in finite and semi-infinite free Fermionic chains is studied. A parallel is drawn with the analysis of time and band limiting in signal processing. It is shown that a tridiagonal matrix commuting with the entanglement Hamiltonian can be found using the algebraic Heun operator construct in instances when there is an underlying bispectral problem. Cases corresponding to the Lie algebras $\mathfrak{su}(2)$ and $\mathfrak{su}(1,1)$ as well as to the q-deformed algebra $\mathfrak{so}_q(3)$ at $q$ a root of unity are presented.
... As explained in [37, Appendix A], the Leonard pairs provide a linear algebra interpretation of a theorem of Doug Leonard [31], [3, p. 260] concerning the q-Racah polynomials and their relatives in the Askey scheme. The Leonard pairs are classified up to isomorphism [37, Theorem 1.9] and described further in [33,34,[38][39][40][41]43]. For a survey see [42]. ...
Let $\mathbb F$ denote a field, and let $V$ denote a vector space over $\mathbb F$ with finite positive dimension. Pick a nonzero $q \in \mathbb F$ such that $q^4 \not=1$, and let $A,B,C$ denote a Leonard triple on $V$ that has $q$-Racah type. We show that there exist invertible $W, W', W'' $ in ${\rm End}(V)$ such that (i) $A$ commutes with $W$ and $W^{-1}BW-C$; (ii) $B$ commutes with $W'$ and $(W')^{-1}CW'-A$; (iii) $C$ commutes with $W''$ and $(W'')^{-1}AW''-B$. Moreover each of $W,W', W''$ is unique up to multiplication by a nonzero scalar in $\mathbb F$. We show that the three elements $W'W, W''W', WW''$ mutually commute, and their product is a scalar multiple of the identity. A number of related results are obtained.
... This notion was introduced by Terwilliger [19][20][21] as an abstraction of some work of Leonard [16] concerning the orthogonal polynomials in the terminating branch of the Askey scheme [13]. Leonard pairs are closely related to quantum groups [1,24,25,[27][28][29][30]32]. Leonard pairs are also related to Q-polynomial distance-regular graphs [15,31,36]. ...
In this paper, we study the incidence algebra $T$ of the attenuated space poset $\mathcal{A}_q(N, M)$. We consider the following topics. We consider some generators of $T$: the raising matrix $R$, the lowering matrix $L$, and a certain diagonal matrix $K$. We describe some relations among $R, L, K$. We put these relations in an attractive form using a certain matrix $S$ in $T$. We characterize the center $Z(T)$. Using $Z(T)$, we relate $T$ to the quantum group $U_\tau({\mathfrak{sl}}_2)$ with $\tau^2=q$. We consider two elements $A, A^*$ in $T$ of a certain form. We find necessary and sufficient conditions for $A, A^*$ to satisfy the tridiagonal relations. Let $W$ denote an irreducible $T$-module. We find necessary and sufficient conditions for the above $A, A^*$ to act on $W$ as a Leonard pair.
... We refer the reader to [3], [10], [13], [14], [15], [16], [17], [18], [20], [21], [22], [23], [24], [25], [26], [27], [29], [30] for background on Leonard pairs. We especially recommend the survey [27]. ...
Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A: V → V and A*: V → V that satisfy (i), (ii) below:. (i)There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A* is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. In this paper we investigate the commutator AA* - A*A. Our results are as follows. Abbreviate d=dim V - 1 and first assume d is odd. We show AA* - A*A is invertible and display several attractive formulae for the determinant. Next assume d is even. We show that the null space of AA* - A*A has dimension 1. We display a nonzero vector in this null space. We express this vector as a sum of eigenvectors for A and as a sum of eigenvectors for A*.
... We remark that this classification amounts to a linear algebraic version of a theorem of D. Leonard [2], [10] concerning the q-Racah polynomials. See [8], [12], [13], [14], [15], [16], [17], [19], [20] for more information about Leonard pairs. ...
Let \({\mathbb K}\) denote an algebraically closed field and let q denote a nonzero scalar in \({\mathbb K}\) that is not a root of unity. Let V denote a vector space over \({\mathbb K}\) with finite positive dimension and let A,A* denote a tridiagonal pair on V. Let θ0, θ1,…, θd
(resp. θ*0, θ*1,…, θ*d
) denote a standard ordering of the eigenvalues of A (resp. A*). We assume there exist nonzero scalars a, a* in \({\mathbb K}\) such that θi
= aq
2i−d
and θ*i
= a*q
d−2i
for 0 ≤ i ≤ d. We display two irreducible \({\boldmath U_q({\widehat {sl}}_2)}\)-module structures on V and discuss how these are related to the actions of A and A*.
... By (19), (24) and (25) we find A * E r V ⊆ E r−1 V +E r+1 V . By this and since W has endpoint r we find A * E r W ⊆ E r+1 W . Therefore A * u ∈ E r+1 W . Now (A−(d−2)I)A * u = 0 since θ r+1 = d−2. ...
Let V denote a vector space over ℂ with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear operators on V such that for each of these operators there exists a basis of V with respect to which the matrix representing that operator is diagonal and the matrices representing the other two operators
are irreducible tridiagonal.
Let D denote a positive integer and let Q
D
denote the graph of the D-dimensional hypercube. Let X denote the vertex set of Q
D
and let
A Î MatX(\mathbbC)A\in {\rm Mat}_{X}({\mathbb{C}})
denote the adjacency matrix of Q
D
. Fix x∈X and let
A* Î MatX(\mathbbC)A^{*}\in {\rm Mat}_{X}({\mathbb{C}})
denote the corresponding dual adjacency matrix. Let T denote the subalgebra of
MatX(\mathbbC){\rm Mat}_{X}({\mathbb{C}})
generated by A,A
*. We refer to T as the Terwilliger algebra of
Q
D
with respect to
x. The matrices A and A
* are related by the fact that 2i
A=A
*
A
ε
−A
ε
A
* and 2i
A
*=A
ε
A−AA
ε
, where 2i
A
ε
=AA
*−A
*
A and i
2=−1.
