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Two Linear Transformations each Tridiagonal with Respect to an Eigenbasis of the other; Comments on the Parameter Array
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 $A:V\to V$ and $A^*:V\to V$ 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 $A^*$ 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 irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. The structure of any given Leonard pair is deterined by a certain sequence of scalars called its {\it parameter array}. The set of parameter arrays is an affine algebraic variety. We give two characterizations of this variety. One involves bidiagonal matrices and the other involves orthogonal polynomials.
... This version gives a complete classification of the orthogonal polynomial systems that satisfy Askey-Wilson duality. It shows that the orthogonal polynomial systems that satisfy Askey-Wilson duality are from the terminating branch of the Askey-tableau, except for one family with q ¼ À1 now called the Bannai-Ito polynomials [2, p. 271], [21,Example 5.14]. In our view, the terminating branch of the Askeytableau should include the Bannai-Ito polynomials. ...
... The classification [18,Theorem 1.9] shows that the ground field does not matter in a substantial way, unless it has characteristic 2. In this case, the theory admits an additional family of polynomials called the orphans. The orphans have diameter d ¼ 3 only; they are described in [21,Example 5.15] and Example 20.13 below. ...
... of scalars in F. Then there exists a Leonard system U over F with parameter array (21) if and only if the following conditions (PA1)-(PA5) hold: ...
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.
... In [18], the parameters arrays are classified into 13 families, each named for certain associated sequences of orthogonal polynomials. e four families, which arise in this paper, share certain property. ...
... Theorem 6 (see [18], Example 5.10). Fix nonzero h, h * ∈ F and s, s * , ...
... Theorem 7 (see [18], Example 5.11). Fix nonzero s, h * ∈ F and some s * , r, ...
Let denote an algebraically closed field with a characteristic not two. Fix an integer ; let , , and be the equitable basis of over . Let denote an irreducible -module with dimension ; let . In this paper, we show that if each of the pairs , , and acts on as a Leonard pair, then these pairs are of Krawtchouk type. Moreover, is a linear combination of 1, , , and .
1. Introduction
In this section, we recall some facts concerning Leonard pairs. Leonard pairs were introduced by P. Terwilliger [1] to extend the algebraic approach of Bannai and Ito [2] 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 [3, 4]. 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 -modules [5, 6]. Fix an integer . Throughout this paper shall denote an algebraically closed field with characteristic not two. Also, shall denote an -vector space of dimension , and shall denote the -algebra of matrices with entries in having rows and columns indexed by 1, 2, …, . A square matrix is said to be tridiagonal whenever every nonzero entry appears on, immediately above, or immediately below the main diagonal. A tridiagonal matrix is said to be irreducible whenever all entries immediately above and below the main diagonal are nonzero. A square matrix is said to be upper (resp. lower) bidiagonal whenever every nonzero entry appears on or immediately above (resp. below) the main diagonal.
Definition 1. Let denote a vector space over with finite positive dimension. By a Leonard pair on , we mean an ordered pair , , where and are linear transformations that satisfy both (i) and (ii) below:(i)There exists a basis for with respect to which the matrix representing is diagonal and the matrix representing is irreducible tridiagonal(ii)There exists a basis for with respect to which the matrix representing is diagonal and the matrix representing is irreducible tridiagonalWe remark that if , is a Leonard pair on , then , is a Leonard pair on , and for any scalars , , the pair , is also a Leonard pair on .
For more details about Leonard pairs, see [7–14].
Definition 2. Let denote a vector space over with dimension . let denote a basis for , which satisfies condition (ii) of Definition 1. For , the vector is an eigenvector of ; let denote the corresponding eigenvalues. Let denote a basis for , which satisfies condition (i) of Definition 1. For , the vector is an eigenvector of ; let denote the corresponding eigenvalues. Let the sequence denote the diagonal of the matrix, which represents with respect to . Let the sequence denote the diagonal of the matrix, which represents with respect to .
The ordering of in Definition 2 is said to be standard. For a standard ordering of eigenvalues of, the ordering is also standard and no further ordering is standard. A similar result applies for .
Theorem 1 (see [15]). Let , be a Leonard pair on ; let (resp. ) be standard ordering of the eigenvalues of (resp. ). Then, there exists a basis of such that the matrices representing , with respect to this basis are, respectively,for some sequence of scalars , , …, in , which we refer to as the first split sequence of , .
Definition 3 (see [16]). Let denote a nonnegative integer. By a parameter array over of diameter , we mean a sequence of scalars (, ; , ) taken from that satisfy the following conditions:
Theorem 2 (see [15]). Let denote a nonnegative integer; let and denote matrices in . Assume is lower bidiagonal and is upper bidiagonal. Then, the following are equivalent:(i)The pair , is a Leonard pair in .(ii)There exists a parameter array (, ; , ) over such thatSuppose (i) and (ii) hold. Then, the parameter array in (ii) is uniquely determined by , .
Theorem 3 (see [15]). With reference to Definition 2, let , be a Leonard pair on ; let (resp. ) be standard ordering of the eigenvalues of (resp. ). Via a standard basis, the matrix representing and areand , wherewhere and .
Theorem 4 (see [17]). Let denote a vector space over with finite positive dimension. Let , denote a Leonard pair on . Then, there exists a sequence of scalars , , , , , , , and taken from such thatThe sequence is uniquely determined by the pair , provided the dimension of is at least 4.
Relations (12) and (13) are called the Askey–Wilson relations.
2. Leonard Pairs of Classical Type
In [18], the parameters arrays are classified into 13 families, each named for certain associated sequences of orthogonal polynomials. The four families, which arise in this paper, share certain property. Given a parameter array, let be the common value of (8) minus one.
Definition 4. A parameter array is of classical type whenever .
Theorem 5 (see [18]). A parameter array is of classical type if and only if it is of Racah, Hahn, dual Hahn, or Krawtchouk type.
Theorem 6 (see [18], Example 5.10). Fix nonzero , and , , , , , such that and none of , , , is equal to for and that neither of , is equal to for . LetThen, is a parameter array of Racah type. We refer to the scalars , , , , , , , and as hypergeometric parameters of .
Theorem 7 (see [18], Example 5.11). Fix nonzero , and some , , , such that neither of , is equal to for and that is not equal for . LetThen, is a parameter array of Hahn type. We refer to the scalars , , , , , and as hypergeometric parameters of .
Theorem 8 (see [18], Example 5.12). Fix nonzero , and , , , such that neither of , is equal to for , and that is not equal for . LetThen, is a parameter array of dual Hahn type. We refer to the scalars , , , , , and as hypergeometric parameters of .
Theorem 9 (see [18]. Example 5.13). Fix nonzero , , , and some , such that :Then, is a parameter array of Krawtchouk type. We refer to the scalars , , , , and as hypergeometric parameters of .
3. The Lie Algebra
In this section, we recall some facts concerning the Lie algebra .
Definition 5. (see [11]). The Lie algebra is the -algebra that has a basis satisfying the following conditions:where denotes the Lie bracket.
