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 .

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 ,