We show that the triple A, A
*, A
ε
acts on each irreducible T-module as a Leonard triple. We give a detailed description of these Leonard triples.
... To get (54), multiply each side of (61) on the left by X r E * d E d and on the right by ξ * 0 ; now sum the resulting equation over r = 0, . . . , d and simplify using E * 0 ξ * 0 = ξ * 0 , the equations on the right in (44), (47), equations (35), (51), the equation on the right in (40), and (21). To get (55), in line (53) replace i by d − i and X 0 , . . . ...
Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A:V→V and A∗:V→V that satisfy (i) and (ii) below:1.[(i)] There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal.2.[(ii)] There exists a basis for V with respect to which the matrix representing A∗ is irreducible tridiagonal and the matrix representing A is diagonal.We call such a pair a Leonard pair on V. In an earlier paper we described 24 special bases for V. One feature of these bases is that with respect to each of them the matrices that represent A and A∗ are (i) diagonal and irreducible tridiagonal or (ii) irreducible tridiagonal and diagonal or (iii) lower bidiagonal and upper bidiagonal or (iv) upper bidiagonal and lower bidiagonal. For each ordered pair of bases among the 24, there exists a unique linear transformation from V to V that sends the first basis to the second basis; we call this the transition map. In this paper we find each transition map explicitly as a polynomial in A,A∗.
... We refer the reader to [1-3, 28, 37-40, 45-49, 51, 63-65, 73-78, 96, 98] for background information about tridiagonal pairs. See [29][30][31][32][33][34][35][36][41][42][43][44]50,[52][53][54][55][56][57][58][59][60][61][62][66][67][68][69][70][71][72][79][80][81][82][83][84][85][86][87][88][89][90][91][92][93][94][95][96][97][99][100][101][102][103] for related topics. ...
Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A:V→V and A∗:V→V that satisfy the following conditions: (i) each of A,A∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A∗Vi⊆Vi-1+Vi+Vi+1 for 0⩽i⩽d, where V-1=0 and Vd+1=0; (iii) there exists an ordering of the eigenspaces of A∗ such that for 0⩽i⩽δ, where and ; (iv) there is no subspace W of V such that AW⊆W, A∗W⊆W, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0⩽i⩽d the dimensions of Vi, , Vd-i, coincide; we denote this common dimension by ρi. In this paper we prove that for 0⩽i⩽d. It is already known that ρ0=1 if K is algebraically closed.
... We refer the reader to [3], [9], [12], [13], [14], [15], [16], [17], [18], [20], [21], [22], [23], [24], [25], [26], [27], [29], [30] for background on Leonard pairs. We especially recommend the survey [27]. ...
Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A∗ : V → V that satisfy (i) and (ii) below:(i)There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A∗ is irreducible tridiagonal and the matrix representing A is diagonal.We call such a pair a Leonard pair on V. It is known that there exists a basis for V with respect to which the matrix representing A is lower bidiagonal and the matrix representing A∗ is upper bidiagonal. In this paper we give some formulae involving the matrix units associated with this basis.
... Remark 9. 7 We discovered the equitable presentation for U q (sl 2 ) during our recent study of tridiagonal pairs [17], [18], [19], and the closely related Leonard pairs [32], [33], [34], [35], [36], [37], [38], [39], [41]. A Leonard pair is a pair of semi-simple linear transformations on a finite-dimensional vector space, each of which acts tridiagonally on an eigenbasis for the other [32, Definition 1.1]. ...
We show that the quantum algebra Uq(sl2) has a presentation with generators x±1,y,z and relations xx−1=x−1x=1, We call this the equitable presentation. We show that y (respectively z) is not invertible in Uq(sl2) by displaying an infinite-dimensional Uq(sl2)-module that contains a nonzero null vector for y (respectively z). We consider finite-dimensional Uq(sl2)-modules under the assumption that q is not a root of 1 and char(K)≠2, where K is the underlying field. We show that y and z are invertible on each finite-dimensional Uq(sl2)-module. We display a linear operator Ω that acts on finite-dimensional Uq(sl2)-modules, and satisfies on these modules. We define Ω using the q-exponential function.
A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let M denote a Hessenberg matrix. Then M is called circular whenever the upper-right corner entry of M is nonzero and every other entry above the superdiagonal is zero. A circular Hessenberg pair consists of two diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a circular Hessenberg fashion. Let A,A⁎ denote a circular Hessenberg pair. We investigate six bases for the underlying vector space that we find attractive. We display the transition matrices between certain pairs of bases among the six. We also display the matrices that represent A and A⁎ with respect to the six bases. We introduce a special type of circular Hessenberg pair, said to be recurrent. We show that a circular Hessenberg pair A,A⁎ is recurrent if and only if A,A⁎ satisfy the tridiagonal relations. For a circular Hessenberg pair, there is a related object called a circular Hessenberg system. We classify up to isomorphism the recurrent circular Hessenberg systems. To this end, we construct four families of recurrent circular Hessenberg systems. We show that every recurrent circular Hessenberg system is isomorphic to a member of one of the four families.
Entanglement in finite and semi-infinite free Fermionic chains is studied. A parallel is drawn with the analysis of time and band limiting in signal processing. It is shown that a tridiagonal matrix commuting with the entanglement Hamiltonian can be found using the algebraic Heun operator construct in instances when there is an underlying bispectral problem. Cases corresponding to the Lie algebras [Formula: see text] and [Formula: see text] as well as to the q-deformed algebra [Formula: see text] at [Formula: see text] a root of unity are presented.
Around 2001 we classified the Leonard systems up to isomorphism. The proof was lengthy and involved considerable computation. In this paper we give a proof that is shorter and involves minimal computation. We also give a comprehensive description of the intersection numbers of a Leonard system.
Around 2001 we classified the Leonard systems up to isomorphism. The proof was lengthy and involved considerable computation. In this paper we give a proof that is shorter and involves minimal computation. We also give a comprehensive descriptionof the intersection numbers of a Leonard system.