Lemma 1 (see [10]). With reference to Definition 5, letThen, is a basis for andWe call , , the equitable basis for the Lie algebra .
Lemma 2 (see [11]). For each nonnegative integer , there is an irreducible finite-dimensional -module with basis , , …, and action , , , , and . Moreover, up to isomorphism, is the unique irreducible -module of dimension .
Lemma 3. (see [10]). With reference to Lemmas 1 and 2,
Definition 6. With reference to Lemmas. 1 and 2, fix . A basis of is said to be a standard-eigenbasis whenever , , and act as
Lemma 4 (see [19]). With reference to Lemmas 1 and 2,(i)For , let . Then, is a standard -eigenbasis of .(ii)For , let . Then, is a standard -eigenbasis of .(iii)Let , , …, be as in Lemma 2. Then, is a standard -eigenbasis of [10].The main results of this paper are the following theorems.
Theorem 10. Fix an integer. ; let , , and be the equitable basis of over . Let denote an irreducible -module with dimension ; let . Then, the following are equivalent:(i)Any two of the pairs , , and are Leonard pairs(ii)All the pairs , , and are Leonard pairs
Theorem 11. Fix an integer ; let , , and be the equitable basis of over . Let denote an irreducible -module with dimension ; let such that each of the pairs , , and is a Leonard pair; then, all the pairs are of Krawtchouk type. Moreover, is a linear combination of 1, , , and .
We remark here that the authors in [12] proved similar result with the generators of the quantum algebra .
4. The Type of Leonard Pair
We start recalling some facts which help us to determine the type of the Leonard pairs.
Lemma 5 (see [17]). Let denote a nonnegative integer, and let denote a vector space over with dimension . Let , denote a Leonard pair on with fundamental parameter . Let the scalars and be as in Definition 2; then, there exist scalars , , , and such that(i)(ii)(iii)(iv)
Theorem 12. (see [17]). Let denote a nonnegative integer, and let denote a vector space over with dimension . Let , denote a Leonard pair on . Let , , , , , , , and denote a sequence of scalars taken from , which satisfy (12) and (13). Let the scalars , , , and be as in Definition 2. Then, the following hold:(i)(ii)(iii)(iv)
Lemma 6. With reference to Lemma 1, if each of the pairs , , and is a Leonard pair, then all the pairs are of dual Hahn type or all the pairs of Krawtchouk type.
Proof. Note that, from Definition 4 and Lemma 3, all the pairs are of classical type. Now, the result holds by Theorem 5 and Theorem 6–Theorem 9.
We have two cases to check, the first case, is there such that all the pairs , , and are Leonard pairs of dual Hahn type?, and the second case, is there such that all the pairs , , and are Leonard pairs of Krawtchouk type? We start with the dual Hahn case.
For the rest of the paper, fix an integer ; let denote an irreducible -module with dimension ; let ; Let be the matrix that represents the linear map with respect to the basis .
Lemma 7. With reference to Lemma 3, let , assume that , is a Leonard pair of dual Hahn type; let (, ; , ) be the parameter array associated with the pair , ; then,where and are nonzero scalars, none of , equal to for , and for .
Proof. Clear from Theorem 8.
Lemma 8. Let such that , is a Leonard pair of dual Hahn type; then, the basis for with respect to which the matrix representing is irreducible tridiagonal and the matrix representing is diagonal is or for some nonzero scalars .
Proof. Clear from Lemma 3 and the paragraph after Definition 2.
Since the case with basis can be treated similar to the case with basis , we shall prove our results only for the case . Let .
Lemma 9. With reference to Lemmas 2 and 8,
Lemma 10. With reference to Lemmas 7 and 8, let , be a Leonard pair of dual Hahn type; then,where
Proof. The result is held by Theorem 3 and Lemmas 7 and 9.
Lemma 11. With reference to Lemma 5 and Theorem 12, let ; assume , is a Leonard pair of dual Hahn type; let and ; then,
Proof. Clear from Lemma 7.
Lemma 12. Let ; assume , is a Leonard pair of dual Hahn type; then, the pair , is not a Leonard pair of dual Hahn type.
Proof. LetSince the pair , is a Leonard pair, the action of on the basis is given in Lemma 10, and the action of is given in Lemma 9. Assume that the pair , is a Leonard pair and let ; hence, by Theorem 4, there exists a sequence of scalars , , , , , , , and as in Lemma 11 such that . Now, if we solve for , we find that . Substitute ; then, for , solve the -entries of for ; we find that . Substitute to find thatBy Lemmas 7 and 8, , , and , so . Hence, the pair , is not a Leonard pair.
Lemma 13. Let , be a Leonard pair of dual Hahn type; then, the pair , is not a Leonard pair of dual Hahn type.
Proof. Similar to proof of Lemma 12.
5. Leonard Pair of Krawtchouk Type
In this section, we describe a linear map such that all the pairs , , and are Leonard pairs of Krawchouch type.
Lemma 14. With reference to Lemma 3, let , assume that , is a Leonard pair of Krawtchouk type; let (, ; , ) be the parameter array associated with the pair , ; then, there exist a nonzero scalars , , and such that , and
Proof. Clear from Theorem 9.
Lemma 15. With reference to Lemma 5 and Theorem 12, assume , is a Leonard pair of Krawtchouk type; let and ; then,
Proof. Clear from Lemma 14.
Lemma 16. With reference to Lemmas 14 and 8, Let , be a Leonard pair of Krawtchouk type; then, , , and , where
Proof. The result holds by Theorem 3 and Lemmas 3 and 14.
Lemma 17. With reference to Lemma 16, the pair , is a Leonard pair of Krawtchouk type in if and only if , , and .
Proof. Let be matrix indexed such that the -entry isThen, the matrices represent and as in (1), where , ; and are as in Lemma 14. Hence, by Theorem 2, , is a Leonard pair of Krawtchouk type if and only if , , and hold, which implies that the pair , is a Leonard pair of Krawtchouk type if and only if , , and .
Lemma 18. With reference to Lemmas 8 and 14, assume , is a Leonard pair of Krawtchouk type. If the pair , is a Leonard pair, then there exists a nonzero scalar such that . Moreover, .
Proof. Routine calculations using Theorem 4 and Lemma 15.
Lemma 19. With reference to Lemmas 8 and 14, assume , is a Leonard pair of Krawtchouk type. If the pair , is a Leonard pair, then there exists a nonzero scalar such that . Moreover, .
Proof. Routine calculations using Theorem 4 and Lemma 15.
Lemma 20. Suppose that the pairs , and , are Leonard pairs of Krawtchouk type; let , and , be the parameter array associated with the pair , . Then, there exist a nonzero scalars , , and such that , , , and
Proof. Let in Lemma 8, so the basis . Hence, the result holds by Lemmas 14 and 18.
Lemma 21. Suppose that the pairs , and , are Leonard pairs of Krawtchouk type, let , and , be the parameter array associated with the pair , . Then, there exist nonzero scalars , , and such that , , , and
Proof. Let in Lemma 8, so the basis . Hence, the result holds by Lemmas 14 and 19.
