International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 3593296 | https://doi.org/10.1155/2020/3593296

Hasan Alnajjar, "A Linear Map Acts as a Leonard Pair with Each of the Generators of ", International Journal of Mathematics and Mathematical Sciences, vol. 2020, Article ID 3593296, 9 pages, 2020. https://doi.org/10.1155/2020/3593296

A Linear Map Acts as a Leonard Pair with Each of the Generators of

Academic Editor: Luca Vitagliano
Received27 May 2020
Accepted08 Jul 2020
Published25 Aug 2020

Abstract

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 [714].

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