Let denote a field, and let V denote a vector space over with finite positive dimension. Pick a nonzero such that , and let denote a Leonard triple on V that has q-Racah type. We show that there exist invertible in such that (i) A commutes with W and ; (ii) B commutes with and ; (iii) C commutes with and . Moreover each of is unique up to multiplication by a nonzero scalar in . We show that the three elements mutually commute, and their product is a scalar multiple of the identity. A number of related results are obtained. We call the pseudo intertwiners for .
In this paper, we study the incidence algebra T of the attenuated space poset . We consider the following topics. We consider some generators of T: the raising matrix R, the lowering matrix L, and a certain diagonal matrix K. We describe some relations among . We put these relations in an attractive form using a certain matrix S in T. We characterize the center . Using , we relate T to the quantum group with . We consider two elements in T of a certain form. We find necessary and sufficient conditions for to satisfy the tridiagonal relations. Let W denote an irreducible T-module. We find necessary and sufficient conditions for the above to act on W as a Leonard pair.
Let F denote a field, and fix a nonzero q ∈ F such that q 4 ̸ = 1. The universal Askey–Wilson algebra is the associative F-algebra ∆ = ∆q defined by generators and relations in the following way. The generators are A, B, C. The relations assert that each of A + qBC − q−1CB q2 − q−2, B + qCA − q−1AC q2 − q−2, C + qAB − q−1BA q2 − q−2 is central in ∆. In this paper we discuss a connection between ∆ and the F-algebra U = Uq(sl2). To summarize the connection, let a, b, c denote mutually commuting indeterminates and let F[a ±1, b ±1, c ±1] denote the F-algebra of Laurent polynomials in a, b, c that have all coefficients in F. We display an injection of F-algebras ∆ → U ⊗F F[a ±1, b ±1, c ±1]. For this injection we give the image of A, B, C and the above three central elements, in terms of the equitable generators for U. The algebra ∆ has another central element of interest, called the Casimir element Ω. One significance of Ω is the following. It is known that the center of ∆ is generated by Ω and the above three central elements, provided that q is not a root of unity. For the above injection we give the image of Ω in terms of the equitable generators for U. We also use the injection to show that ∆ contains no zero divisors.
We introduce a linear algebraic object called a bidiagonal pair. Roughly speaking, a bidiagonal pair is a pair of diagonalizable linear transformations on a finite-dimensional vector space, each of which acts in a bidiagonal fashion on the eigenspaces of the other. We associate to each bidiagonal pair a sequence of scalars called a parameter array. Using this concept of a parameter array we present a classification of bidiagonal pairs up to isomorphism. The statement of this classification does not explicitly mention the Lie algebra 𝔰𝔩2 or the quantum group Uq(𝔰𝔩2). However, its proof makes use of the finite-dimensional representation theory of 𝔰𝔩2 and Uq(𝔰𝔩2). In addition to the classification we make explicit the relationship between bidiagonal pairs and modules for 𝔰𝔩2 and Uq(𝔰𝔩2).
Recently Brian Hartwig and the second author found a presentation for the three-point 𝔰𝔩2 loop algebra by generators and relations. To obtain this presentation they defined a Lie algebra ⊠ by generators and relations, and displayed an isomorphism from ⊠ to the three-point 𝔰𝔩2 loop algebra. In this article, we describe the finite-dimensional irreducible ⊠-modules from multiple points of view.
We study Poisson and operator algebras with the ''quasi-linear property'' from the Heisenberg picture point of view. This means that there exists a set of one-parameter groups yielding an explicit expression of dynamical variables (operators) as functions of ''time'' $t$. We show that many algebras with nonlinear commutation relations such as the Askey-Wilson, $q$-Dolan-Grady and others satisfy this property. This provides one more (explicit Heisenberg evolution) interpretation of the corresponding integrable systems.
We construct Leonard pairs from finite-dimensional irreducible sl2-modules, using the equitable basis for sl2. We show that our construction yields all Leonard pairs of Racah, Hahn, dual Hahn, and Krawtchouk type, and no other types of Leonard pairs. 1. Introduction. In this paper, we construct Leonard pairs from each finite- dimensional irreducible sl2-module. We show that this construction yields all Leonard pairs of Racah, Hahn, dual Hahn, and Krawtchouk type, and no other types of Leonard pairs. Leonard pairs were introduced by P. Terwilliger (9) to abstract Bannai and Ito's (1) algebraic approach to a result of D. Leonard concerning the sequences of orthogonal polynomials with finite support for which the dual sequence of polynomials is also a sequence of orthogonal polynomials (7, 8). These polynomials arise in connection with the finite-dimensional representations of certain Lie algebras and quantum groups, so one expects Leonard pairs to arise as well. Leonard pairs of Krawtchouk type have been constructed from finite-dimensional irreducible sl2-modules (12). In this paper, we give a more general construction based upon the equitable basis for sl2 (2, 5). The equitable basis of sl2 arose in the study of the Tetrahedron algebra and the 3-point loop algebra of sl2 (3)-(5). These references consider the modules of these algebras and their connections with a generalization of Leonard pairs called tridiagonal. Here, we consider only Leonard pairs and sl2, which has not been considered elsewhere. 2. Leonard pairs. We recall some facts concerning Leonard pairs; see (10)-(14) for more details. Fix an integer d ≥ 1. Throughout this paper F shall denote a field whose characteristic is either zero or an odd prime greater than d. Also, V shall denote an F-vector space of dimension d+1, and End(V ) shall denote the F-algebra
Let G be a connected strongly regular graph of order n and degree r. Let λ
1=r, λ
2 and λ
3 denote the distinct eigenvalues of G. We demonstrate that the Krein parameters of the strongly regular graph G can be expressed in terms of its eigenvalues r, λ
2 and λ
3. Due to this fact, we prove that there exists no strongly regular graph of order n=4k+3, provided that n is a prime number. Besides, we prove that the only strongly regular graphs of order n=4k+1 (n is a prime number) are conference graphs. In this work we also establish a characterization of strongly regular graphs of
order n=2(2k+1), where 2k+1 is a prime number.