Lemma 22. With reference to Lemma 2, let be a nonzero scalar; let . Let be as in Lemma 16; assume that , is a Leonard pair of Krawtchouk type; then, the pair , is a Leonard pair of Krawtchouk type if and only if and .
Proof. The pair , is a Leonard pair of Krawtchouk type; then, by Lemma 17, , , and .
Let be matrix indexed such that the -entry isThen, the matrices represent , as in (1), where , and , are as in Lemma 20. Hence, by Theorem 2, , is a Leonard pair of Krawtchouk type, if and only if and hold, which implies that the pair , is a Leonard pair of Krawtchouk type if and only if and .
Lemma 23. With reference to Lemma 2, let be a nonzero scalar; let . Let be as in Lemma 16; assume, that , is a Leonard pair of Krawtchouk type; then ,the pair , is a Leonard pair of Krawtchouk type if and only if and .
Proof. The pair , is a Leonard pair of Krawtchouk type; then, by Lemma 17, , , and .
Let be matrix indexed such that the -entry isThen,where , and , are as in Lemma 21. Let , where , , and are in such that , for , and if and 0 otherwise. Then, the matrices represent , as in (1). The rest of the proof will be similar to proof of Lemma 22.
We remark here that the existence of that appears in proof of Lemma 23 was proved by Terwilliger in [18].
Lemma 24. With reference to Lemma 2, let be a nonzero scalar; let . Let be as in Lemma 16, assume that , is a Leonard pair of Krawtchouk type; then, , is a Leonard pair of Krawtchouk type if and only if , is a Leonard pair of Krawtchouk type.
Proof. Clear from Lemmas 22 and 23.
Lemma 25. Fix an integer , let be a nonzero scalar; let , , denote the equitable basis for the Lie algebra . Let denote a finite-dimensional irreducible -module, let be a basis of as in Lemma 3, let , and let such that acts on as in Lemma 16. Then, each of the pairs , , is a Leonard pair of Krawtchouk type if and only if , , , and .
Proof. The result holds from Lemmas 17, 22, and 23.
6. A Basis of
Let denote the universal enveloping algebra of . Thus, is the associative -algebra with generators , , and relations , , and , where is commutator of and . In this section, we write as a linear combination of the generators of [16, 20–23].
Definition 7. Let denote any linear combination of 1, , , and , and writewhere , , , .
Lemma 26. With reference to Lemma 3 and Definition 7, let be a nonzero scalar in and let ; then,where
Proof. Expand the action of , , and on using Lemma 8.
Lemma 27. Fix an integer ; let , , denote the generators of , let such that , let denote a finite-dimensional irreducible -module, and let be a basis of , assume that acts on as in Lemma 16. Then,
Proof. Compare the action of on with the action of in Lemma 26 to get the result.
Lemma 28. With reference to Lemma 27, .
Proof. Routine calculations show that
Lemma 29. Fix an integer ; let , , denote the generators of , let such that , where , , and in , and let denote a finite-dimensional irreducible -module. Then, each of the pairs , , and is a Leonard pair of Krawtchouk type if and only if , , are nonzero scalars in and there exists a nonzero scalar such that and .
Proof. Assume all the pairs , , and are Leonard pairs of Krawtchouk type; then, by Lemma 25, there exist nonzero scalars , , , such that acts on the basis as in Lemma 16. Moreover, and . Hence, acts on as , where , , and are as in Lemma 27, and .
Note that , , and are nonzero scalars if and only if , , and are nonzero scalars, respectively, and since , then, by Lemma 28, if and only if .
In the another direction, if for , , , are nonzero scalars in , and for nonzero scalar , then implies ; hence, we can find , nonzero scalars such that acts on as in Lemma 16. Hence, the pairs , , and are Leonard pairs of Krawtchouk type by Lemma 25.
Proof of Theorem 10. Note that, from Lemma 4, . Hence, the result holds from Lemmas 16, 17, and 24.
Proof of Theorem 11. Clear from Lemmas 12, 25, and 29.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
... This version gives a complete classification of the orthogonal polynomial systems that satisfy Askey-Wilson duality. It shows that the orthogonal polynomial systems that satisfy Askey-Wilson duality are from the terminating branch of the Askey-tableau, except for one family with q = −1 now called the Bannai-Ito polynomials [2, p. 271], [23,Example 5.14]. In our view, the terminating branch of the Askey-tableau should include the Bannai-Ito polynomials. ...
... The classification [19,Theorem 1.9] shows that the ground field does not matter in a substantial way, unless it has characteristic 2. In this case, the theory admits an additional family of polynomials called the orphans. The orphans have diameter d = 3 only; they are described in [23,Example 5.15] and Example 20.13 below. ...
... The parameter arrays are listed in [23,Section 5]. In this appendix we go through the list, and for each parameter array we give the corresponding intersection numbers and dual intersection numbers. ...
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.
... 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]. ...
... 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]. The Leonard pairs are classified [25,29] and correspond to the orthogonal polynomials that make up the terminating branch of the Askey scheme [20]. A tridiagonal pair is a mild generalization of a Leonard pair [5, Definition 1.1]. ...
... We now turn our attention to Leonard pairs. There is a general family of Leonard pairs said to have q-Racah type [4, Section 5], [29,Example 5.3]. In this section, we show that for any Leonard pair of q-Racah type, the underlying vector space becomes an evaluation module for ⊠ q in a natural way. ...
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.
... For a parameter array of A, A * , the entries satisfy numerous relations [47,Theorem 1.9]. In [52], the solutions are given in closed form, in terms of seven free variables in addition to d and β. These seven variables are called basic. ...
... The Leonard pair A, A * is said to have type I whenever β ̸ = ±2; type II whenever β = 2 and Char(F) ̸ = 2; type III + whenever β = −2, Char(F) ̸ = 2, and d is even; type III − whenever β = −2, Char(F) ̸ = 2, and d is odd; type IV whenever β = 2 and Char(F) = 2. For each type, the solutions are given in [52,Section 5]. ...
In this paper, we introduce the concepts of compatibility and companion for Leonard pairs. These concepts are roughly described as follows. Let $\mathbb{F}$ denote a field, and let $V$ denote a vector space over $\mathbb{F}$ with finite positive dimension.A Leonard pair on $V$ is an ordered pair of diagonalizable $\mathbb{F}$-linear maps $A : V \to V$ and $A^* : V \to V$ that each act in an irreducible tridiagonal fashion on an eigenbasis for the other one. Leonard pairs $A,A^*$ and $B,B^*$ on $V$ are said to be compatible whenever $A^* = B^*$ and $[A,A^*] = [B,B^*]$, where $[r,s] = r s - s r$. For a Leonard pair $A,A^*$ on $V$, by a companion of $A,A^*$ we mean an $\mathbb{F}$-linear map $K: V \to V$ such that $K$ is a polynomial in $A^*$ and $A-K, A^*$ is a Leonard pair on $V$. The concepts of compatibility and companion are related as follows. For compatible Leonard pairs $A,A^*$ and $B,B^*$ on $V$, define $K = A-B$. Then $K$ is a companion of $A,A^*$. For a Leonard pair $A,A^*$ on $V$ and a companion $K$ of $A,A^*$,define $B = A-K$ and $B^* = A^*$. Then $B,B^*$ is a Leonard pair on $V$ that is compatible with $A,A^*$. Let $A,A^*$ denote a Leonard pair on $V$. We find all the Leonard pairs $B, B^*$ on $V$ that are compatible with $A,A^*$.For each solution $B, B^*$, we describe the corresponding companion $K = A-B$.