A one-parameter family of operators that have the complementary Bannai-Ito
(CBI) polynomials as eigenfunctions is obtained. The CBI polynomials are the
kernel partners of the Bannai-Ito polynomials and also correspond to a
$q\rightarrow-1$ limit of the Askey-Wilson polynomials. The eigenvalue
equations for the CBI polynomials are found to involve second order Dunkl shift
operators with reflections and exhibit quadratic spectra. The algebra
associated to the CBI polynomials is given and seen to be a deformation of the
Askey-Wilson algebra with an involution. The relation between the CBI
polynomials and the recently discovered dual -1 Hahn and para-Krawtchouk
polynomials, as well as their relation with the symmetric Hahn polynomials, is
also discussed.
We introduce and discuss an Erd\H{o}s-Ko-Rado basis for the underlying vector
space of a Leonard system $\Phi = (A; A^*; \{E_i\}_{i=0}^d ; \{E_i^*
\}_{i=0}^d)$ that satisfies a mild condition on the eigenvalues of $A$ and
$A^*$. We describe the transition matrices to/from other known bases, as well
as the matrices representing $A$ and $A^*$ with respect to the new basis. We
also discuss how these results can be viewed as a generalization of the linear
programming method used previously in the proofs of the "Erd\H{o}s-Ko-Rado
theorems" for several classical families of $Q$-polynomial distance-regular
graphs, including the original 1961 theorem of Erd\H{o}s, Ko, and Rado.
We describe Poisson algebras associated with classical analogs of some self-dual generalized eigenvalue problems. These algebras are related in a natural way to various elliptic curves.
Let K denote a field, and let V denote a vector space over K with finite. positive dimension. We consider a pair of linear transformations A : V → V and A?: V → V that satisfy the following two conditions: 1. There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A? is diagonal. 2. There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V . We give a correspondence between Leonard pairs and a class of orthogonal polynomials. This class coincides with the terminating branch of the Askey scheme and consists of the q-Racah, q-Hahn, dual q-Hahn q-Krawtchouk, dual q-Krawtchouk, quantum q-Krawtchouk, affine q-Krawtchouk Racah, Hahn, dual Hahn, Krawtchouk, Bannai/Ito, and orphan polynomials. We describe the above correspondence in detail. We show how, for the listed polynomials the 3-term recurrence, difference equation, Askey-Wilson duality, and orthogonality can be expressed in a uniform and attractive manner using the corresponding Leonard pair. We give some examples that indicate how Leonard pairs arise in representation theory and algebraic combinatorics. We discuss a mild generalization of a Leonard pair called a tridiagonal pair. At the end we list some open problems. Throughout these notes our argument is elementary and uses only linear algebra. No prior exposure to the topic is assumed.
Let \({\mathbb K}\) denote a field, and let V denote a vector space over \({\mathbb K}\) of finite positive dimension. A pair A, A* of linear operators on V is said to be a Leonard pair on V whenever for each B∈{A, A*}, there exists a basis of V with respect to which the matrix representing B is diagonal and the matrix representing the other member of the pair is irreducible tridiagonal. A Leonard pair A, A* on V is said to be a spin Leonard pair whenever there exist invertible linear operators U, U* on V such that UA = A U, U*A* = A*U*, and UA* U
−1 = U*−1 AU*. In this case, we refer to U, U* as a Boltzmann pair for A, A*. We characterize the spin Leonard pairs. This characterization involves explicit formulas for the entries of the matrices that represent A and A* with respect to a particular basis. The formulas are expressed in terms of four algebraically independent parameters. We describe all Boltzmann pairs for a spin Leonard pair in terms of these parameters. We then describe all spin Leonard pairs associated with a given Boltzmann pair. We also describe the relationship between spin Leonard pairs and modular Leonard triples. We note a modular group action on each isomorphism class of spin Leonard pairs.
In this paper we consider how the following three objects are related: (i)
the dual polar graphs; (ii) the quantum algebra U_q(sl_2); (iii) the Leonard
systems of dual q-Krawtchouk type. For convenience we first describe how (ii)
and (iii) are related. For a given Leonard system of dual q-Krawtchouk type, we
obtain two U_q(sl_2)-module structures on its underlying vector space. We now
describe how (i) and (iii) are related. Let \Gamma denote a dual polar graph.
Fix a vertex x of \Gamma and let T = T(x) denote the corresponding
subconstituent algebra. By definition T is generated by the adjacency matrix A
of \Gamma and a certain diagonal matrix A* = A*(x) called the dual adjacency
matrix that corresponds to x. By construction the algebra T is semisimple. We
show that for each irreducible T-module W the restrictions of A and A* to W
induce a Leonard system of dual q-Krawtchouk type. We now describe how (i) and
(ii) are related. We obtain two U_q(sl_2)-module structures on the standard
module of \Gamma. We describe how these two U_q(sl_2)-module structures are
related. Each of these U_q(sl_2)-module structures induces a
$\mathbb{C}$-algebra homomorphism U_q(sl_2) \rightarrow T. We show that in each
case T is generated by the image together with the center of T. Using the
combinatorics of \Gamma we obtain a generating set L, F, R, K of T along with
some attractive relations satisfied by these generators.
Let V denote a nonzero finite dimensional vector space over a field K, and let (A, A∗) denote a tridiagonal pair on V of diameter d. Let V = U0 + ⋯ + Ud denote the split decomposition, and let ρi denote the dimension of Ui. In this paper, at first we show there exists a unique integer h (0 ⩽ h ⩽ d/2) such that ρi−1 < ρi for 1 ⩽ i ⩽ h, ρi−1 = ρi for h < i ⩽ d − h and ρi−1 > ρi for d − h < i ⩽ d. We call h the height of the tridiagonal pair. For 0 ⩽ r ⩽ h, we define subspaces (r ⩽ i ⩽ d − r) by , where R denotes the rasing map. We show V is decomposed as a direct sum . This gives a refinement of the split decomposition. Define , and observe . We show LU(r)⊆U(r-1)+U(r)+U(r+1) for 0 ⩽ r ⩽ h, where we set U(−1) = U(h+1) = 0. Let F(r):V→U(r) denote the projection. We show the lowering map L is decomposed as L = L(−) + L(0) + L(+), where , , and . These maps satisfy L(-)U(r)⊂U(r-1),L(0)U(r)⊆U(r), and L(+)U(r)⊆U(r+1) for 0 ⩽ r ⩽ h. The main results of this paper are the following: (i) For 0 ⩽ r ⩽ h − 1 and r + 2 ⩽ i ⩽ d − r − 1, RL(+) = αL(+)R holds on for some scalar α; (ii) For 1 ⩽ r ⩽ h and r ⩽ i ⩽ d − r − 1, RL(−) = βL(−)R holds on for some scalar β; (iii) For 0 ⩽ r ⩽ h and r + 1 ⩽ i ⩽ d − r − 1, RL(0) = βL(0)R + γI holds on for some scalars γ, δ. Moreover we give explicit expressions of α, β, γ, δ.
Let (A, A*) denote a tridiagonal pair on a vector space V over a field K. Let V0, … , Vd denote a standard ordering of the eigenspaces of A on V, and let θ0, … , θd denote the corresponding eigenvalues of A. We assume d ⩾ 3. Let q denote a scalar taken from the algebraic closure of K such that q2 + q−2 + 1 = (θ3 − θ0)/(θ2 − θ1). We assume q is not a root of unity. Let ρi denote the dimension of Vi. The sequence ρ0, ρ1, … , ρd is called the shape of the tridiagonal pair. It is known there exists a unique integer h (0 ⩽ h ⩽d/2) such that ρi−1 < ρi for 1 ⩽ i ⩽ h, ρi−1 = ρi for h < i ⩽ d − h, and ρi−1 > ρi for d − h < i ⩽ d. The integer h is known as the height of the tridiagonal pair. In this paper we show that the shape of a tridiagonal pair of height one with ρ0 = 1 is either 1, 2, 2, … , 2, 1 or 1, 3, 3, 1. In each case, we display a basis for V and give the action of A, A* on this basis.
In this survey paper we give an elementary introduction to the theory of Leonard pairs. A Leonard pair is defined as follows. Let denote a field and let V denote a vector space over with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A:V→V and B:V→V that satisfy conditions (i), (ii) below. (i)There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing B is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing B is irreducible tridiagonal. We give several examples of Leonard pairs. Using these we illustrate how Leonard pairs arise in representation theory, combinatorics, and the theory of orthogonal polynomials.
Let denote a field, and let V denote a vector space over with finite positive dimension. We consider a pair of linear transformations A:V→V and A*:V→V satisfying both conditions below:1.[(i)] There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A* is irreducible tridiagonal.2.[(ii)] There exists a basis for V with respect to which the matrix representing A* is diagonal and the matrix representing A is irreducible tridiagonal.We call such a pair a Leonard pair on V. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. Integral to our proof is the following result. We show that for any Leonard pair A,A* on V, there exists a sequence of scalars β,γ,γ*,ϱ,ϱ* taken from such that bothwhere [r,s] means rs−sr. The sequence is uniquely determined by the Leonard pair if the dimension of V is at least 4. We conclude by showing how Leonard systems correspond to q-Racah and related polynomials from the Askey scheme.
As part of our study of the q-tetrahedron algebra boxed times(q) we introduce the notion of a q-inverting pair. Roughly speaking, this is a pair of invertible semisimple linear transformations on a finite-dimensional vector space, each of which acts on the eigenspaces of the other according to a certain rule. Our main result is a bijection between the following two sets: (i) the isomorphism classes of finite-dimensional irreducible boxed times q-modules of type 1; (ii) the isomorphism classes of q-inverting pairs. (c) 2007 Elsevier Inc. All rights reserved.
Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the ``commutative case,'' has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.
The notion of a tridiagonal pair was introduced by Ito, Tanabe and Terwilliger. Let V denote a nonzero finite dimensional vector space over a field F. A tridiagonal pair on V is a pair (A, A*), where A : V → V and A* : V → V are linear transformations that satisfy some conditions. Assume (A, A*) is a tridiagonal pair on V. Recently Terwilliger and Vidunas showed that if A is multiplicity-free on V, then (A, A*) satisfy the following “Askey–Wilson relation” for some scalars β, γ, γ*, ϱ, ϱ*, ω, η, η*.In the present paper, we show that, if a tridiagonal pair (A, A*) satisfy the Askey–Wilson relations, then the eigenspaces of A and the eigenspaces of A* have one common dimension, and moreover if F is algebraically closed then that common dimension is 1.
Let $\Phi$, $\Phi'$ be Leonard systems over a field $\mathbb{K}$, and $V$, $V'$ the vector spaces underlying $\Phi$, $\Phi'$, respectively. In this paper, we introduce and discuss a \emph{balanced bilinear form} on $V\times V'$. Such a form naturally arises in the study of $Q$-polynomial distance-regular graphs. We characterize a balanced bilinear form from several points of view.
Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A:V→V and A∗:V→V that satisfy the following conditions: (i) each of is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A∗Vi⊆Vi-1+Vi+Vi+1 for 0⩽i⩽d, where V-1=0 and Vd+1=0; (iii) there exists an ordering of the eigenspaces of A∗ such that for 0⩽i⩽δ, where and ; (iv) there is no subspace W of V such that . We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0⩽i⩽d the dimensions of , coincide. We say the pair A,A∗ is sharp whenever dimV0=1. It is known that if K is algebraically closed then is sharp. A conjectured classification of the sharp tridiagonal pairs was recently introduced by Ito and the second author. Shortly afterwards we introduced a conjecture, called the μ-conjecture, which implies the classification conjecture. In this paper we show that the μ-conjecture holds in a special case called q-Racah.