... 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]. ...
... By (39) and the construction, the matrix W 2 is lower triangular with (i, i)-entry a −2i q 2i(d−i) for 0 ≤ i ≤ d. By (40) and the construction, the matrix (W ′ ) 2 is upper triangular with ...
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.
... for 0 ≤ i ≤ d. In both equations above the coefficients of q 2i−d and q d−2i are nonzero; therefore the action of A, A * on W is a Leonard pair of q-Racah type in the sense of [36,Example 5.3]. Referring to this Leonard pair, let ...
... By [35,Section 7] each of ϕ i , φ i is nonzero for 1 ≤ i ≤ d. By [36,Example 5.3] there exists a nonzero r ∈ C such that ...
In this paper we discuss a relationship between the following two
algebras: (i) the subconstituent algebra $T$ of a distance-regular graph
that has $q$-Racah type; (ii) the $q$-tetrahedron algebra $\boxtimes_q$
which is a $q$-deformation of the three-point $sl_2$ loop algebra.
Assuming that every irreducible $T$-module is thin, we display an
algebra homomorphism from $\boxtimes_q$ into $T$ and show that $T$ is
generated by the image together with the center $Z(T)$.
... A tridiagonal pair for which the V i , V * i all have dimension 1 is called a Leonard pair [11]. There is a natural correspondence between the Leonard pairs and a family of orthogonal polynomials consisting of the q-Racah polynomials [1], [6] and some related polynomials in the Askey-scheme [9], [18]. This correspondence follows from the classification of Leonard pairs [11], [18]. ...
... There is a natural correspondence between the Leonard pairs and a family of orthogonal polynomials consisting of the q-Racah polynomials [1], [6] and some related polynomials in the Askey-scheme [9], [18]. This correspondence follows from the classification of Leonard pairs [11], [18]. We remark that this classification amounts to a linear algebraic version of a theorem of D. Leonard [2], [10] concerning the q-Racah polynomials. ...
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*.
... Other relevant references include [16][17][18][19][20][21]. ...
Abstract: Let F denote an algebraically closed field; let q be a nonzero scalar in F such that q is not
a root of unity; let d be a nonnegative integer; and let X, Y, Z be the equitable generators of Uq(sl2)
over F. Let V denote a finite-dimensional irreducible Uq(sl2)
-module with dimension d + 1, and let
R denote the set of all linear maps from V to itself that act tridiagonally on the standard ordering
of the eigenbases for each of X, Y, and Z. We show that R has dimension at most seven. Indeed,
we show that the actions of 1, X, Y, Z, XY, YZ, and ZX on V give a basis for R when d >= 3.
... The tridiagonal pair concept originated in the theory of Q-polynomial distance-regular graphs [4,11] where it is used to describe how the adjacency matrix is related to each dual adjacency matrix [17,Example 1.4], [29,Section 3]. Since that origin, the tridiagonal pair concept has found applications to all sorts of topics in special functions (orthogonal polynomials of the Askey-scheme [3,25,27,32,33], the Askey-Wilson algebra [14,35,40,42]), Lie theory (the sl 2 loop algebra [20], the tetrahedron algebra [15,21]), statistical mechanics (the Onsager algebra [12,16] and q-Onsager algebra [5,6,23,30,37,39]), and quantum groups (the equitable presentation [1,24,36], the quantum affine sl 2 algebra [2,18,19,22], L-operators [26,38]). For more information about the above topics, see [28,34] and the references therein. ...
Recently Sarah Bockting-Conrad introduced the double lowering operator $\psi$ for a tridiagonal pair. Motivated by $\psi$ we consider the following problem about polynomials. Let $\mathbb F$ denote a field. Let $x$ denote an indeterminate, and let $\mathbb F\lbrack x \rbrack$ denote the algebra consisting of the polynomials in $x$ that have all coefficients in $\mathbb F$. Let $N$ denote a positive integer or $\infty$. Let $\lbrace a_i\rbrace_{i=0}^{N-1}$, $\lbrace b_i\rbrace_{i=0}^{N-1}$ denote scalars in $\mathbb F$ such that $\sum_{h=0}^{i-1} a_h \not= \sum_{h=0}^{i-1} b_h$ for $1 \leq i \leq N$. For $0 \leq i \leq N$ define polynomials $\tau_i, \eta_i \in \mathbb F\lbrack x \rbrack$ by $\tau_i = \prod_{h=0}^{i-1} (x-a_h)$ and $\eta_i = \prod_{h=0}^{i-1} (x-b_h)$. Let $V$ denote the subspace of $\mathbb F\lbrack x \rbrack$ spanned by $\lbrace x^i\rbrace_{i=0}^N$. An element $\psi \in {\rm End}(V)$ is called double lowering whenever $\psi \tau_i \in \mathbb F \tau_{i-1}$ and $\psi \eta_i \in \mathbb F \eta_{i-1}$ for $0 \leq i \leq N$, where $\tau_{-1}=0$ and $\eta_{-1}=0$. We give necessary and sufficient conditions on $\lbrace a_i\rbrace_{i=0}^{N-1}$, $\lbrace b_i\rbrace_{i=0}^{N-1}$ for there to exist a nonzero double lowering map. There are four families of solutions, which we describe in detail.
... The Leonard system is uniquely determined up to isomorphism by the parameter array, cf. [24,Theorem 1.9], and all families of the parameter arrays of Leonard systems are displayed in [26] as parametric form. We now recall the dual q-Hahn family of Leonard systems. ...
We discuss the Grassmann graph $J_q(N,D)$ with $N \geq 2D$, having as vertices the $D$-dimensional subspaces of an $N$-dimensional vector space over the finite field $\mathbb{F}_q$. This graph is distance-regular with diameter $D$; to avoid trivialities we assume $D\geq 3$. Fix a pair of a maximal clique $C$ of $J_q(N,D)$ and a vertex $x$ in $C$. Using the pair $x, C$, we construct a $2D$-dimensional irreducible module $\mathbf{W}$ for the confluent Cherednik algebra $\mathcal{H}_\mathrm{V}$ and show its connection to the generalized Terwilliger algebra associated with the pair $x$, $C$. We discuss how $\mathcal{H}_\mathrm{V}$ is related to a nil-DAHA of type $(C^\vee_1, C_1)$. We note that $\mathbf{W}$ becomes a module for the nil-DAHA if $N=2D$. Using the $\mathcal{H}_\mathrm{V}$-module $\mathbf{W}$, we obtain non-symmetric dual $q$-Hahn polynomials and prove their recurrence and orthogonality relations from a combinatorial viewpoint.