Let F denote an algebraically closed field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A:V→V and A∗:V→V that satisfy the following conditions: (i) each of A,A∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A∗Vi⊆Vi−1+Vi+Vi+1 for 0⩽i⩽d, where V−1=0 and Vd+1=0; (iii) there exists an ordering of the eigenspaces of A∗ such that for 0⩽i⩽δ, where and ; (iv) there is no subspace W of V such that AW⊆W, A∗W⊆W, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ. For 0⩽i⩽d let θi (resp. ) denote the eigenvalue of A (resp. A∗) associated with Vi (resp. ). The pair A,A∗ is said to have q-Racah type whenever θi=a+bq2i−d+cqd−2i and for 0⩽i⩽d, where q,a,b,c,a∗,b∗,c∗ are scalars in F with q,b,c,b∗,c∗ nonzero and q2∉{1,−1}. This type is the most general one. We classify up to isomorphism the tridiagonal pairs over F that have q-Racah type. Our proof involves the representation theory of the quantum affine algebra .
A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let V denote a nonzero finite-dimensional vector space over a field K. We consider an ordered pair of linear transformations A:V→V and A∗:V→V which satisfy both (i) and (ii) below.(i)There exists a basis for V with respect to which the matrix representing A is Hessenberg and the matrix representing A∗ is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A∗ is Hessenberg.We call such a pair a thin Hessenberg pair (or TH pair). This is a special case of a Hessenberg pair which was introduced by the author in an earlier paper. We investigate several bases for V with respect to which the matrices representing A and A∗ are attractive. We display these matrices along with the transition matrices relating the bases. We introduce an “oriented” version of called a TH system. We classify the TH systems up to isomorphism.
Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations and which satisfy both (i), (ii) below. (i)There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing B is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing B is irreducible tridiagonal. We call such a pair a Leonard pair on V. We introduce two canonical forms for Leonard pairs. We call these the TD–D canonical form and the LB–UB canonical form. In the TD–D canonical form the Leonard pair is represented by an irreducible tridiagonal matrix and a diagonal matrix, subject to a certain normalization. In the LB–UB canonical form the Leonard pair is represented by a lower bidiagonal matrix and an upper bidiagonal matrix, subject to a certain normalization. We describe the two canonical forms in detail. As an application we obtain the following results. Given square matrices A,B over K, with A tridiagonal and B diagonal, we display a necessary and sufficient condition for A,B to represent a Leonard pair. Given square matrices A,B over K, with A lower bidiagonal and B upper bidiagonal, we display a necessary and sufficient condition for A,B to represent a Leonard pair. We briefly discuss how Leonard pairs correspond to the q-Racah polynomials and some related polynomials in the Askey scheme. We present some open problems concerning Leonard pairs.
Let K denote a field with characteristic 0 and let T denote an indeterminate. We give a presentation for the three-point loop algebra sl2⊗K[T,T−1,−1(T−1)] via generators and relations. This presentation displays S4-symmetry. Using this presentation we obtain a decomposition of the above loop algebra into a direct sum of three subalgebras, each of which is isomorphic to the Onsager algebra.
Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A:V→V and A∗:V→V that satisfies the following conditions: (i) each of A,A∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A∗Vi⊆Vi-1+Vi+Vi+1 for 0⩽i⩽d, where V-1=0 and Vd+1=0; (iii) there exists an ordering of the eigenspaces of A∗ such that for 0⩽i⩽δ, where and ; (iv) there is no subspace W of V such that AW⊆W, A∗W⊆W, W≠0,W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and that for 0⩽i⩽d the dimensions of coincide; we denote this common value by ρi. The sequence is called the shape of the pair. In this paper we assume the shape is (1,2,1) and obtain the following results. We describe six bases for V; one diagonalizes A, another diagonalizes A∗, and the other four underlie the split decompositions for A,A∗. We give the action of A and A∗ on each basis. For each ordered pair of bases among the six, we give the transition matrix. At the end we classify the tridiagonal pairs of shape (1,2,1) in terms of a sequence of scalars called the parameter array.
Let K denote a field and let V denote a vector space over K with finite positive dimension.We consider a pair of K-linear transformations A:V→V and A∗:V→V that satisfy the following conditions: (i) each of A,A∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A∗Vi⊆Vi-1+Vi+Vi+1 for 0⩽i⩽d, where V-1=0 and Vd+1=0; (iii) there exists an ordering of the eigenspaces of A∗ such that for 0⩽i⩽δ, where and ; (iv) there is no subspace W of V such that AW⊆W,A∗W⊆W,W≠0,W≠V.We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0⩽i⩽d the dimensions of coincide.In this paper we show that the following (i)–(iv) hold provided that K is algebraically closed: (i) Each of has dimension 1.(ii) There exists a nondegenerate symmetric bilinear form 〈,〉 on V such that 〈Au,v〉=〈u,Av〉 and 〈A∗u,v〉=〈u,A∗v〉 for all u,v∈V.(iii) There exists a unique anti-automorphism of End(V) that fixes each of A,A∗.(iv) The pair A,A∗ is determined up to isomorphism by the data , where θi (resp.) is the eigenvalue of A (resp.A∗) on Vi (resp.), and is the split sequence of A,A∗ corresponding to and .
Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A∗ : V → V that satisfy (i)–(iv) below:(i)Each of A, A∗ is diagonalizable.(ii)There exists an ordering V0, V1, …, Vd of the eigenspaces of A such that A∗Vi⊆Vi-1+Vi+Vi+1 for 0 ⩽ i ⩽ d, where V-1 = 0, Vd+1 = 0.(iii)There exists an ordering of the eigenspaces of A∗ such that for 0 ⩽ i ⩽ δ, where , .(iv)There is no subspace W of V such that both AW⊆W,A∗W⊆W, other than W = 0 and W = V.We call such a pair a tridiagonal pair on V. In this note we obtain two results. First, we show that each of A, A∗ is determined up to affine transformation by the Vi and . Secondly, we characterize the case in which the Vi and all have dimension one. We prove both results using a certain decomposition of V called the split decomposition.
Let K denote a field, and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A : V → V and A∗ : V → V that satisfy the following two conditions:(i)There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A∗ is irreducible tridiagonal and the matrix representing A is diagonal.Let (respectively v0, v1, … , vd) denote a basis for V that satisfies (i) (respectively (ii)). For 0 ⩽ i ⩽ d, let ai denote the coefficient of , when we write as a linear combination of , and let denote the coefficient of vi, when we write A∗vi as a linear combination of v0, v1, … , vd.In this paper we show a0 = ad if and only if . Moreover we show that for d ⩾ 1 the following are equivalent; (i) a0 = ad and a1 = ad−1; (ii) and ; (iii) ai = ad−i and for 0 ⩽ i ⩽ d. These give a proof of a conjecture by the second author. We say A, A∗ is balanced whenever ai = ad−i and for 0 ⩽ i ⩽ d. We say A,A∗ is essentially bipartite (respectively essentially dual bipartite) whenever ai (respectively ) is independent of i for 0 ⩽ i ⩽ d. Observe that if A, A∗ is essentially bipartite or dual bipartite, then A, A∗ is balanced. For d ≠ 2, we show that if A, A∗ is balanced then A, A∗ is essentially bipartite or dual bipartite.
We have shown that the Askey-Wilson polynomials of general form are generated by the algebra AW(3), which has a fairly simple structure and is the q-analog of a Lie algebra with three generators. The main properties of these polynomials (weight function, recursion relation, etc.) can be obtained directly from analysis of the representations of the algebra.
In this paper, we have considered finite-dimensional representations of the algebra AW(3) and the Aksey-Wilson polynomials of discrete argument corresponding to these representations. A separate analysis is required for the infinite-dimensional representations, which generate polynomials of a continuous argument (these polynomials were investigated in detail in the review [2]). Also of interest is investigation of representations of the algebra AW(3) for complex values of the basic parameter ω and of the structure parameters.
In our view, the algebra AW(3) by itself warrants careful study on account of several remarkable properties (in the first place, the duality with respect to the operators K0, K1) not present in the currently very popular quantum algebras of the type SUq(2).
We assume that the algebra AW(3) is an algebra of dynamical or “hidden” symmetry in all problems in which exponential or hyperbolic spectra and the corresponding q-polynomials arise. We hope that in time the algebra AW(3) will come to play the same role in “q-problems” as Lie algebras play in exactly solvable problems of quantum mechanics.
Askey and Wilson (1985) found a family of orthogonal polynomials in the variable that satisfy a q-difference equation of the form , n = 0, 1, …. We show here that this property characterizes the Askey-Wilson polynomials. The proof is based on an “operator identity” of independent interest. This identity can be adapted to prove other characterization results. Indeed it was used in (Grünbaum and Haine, 1996) to give a new derivation of the result of Bochner alluded to in the title of this paper. We give the appropriate identity for the case of difference equations (leading to the Wilson polynomials), but pursue the consequences only in the case of q-difference equations leading to the Askey-Wilson and big q-Jacobi polynomials. This approach also works in the discrete case and should yield the results in (Leonard, 1982).
The concept of mutually integrable dynamical variables is proposed. This concept leads to the quadratic Askey-Wilson algebra QAW(3) which is the dynamical symmetry algebra for all problems where the most general “classical” polynoials arise. In classical mechanics the algebra of the same structure describes the time evolution of dynamical variables in terms of elementary functions. We apply the special case of QAW(3)—Jacobi algebra—to describe the dynamical symmetry of exactly solvable potentials and to resolve the “Manning mystery”—the intimate relation between classical and quantum exactly solvable potentials.
We study coupling coefficients for a multiple tensor
product of highest weight representations of the SU(1,1) quantum group. These are multivariable generalizations of the q-Hahn polynomials.
This paper reconstructs and characterizes the Askey–Wilson orthogonal polynomials as those having duals (in the sense of Delsarte) which are also orthogonal. It introduces the concepts of eigenvalues and Delsarte’s duality to the study of orthogonal polynomials and provides those interested in P- and Q-polynomial association schemes with a closed form for their parameters.
We study polynomials of several variables which occur as coupling coefficients for the analytic continuation of the holomorphic discrete series of SU(1, 1). There are three types of such polynomials, one corresponding to each conjugacy class of one-parameter subgroups. They may be viewed as multivariable generalizations of Hahn, Jacobi, and continuous Hahn polynomials and include many orthogonal and biorthogonal families occurring in the literature. We give a simple and unified approach to these polynomials using the group theoretic interpretation. We prove many formal properties, in particular a number of convolution and linearization formulas. We also develop the corresponding theory for the Heisenberg group, leading to multivariable generalizations of Krawtchouk and Hermite polynomials.
Uno de los mejores libros a la hora de hacer un análisis minucioso en el campo de las Series Hipergeométricas Básicas, ó q-Series. Incluye avances recientes en el tema
On the SU(2) quantum group the notion of (zonal) spherical element is generalized by considering left and right invariance in the infinitesimal sense with respect to twisted primitive elements of the sl(2) quantized universal enveloping algebra. The resulting spherical elements belonging to irreducible representations of quantum SU(2) turn out to be expressible as a two-parameter family of Askey-Wilson polynomials. For a related basis change of the representation space a matrix of dual q-Krawtchouk polynomials is obtained. Big and little q-Jacobi polynomials are obtained as limits of Askey-Wilson polynomials. Key words & phrases: quantum groups, SU(2), spherical functions, infinitesimal invariance, Askey-Wilson polynomials, dual q-Krawtchouk polynomials, big q-Jacobi polynomials, little q-Jacobi polynomials.813. In the present revision (August 5, 2005) references [8] and [9] have been updated and the definition of group-like after (3.7) has been slightly improved.
We introduce a method for studying commutative association schemes with
\mathbbC\mathbb{C}
-algebra T = T(x) whose structure reflects the combinatorial structure of Y. We call T the subconstituent algebra of Y with respect to x. Roughly speaking, T is a combinatorial analog of the centralizer algebra of the stabilizer of x in the automorphism group of Y.In general, the structure of T is not determined by the intersection numbers of Y, but these parameters do give some information. Indeed, we find a relation among the generators of T for each vanishing intersection number or Krein parameter.We identify a class of irreducible T-moduIes whose structure is especially simple, and say the members of this class are thin. Expanding on this, we say Y is thin if every irreducible T(y)-module is thin for every vertex y of Y. We compute the possible thin, irreducible T-modules when Y is P- and Q-polynomial. The ones with sufficiently large dimension are indexed by four bounded integer parameters. If Y is assumed to be thin, then sufficiently large dimension means dimension at least four.We give a combinatorial characterization of the thin P- and Q-polynomial schemes, and supply a number of examples of these objects. For each example, we show which irreducible T-modules actually occur.We close with some conjectures and open problems.