... In [30, Theorem 1.9] Terwilliger classified Leonard systems by using parameter arrays, and characterized the set of parameter arrays of Leonard systems with diameter d. Moreover, he displayed all the parameter arrays over C in [33]. We recall the q-Racah family of parameter arrays, that is the most general family. ...
In his famous theorem (1982), Douglas Leonard characterized the $q$-Racah
polynomials and their relatives in the Askey scheme from the duality property
of $Q$-polynomial distance-regular graphs. In this paper we consider a
nonsymmetric (or Laurent) version of the $q$-Racah polynomials in the above
situation. Let $\Gamma$ denote a $Q$-polynomial distance-regular graph that
contains a Delsarte clique $C$. Assume that $\Gamma$ has $q$-Racah type. Fix a
vertex $x \in C$. We partition the vertex set of $\Gamma$ according to the
path-length distance to both $x$ and $C$. The linear span of the characteristic
vectors corresponding to the cells in this partition has an irreducible module
structure for the universal double affine Hecke algebra $\hat{H}_q$ of type
$(C^{\vee}_1, C_1)$. From this module, we naturally obtain a finite sequence of
orthogonal Laurent polynomials. We prove the orthogonality relations for these
polynomials, using the $\hat{H}_q$-module and the theory of Leonard systems.
Changing $\hat{H}_q$ by $\hat{H}_{q^{-1}}$ we show how our Laurent polynomials
are related to the nonsymmetric Askey-Wilson polynomials, and therefore how our
Laurent polynomials can be viewed as nonsymmetric $q$-Racah polynomials.
... In this case, the fundamental parameter β is well-defined (see Definition 2.12 for the definition). In [22] Terwilliger gave a classification of Leonard pairs. By that classification, Leonard pairs are classified into thirteen types. ...
Fix an algebraically closed field $\mathbb{F}$ and an integer $n \geq 1$. Let
$\text{Mat}_n(\mathbb{F})$ denote the $\mathbb{F}$-algebra consisting of the $n
\times n$ matrices that have all entries in $\mathbb{F}$. We consider a pair of
diagonalizable matrices in $\text{Mat}_{n}(\mathbb{F})$, each acting in an
irreducible tridiagonal fashion on an eigenbasis for the other one. Such a pair
is called a Leonard pair in $\text{Mat}_{n}(\mathbb{F})$. In the present paper,
we find all Leonard pairs $A,A^*$ in $\text{Mat}_{n}(\mathbb{F})$ such that
each of $A$ and $A^*$ is irreducible tridiagonal with all diagonal entries $0$.
This solves a problem given by Paul Terwilliger.
... We refer the reader to [3,[5][6][7][8] for background on Leonard pairs. ...
Fix an algebraically closed field $\mathbb{F}$ and an integer $d \geq 3$. Let
$V$ be a vector space over $\mathbb{F}$ with dimension $d+1$. A Leonard pair on
$V$ is an ordered pair of diagonalizable linear transformations $A: V \to V$
and $A^* : V \to V$, each acting in an irreducible tridiagonal fashion on an
eigenbasis for the other one. Let $\{v_i\}_{i=0}^d$ (resp.\
$\{v^*_i\}_{i=0}^d$) be such an eigenbasis for $A$ (resp.\ $A^*$). For $0 \leq
i \leq d$ define a linear transformation $E_i : V \to V$ such that $E_i
v_i=v_i$ and $E_i v_j =0$ if $j \neq i$ $(0 \leq j \leq d)$. Define $E^*_i : V
\to V$ in a similar way. The sequence $\Phi =(A, \{E_i\}_{i=0}^d, A^*,
\{E^*_i\}_{i=0}^d)$ is called a Leonard system on $V$ with diameter $d$. With
respect to the basis $\{v_i\}_{i=0}^d$, let $\{\th_i\}_{i=0}^d$ (resp.\
$\{a^*_i\}_{i=0}^d$) be the diagonal entries of the matrix representing $A$
(resp.\ $A^*$). With respect to the basis $\{v^*_i\}_{i=0}^d$, let
$\{\theta^*_i\}_{i=0}^d$ (resp.\ $\{a_i\}_{i=0}^d$) be the diagonal entries of
the matrix representing $A^*$ (resp.\ $A$). It is known that
$\{\theta_i\}_{i=0}^d$ (resp. $\{\th^*_i\}_{i=0}^d$) are mutually distinct, and
the expressions $(\theta_{i-1}-\theta_{i+2})/(\theta_i-\theta_{i+1})$,
$(\theta^*_{i-1}-\theta^*_{i+2})/(\theta^*_i - \theta^*_{i+1})$ are equal and
independent of $i$ for $1 \leq i \leq d-2$. Write this common value as $\beta +
1$. In the present paper we consider the "end-entries" $\theta_0$, $\theta_d$,
$\theta^*_0$, $\theta^*_d$, $a_0$, $a_d$, $a^*_0$, $a^*_d$. We prove that a
Leonard system with diameter $d$ is determined up to isomorphism by its
end-entries and $\beta$ if and only if either (i) $\beta \neq \pm 2$ and
$q^{d-1} \neq -1$, where $\beta=q+q^{-1}$, or (ii) $\beta = \pm 2$ and
$\text{Char}(\mathbb{F}) \neq 2$.
... 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 refer the reader to [3,[5][6][7][8] for background on Leonard pairs. For the rest of this section, fix an integer d ≥ 0 and a vector space V over F with dimension d + 1. ...
Fix an algebraically closed field $\F$ and an integer $d \geq 3$. Let $V$ be
a vector space over $\F$ with dimension $d+1$. A Leonard pair on $V$ is a pair
of diagonalizable linear transformations $A: V \to V$ and $A^* : V \to V$, each
acting in an irreducible tridiagonal fashion on an eigenbasis for the other
one. There is an object related to a Leonard pair called a Leonard system. It
is known that a Leonard system is determined up to isomorphism by a sequence of
scalars $(\{\th_i\}_{i=0}^d, \{\th^*_i\}_{i=0}^d, \{\vphi_i\}_{i=1}^d,
\{\phi_i\}_{i=1}^d)$, called its parameter array. The scalars
$\{\th_i\}_{i=0}^d$ (resp.\ $\{\th^*_i\}_{i=0}^d$) are mutually distinct, and
the expressions $(\th_{i-2} - \th_{i+1})/(\th_{i-1}-\th_{i})$, $(\th^*_{i-2} -
\th^*_{i+1})/(\th^*_{i-1}-\th^*_{i})$ are equal and independent of $i$ for $2
\leq i \leq d-1$. Write this common value as $\beta+1$. In the present paper,
we consider the "end-parameters" $\th_0$, $\th_d$, $\th^*_0$, $\th^*_d$,
$\vphi_1$, $\vphi_d$, $\phi_1$, $\phi_d$ of the parameter array. We show that a
Leonard system is determined up to isomorphism by the end-parameters and
$\beta$. We display a relation between the end-parameters and $\beta$. Using
this relation, we show that there are up to inverse at most $\lfloor (d-1)/2
\rfloor$ Leonard systems that have specified end-parameters. The upper bound
$\lfloor (d-1)/2 \rfloor$ is best possible.