Generalised matrix elements of the irreducible representations of the quantum SU(2) group are defined using certain orthonormal bases of the representation space. The generalised matrix elements are relatively infinitesimal invariant with respect to Lie algebra like elements of the quantised universal enveloping algebra of sl(2). A full proof of the theorem announced by Noumi and Mimachi [Proc. Japan Acad. Sci. Ser. A
66 (1990), 146–149] describing the generalised matrix elements in terms of the full four-parameter family of Askey-Wilson polynomials is given. Various known and new applications of this interpretation are presented.
This is a continuation of an article from the previous issue. In this section, we determine the structure of a thin, irreducible module for the subconstituent algebra of a P- and Q- polynomial association scheme. Such a module is naturally associated with a Leonard system. The isomorphism class of the module is determined by this Leonard system, which in turn is determined by four parameters: the endpoint, the dual endpoint, the diameter, and an additional parameter f. If the module has sufficiently large dimension, the parameter f takes one of a certain set of values indexed by a bounded integer parameter e.
In this survey paper we give an elementary introduction to the theory of Leonard pairs. A Leonard pair is defined as follows. Let denote a field and let V denote a vector space over with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A:V→V and B:V→V that satisfy conditions (i), (ii) below. (i)There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing B is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing B is irreducible tridiagonal. We give several examples of Leonard pairs. Using these we illustrate how Leonard pairs arise in representation theory, combinatorics, and the theory of orthogonal polynomials.
Let denote a field, and let V denote a vector space over with finite positive dimension. We consider a pair of linear transformations A:V→V and A*:V→V satisfying both conditions below:1.[(i)] There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A* is irreducible tridiagonal.2.[(ii)] There exists a basis for V with respect to which the matrix representing A* is diagonal and the matrix representing A is irreducible tridiagonal.We call such a pair a Leonard pair on V. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. Integral to our proof is the following result. We show that for any Leonard pair A,A* on V, there exists a sequence of scalars β,γ,γ*,ϱ,ϱ* taken from such that bothwhere [r,s] means rs−sr. The sequence is uniquely determined by the Leonard pair if the dimension of V is at least 4. We conclude by showing how Leonard systems correspond to q-Racah and related polynomials from the Askey scheme.
Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations and which satisfy both (i), (ii) below. (i)There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing B is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing B is irreducible tridiagonal. We call such a pair a Leonard pair on V. We introduce two canonical forms for Leonard pairs. We call these the TD–D canonical form and the LB–UB canonical form. In the TD–D canonical form the Leonard pair is represented by an irreducible tridiagonal matrix and a diagonal matrix, subject to a certain normalization. In the LB–UB canonical form the Leonard pair is represented by a lower bidiagonal matrix and an upper bidiagonal matrix, subject to a certain normalization. We describe the two canonical forms in detail. As an application we obtain the following results. Given square matrices A,B over K, with A tridiagonal and B diagonal, we display a necessary and sufficient condition for A,B to represent a Leonard pair. Given square matrices A,B over K, with A lower bidiagonal and B upper bidiagonal, we display a necessary and sufficient condition for A,B to represent a Leonard pair. We briefly discuss how Leonard pairs correspond to the q-Racah polynomials and some related polynomials in the Askey scheme. We present some open problems concerning Leonard pairs.
We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.
Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy both conditions below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is diagonal and the matrix representing $A$ is irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. Referring to the above Leonard pair, we investigate 24 bases for $V$ on which the action of $A$ and $A^*$ takes an attractive form. With respect to each of these bases, the matrices representing $A$ and $A^*$ are either diagonal, lower bidiagonal, upper bidiagonal, or tridiagonal.
Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. Consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy both conditions below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal, and the matrix representing $A^*$ is irreducible tridiagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is diagonal, and the matrix representing $A$ is irreducible tridiagonal. Such a pair is called a Leonard pair on $V$. In this paper we introduce a mild generalization of a Leonard pair called a tridiagonal pair. A Leonard pair is the same thing as a tridiagonal pair such that for each transformation all eigenspaces have dimension one.
We define an algebra on two generators which we call the Tridiagonal algebra, and we consider its irreducible modules. The algebra is defined as follows. Let K denote a field, and let $\beta, \gamma, \gamma^*, \varrho, \varrho^*$ denote a sequence of scalars taken from K. The corresponding Tridiagonal algebra $T$ is the associative K-algebra with 1 generated by two symbols $A$, $A^*$ subject to the relations (i) \lbrack A,A^2A^*-\beta AA^*A + A^*A^2 -\gamma (AA^*+A^*A)- \varrho A^*\rbrack = 0, (ii) \lbrack A^*,A^{*2}A-\beta A^*AA^* + AA^{*2} -\gamma^* (A^*A+AA^*)- \varrho^* A\rbrack = 0, where $\lbrack r,s\rbrack $ means $rs-sr$. We call these relations the Tridiagonal relations. For $\beta = q+q^{-1}$, $\gamma = \gamma^*=0$, $\varrho=\varrho^*=0$, the Tridiagonal relations are the $q$-Serre relations. For $\beta = 2$, $\gamma = \gamma^*=0$, $\varrho=b^2$, $\varrho^*=b^{*2}$, the Tridiagonal relations are the Dolan-Grady relations. In the first part of this paper, we survey what is known about irreducible finite dimensional $T$-modules. We focus on how these modules are related to the Leonard pairs recently introduced by the present author, and the more general Tridiagonal pairs recently introduced by Ito, Tanabe, and the present author. In the second part of the paper, we construct an infinite dimensional irreducible $T$-module based on the Askey-Wilson polynomials.
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