... In fact, we have α = d i=1 c i k i .5 In the terminology of[33], these are of q-Racah, affine q-Krawtchouk, Racah, Krawtchouk and Bannai/Ito types, respectively. ...
Motivated by the similarities between the theory of spherical $t$-designs and
that of $t$-designs in $Q$-polynomial association schemes, we study two
versions of relative $t$-designs, the counterparts of Euclidean $t$-designs for
$P$- and/or $Q$-polynomial association schemes. We develop the theory based on
the Terwilliger algebra, which is a noncommutative associative semisimple
$\mathbb{C}$-algebra associated with each vertex of an association scheme. We
compute explicitly the Fisher type lower bounds on the sizes of relative
$t$-designs, assuming that certain irreducible modules behave nicely. The two
versions of relative $t$-designs turn out to be equivalent in the case of the
Hamming schemes. From this point of view, we establish a new algebraic
characterization of the Hamming schemes.
... To verify (12), in the right-hand side, replace δ * by (11) and eliminate both occurrences of γ * in the resulting expression using γ * = θ * i−1 − βθ * i + θ * i+1 . We have now verified (12). ...
Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A:V→VA:V→V and A*:V→VA*: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. Pascasio recently obtained a characterization of the Q-polynomial distance-regular graphs using the intersection numbers ai. In this paper, we extend her results to a linear algebraic level and obtain a characterization of Leonard pairs. Pascasio’s argument appears to rely on the underlying combinatorial assumptions, so we take a different approach that is algebraic in nature.
... A Leonard pair is a pair of semisimple linear transformations on a finite-dimensional vector space, each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other [35, Definition 1.1]. There is a close connection between the Leonard pairs and the orthogonal polynomials that make up the terminating branch of the Askey scheme [25], [35], [38]. A tridiagonal pair is a mild generalization of a Leonard pair [19, Definition 1.1]. ...
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.
In this paper we resolve a conjecture of Kresch and Tamvakis [Duke Math. J. 110 (2001), pp. 359–376]. Our result is the following.
Theorem : For any positive integer D D and any integers i , j i,j ( 0 ≤ i , j ≤ D ) , (0\leq i,j \leq D), \; the absolute value of the following hypergeometric series is at most 1: 4 F 3 [ − i , i + 1 , − j , j + 1 1 , D + 2 , − D ; 1 ] . \begin{equation*} {_4F_3} \left [ \begin {array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right ]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem.
Following Verde-Star [Linear Algebra Appl. 627 (2021), pp. 242–274] we label families of orthogonal polynomials in the q q -Askey scheme together with their q q -hypergeometric representations by three sequences x k , h k , g k x_k, h_k, g_k of Laurent polynomials in q k q^k , two of degree 1 and one of degree 2, satisfying certain constraints. This gives rise to a precise classification and parametrization of these families together with their limit transitions. This is displayed in a graphical scheme. We also describe the four-manifold structure underlying the scheme.
Let F denote a field, and let V denote a vector space over F with finite positive dimension. We consider an ordered pair of F-linear maps A:V→V and A⁎:V→V such that (i) each of A,A⁎ is diagonalizable; (ii) there exists an ordering {Vi}i=0d 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 {Vi⁎}i=0δ of the eigenspaces of A⁎ such that AVi⁎⊆Vi−1⁎+Vi⁎+Vi+1⁎ for 0≤i≤δ, where V−1⁎=0 and Vδ+1⁎=0; (iv) there does not exist a subspace U of V such that AU⊆U, A⁎U⊆U, U≠0, U≠V. We call such a pair a tridiagonal pair on V. We assume that A,A⁎ belongs to a family of tridiagonal pairs said to have q-Racah type. There is an infinite-dimensional algebra ⊠q called the q-tetrahedron algebra; it is generated by four copies of Uq(sl2) that are related in a certain way. Using A,A⁎ we construct two ⊠q-module structures on V. In this construction the two main ingredients are the double lowering map ψ:V→V due to Sarah Bockting-Conrad, and a certain invertible map W:V→V motivated by the spin model concept due to V. F. R. Jones.
Let \(\mathbb {K}\) denote an algebraically closed field and let V denote a vector space over \(\mathbb {K}\) with finite positive dimension. Let A, A∗ denote a tridiagonal pair on V . We assume that A, A∗ belongs to a family of tridiagonal pairs said to have q-Racah type. Let \(\{U_i\}_{i=0}^d\) and \(\{U_i^\Downarrow \}_{i=0}^{d}\) denote the first and second split decompositions of V . In an earlier paper we introduced a double lowering operator ψ : V → V with the notable feature that both ψUi ⊆ Ui−1 and \(\psi U_i^\Downarrow \subseteq U_{i-1}^\Downarrow \) for 0 ≤ i ≤ d, where U−1 = 0 and \(U_{-1}^\Downarrow =0\). In the same paper, we showed that there exists a unique linear transformation Δ : V → V such that \(\Delta (U_i)\subseteq U_i^{\Downarrow }\) and ( Δ − I)Ui ⊆ U0 + U1 + ⋯ + Ui−1 for 0 ≤ i ≤ d. In the present paper, we show that Δ can be expressed as a product of two linear transformations; one is a q-exponential in ψ and the other is a q⁻¹-exponential in ψ. We view Δ as a transition matrix from the first split decomposition of V to the second. Consequently, we view the q⁻¹-exponential in ψ as a transition matrix from the first split decomposition to a decomposition of V which we interpret as a kind of halfway point. This halfway point turns out to be the eigenspace decomposition of a certain linear transformation \(\mathcal {M}\). We discuss the eigenspace decomposition of \(\mathcal {M}\) and give the actions of various operators on this decomposition.
We discuss the Grassmann graph Jq(N,D) with N≥2D, having as vertices the D-dimensional subspaces of an N-dimensional vector space over the finite field Fq. This graph is distance-regular with diameter D; to avoid trivialities we assume D≥3. Fix a pair of a Delsarte clique C of Jq(N,D) and a vertex x in C. We construct a 2D-dimensional irreducible module W for the Terwilliger algebra T of Jq(N,D) associated with the pair x, C. We show that W is an irreducible module for the confluent Cherednik algebra HV and describe how the T-action on W is related to the HV-action on W. Using the HV-module W, we define non-symmetric dual q-Hahn polynomials and prove their recurrence and orthogonality relations from a combinatorial viewpoint.
Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A∗:V→V that satisfy (i) and (ii) below.
• There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal.
• 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 the literature, there are some parameters that are used to describe Leonard pairs called the intersection numbers {ai}i=0d, {bi}i=0d−1, {ci}i=1d, and the dual eigenvalues {θi∗}i=0d. In this paper, we provide two characterizations of Leonard pairs. For the first characterization, the focus is on the {ai}i=0d and {θi∗}i=0d. For the second characterization, the focus is on the {bi}i=0d−1, {ci}i=1d, and {θi∗}i=0d.
Let K denote an algebraically closed field of characteristic zero and let d0,e1,e2 be some scalars in K. By the Racah algebra A associated with d0,e1,e2, we mean the most general quadratic algebra with two algebraically independent generators x,y, which possesses presentations with ladder relations. In this paper, we classify the finite-dimensional irreducible A-modules up to isomorphism by using the theory of the Leonard pairs. For a given irreducible A-module V with dimension d + 1, we give its corresponding isomorphism classes of Leonard pairs on V that have Racah type.
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.
Let K denote a field of characteristic zero and let d denote an integer at least 3. LetA=(−1)d(02d+10102d202d−13.......d+3d−10d+20dd+1)
andA⁎=diag((−1)d(2d+1),…,−7,5,−3,1)
be two matrices in Matd+1(K). Then A,A⁎ is a totally almost bipartite Leonard pair on Kd+1 of Bannai/Ito type. In this paper, we determine all the matrices Aε∈Matd+1(K) such that A,A⁎,Aε form a Leonard triple on Kd+1.
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 .
The concept of cyclic tridiagonal pairs is introduced, and explicit examples are given. For a fairly general class of cyclic tridiagonal pairs with cyclicity N, we associate a pair of `divided polynomials'. The properties of this pair generalize the ones of tridiagonal pairs of Racah type. The algebra generated by the pair of divided polynomials is identified as a higher-order generalization of the Onsager algebra. It can be viewed as a subalgebra of the q-Onsager algebra for a proper specialization at q the primitive 2Nth root of unity. Orthogonal polynomials beyond the Leonard duality are revisited in light of this framework. In particular, certain second-order Dunkl shift operators provide a realization of the divided polynomials at N=2 or q=i.
Let denote an algebraically closed field of characteristic zero. Let V denote a vector space over with finite positive dimension. A Leonard pair on V is an ordered pair of linear transformations in such that for each of these transformations there exists a basis for V with respect to which the matrix representing that transformation is diagonal and the matrix representing the other transformation is irreducible tridiagonal. Whenever these two tridiagonal matrices are almost bipartite, the Leonard pair is said to be totally almost bipartite. The notion of a Leonard triple and the corresponding notion of totally almost bipartite are similarly defined. Let q denote a quantum parameter of a Leonard pair and let ‘TAB’ be an abbreviation for ‘totally almost bipartite’. In this paper we show that a TAB Leonard pair with q equal to is of Bannai/Ito type, and a TAB Leonard pair with q being not a root of unity is of q-Racah type. Under the assumption that q is not a root of unity, we classify, up to isomorphism, the TAB Leonard pairs of q-Racah type and the TAB Leonard triples of q-Racah type.
We introduce the notion of a lowering-raising (or LR) triple of linear transformations on a nonzero finite-dimensional vector space. We show how to normalize an LR triple, and classify up to isomorphism the normalized LR triples. We describe the LR triples using various maps, such as the reflectors, the inverters, the unipotent maps, and the rotators. We relate the LR triples to the equitable presentation of the quantum algebra Uq(sl2) and Lie algebra sl2.
Let double-struck K denote an algebraically closed field of characteristic zero. Let V denote a vector space over double-struck K with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear transformations A, A∗, Aε in End(V) such that for each B ∈ {A, A∗, A∗} there exists a basis for V with respect to which the matrix representing B is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal. In this paper, we define a family of Leonard triples said to have classical type and show that these Leonard triples consist of two families: the Racah type and the Krawtchouk type. Moreover, we construct all Leonard triples that have classical type from the universal enveloping algebra U (sl2).
Let denote an algebraically closed field of characteristic zero. Let V denote a vector space over with finite positive dimension. By a Leonard triple on V, we mean an ordered triple of linear transformations A, A*, A in End(V) such that for each B ∈ {A, A*, A } there exists a basis for V with respect to which the matrix representing B is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal. The diameter of the Leonard triple (A, A*, A ) is defined to be one less than the dimension of V. In this paper we define a family of Leonard triples said to be Bannai/Ito type and classify these Leonard triples with even diameters up to isomorphism. Moreover, we show that each of them satisfies the 3-symmetric Askey–Wilson relations.
For the class of quantum integrable models generated from the $q-$Onsager
algebra, a basis of bispectral multivariable $q-$orthogonal polynomials is
exhibited. In a first part, it is shown that the multivariable Askey-Wilson
polynomials with $N$ variables and $N+3$ parameters introduced by Gasper and
Rahman [GR1] generate a family of infinite dimensional irreducible modules for
the $q-$Onsager algebra, whose fundamental generators are realized in terms of
the multivariable $q-$difference and difference operators proposed by Iliev
[Il]. Raising and lowering operators extending those of Sahi [Sa2] are also
constructed. In a second part, finite dimensional irreducible modules are
constructed and studied for a certain class of parameters and if the $N$
variables belong to a discrete support. In this case, the bispectral property
finds a natural interpretation within the framework of tridiagonal pairs. In a
third part, eigenfunctions of the $q-$Dolan-Grady hierarchy are considered in
the polynomial basis. In particular, invariant subspaces are identified for
certain conditions generalizing Nepomechie's relations. In a fourth part, the
analysis is extended to the special case $q=1$. This framework provides a
$q-$hypergeometric formulation of quantum integrable models such as the open
XXZ spin chain with generic integrable boundary conditions ($q\neq 1$).
Let denote an algebraically closed field of characteristic zero and let denote a vector space over with finite positive dimension. By a Leonard triple on , we mean an ordered triple of linear transformations in such that for each there exists a basis for with respect to which the matrix representing is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal. In this paper, we consider Leonard triples with quantum parameter , where is not a root of unity. We show these Leonard triples are all of -Racah type.
We construct Leonard triples on the irreducible, finite-dimensional -modules using the equitable basis for . We show that the Leonard triples are of either Racah or Krawtchouk type.
Let denote an algebraically closed field of characteristic 0. Let denote a vector space over of finite positive dimension. A Leonard pair on is an ordered pair of linear transformations in such that for each of these transformations there exists a basis for with respect to which the matrix representing that transformation is diagonal and the matrix representing the other transformation is irreducible tridiagonal. The diameter of the Leonard pair is defined as . Let be a Leonard pair of diameter . The Leonard pair is of Krawtchouk type whenever its eigenvalue sequence and its dual eigenvalue sequence are both . The notions of a Leonard triple and a Leonard triple of Krawtchouk type are similarly defined. In this paper, let be a Leonard pair of Krawtchouk type. We first determine all linear transformations in such that is a Leonard triple of Krawtchouk type. Then we classify up to isomorphism Leonard triples of Krawtchouk type.
We introduce the notion of a lowering-raising (or LR) triple of linear
transformations on a nonzero finite-dimensional vector space. We show how to
normalize an LR triple, and classify up to isomorphism the normalized LR
triples. We describe the LR triples using various maps, such as the reflectors,
the inverters, the unipotent maps, and the rotators. We relate the LR triples
to the equitable presentation of the quantum algebra $U_q(\mathfrak{sl}_2)$ and
Lie algebra
$\mathfrak{sl}_2$.
Let (Formula presented.) denote an algebraically closed field, and let x, y, z be the equitable generators of (Formula presented.) over (Formula presented.). Let V denote a finite-dimensional irreducible (Formula presented.) -module, let (Formula presented.) be a linear map. We show that if any two of the matrices representing (Formula presented.) with respect to a standard x-, y- and z-eigenbasis are tridiagonal, then (Formula presented.) acts as a linear combination of 1, x, y, z, xy, zx, yz and yzx.
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.
Let denote an algebraically closed field of characteristic zero. Let V denote a vector space over with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear transformations A, , in such that for each there exists a basis for V with respect to which the matrix representing B is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal. The diameter of the Leonard triple is defined to be one less than the dimension of V. In this paper we define a family of Leonard triples said to have Bannai/Ito type. We classify up to isomorphism the Leonard triples that have Bannai/Ito type and odd diameter. We show that each of these Leonard triples satisfies the -symmetric Askey–Wilson relations.
Let denote an algebraically closed field of characteristic zero. Let V denote a vector space over with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear transformations A, , in such that for each there exists a basis for V with respect to which the matrix representing B is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal. In this paper we define a family of Leonard triples said to have Racah type and classify them up to isomorphism. Moreover, we show that each of them satisfies the -symmetric Askey–Wilson relations. As an application, we construct all Leonard triples that have Racah type from the universal enveloping algebra .
Let denote an algebraically closed field of characteristic zero. Let V denote a vector space over with finite positive dimension. A Leonard pair on V is an ordered pair of linear transformations in such that for each of these transformations there exists a basis for V with respect to which the matrix representing that transformation is diagonal and the matrix representing the other transformation is irreducible tridiagonal. Whenever these tridiagonal matrices are bipartite, the Leonard pair is said to be totally bipartite. The notion of a Leonard triple and the corresponding notion of totally bipartite are similarly defined. In this paper, we first classify up to isomorphism the totally bipartite Leonard pairs. The classification reveals that these Leonard pairs are of the q-Racah, Krawtchouk, or Bannai/Ito type. Then we determine all Leonard triples extended from given totally bipartite Leonard pairs of q-Racah type. Finally, we classify up to isomorphism the totally bipartite Leonard triples of q-Racah type.
Let [Inline formula] denote an algebraically closed field of characteristic zero and let [Inline formula] denote an even at least [Inline formula]. Let and be [Inline formula] by [Inline formula] matrices. Then [Inline formula] is a Leonard pair on [Inline formula] of Bannai/Ito type. We determine all the matrices [Inline formula] such that [Inline formula] form a Leonard triple on [Inline formula].
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).
Let [Inline formula] denote an algebraically closed field with characteristic not two, and let [Inline formula], [Inline formula] and [Inline formula] be the equitable basis of [Inline formula] over [Inline formula]. Let [Inline formula] denote a finite-dimensional irreducible [Inline formula]-module. We show that a linear map [Inline formula] acts as a linear combination of [Inline formula], [Inline formula], [Inline formula] and [Inline formula] if, and only if, all three of the following conditions hold if, and only if, any two of the following conditions hold: (a) the matrix representing [Inline formula] with respect to a standard [Inline formula]-eigenbasis is tridiagonal, (b) the matrix representing [Inline formula] with respect to a standard [Inline formula]-eigenbasis is upper bidiagonal and (c) the matrix representing [Inline formula] with respect to a standard [Inline formula]-eigenbasis is lower bidiagonal.
Let KK denote an algebraically closed field and let V denote a vector space over KK with finite positive dimension. We consider an ordered pair of linear transformations A:V→VA:V→V and A⁎:V→VA⁎:V→V that satisfy the following four conditions: (i) Each of A,A⁎A,A⁎ is diagonalizable; (ii) there exists an ordering {Vi}i=0d of the eigenspaces of A such that A⁎Vi⊆Vi−1+Vi+Vi+1A⁎Vi⊆Vi−1+Vi+Vi+1 for 0⩽i⩽d0⩽i⩽d, where V−1=0V−1=0 and Vd+1=0Vd+1=0; (iii) there exists an ordering {Vi⁎}i=0δ of the eigenspaces of A⁎A⁎ such that AVi⁎⊆Vi−1⁎+Vi⁎+Vi+1⁎ for 0⩽i⩽δ0⩽i⩽δ, where V−1⁎=0 and Vδ+1⁎=0; (iv) there does not exist a subspace W of V such that AW⊆WAW⊆W, A⁎W⊆WA⁎W⊆W, W≠0W≠0, W≠VW≠V. We call such a pair a tridiagonal pair on V. It is known that d=δd=δ; to avoid trivialities assume d⩾1d⩾1. We assume that A,A⁎A,A⁎ belongs to a family of tridiagonal pairs said to have q-Racah type. This is the most general type of tridiagonal pair. Let {Ui}i=0d and {Ui⇓}i=0d denote the first and second split decompositions of V. In an earlier paper we introduced the double lowering operator ψ:V→Vψ:V→V. One feature of ψ is that both ψUi⊆Ui−1ψUi⊆Ui−1 and ψUi⇓⊆Ui−1⇓ for 0⩽i⩽d0⩽i⩽d, where U−1=0U−1=0 and U−1⇓=0. Define linear transformations K:V→VK:V→V and B:V→VB:V→V such that (K−qd−2iI)Ui=0(K−qd−2iI)Ui=0 and (B−qd−2iI)Ui⇓=0 for 0⩽i⩽d0⩽i⩽d. Our results are summarized as follows. Using ψ, K, B we obtain two actions of Uq(sl2)Uq(sl2) on V. For each of these Uq(sl2)Uq(sl2)-module structures, the Chevalley generator e acts as a scalar multiple of ψ. For each of the Uq(sl2)Uq(sl2)-module structures, we compute the action of the Casimir element on V. We show that these two actions agree. Using this fact, we express ψ as a rational function of K±1,B±1K±1,B±1 in several ways. Eliminating ψ from these equations we find that K and B are related by a quadratic equation.
In this paper three new infinite families of linear binary completely regular codes are constructed. They have covering radius ρ = 3 and 4, and are halves of binary Hamming and binary extended Hamming codes of length n = 2m
−1 and 2m
, where m is even. There are also shown some combinatorial (binomial) identities which are new, to our knowledge.These completely regular codes induce, in the usual way, i.e., as coset graphs, three infinite families of distance-regular graphs of diameter three and four. This description of such graphs is new.
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.
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. {\bf 66}, Ser. A (1990), pp. 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 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 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 the continuation of an article from the previous issue. In this part, we focus on the thin P- and Q-polynomial association schemes. We provide some combinatorial characterizations of these objects and exhibit the known examples with diameter at least 6. For each example, we give the irreducible modules for the subconstituent algebra. We close with some conjectures and open problems.
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 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.
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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