数理解析研究所講究録1109 巻 1999 年 67-79

67

An introduction to Leonard pairs andLeonard systems
Paul Terwilliger*
May 11, 1999

Abstract

Let denote a field, and let denote a finite dimensional vector space over $F$ .
We consider an ordered pair $(A, A^{*})$ , where andare -linear transformations
from to that satisfy conditions (i), (ii) below:$V$$\mathcal{F}$

$A$

$V$

$A^{*}$$\mathcal{F}$$V$(i) There exists a basis for $V$ with respect to which the matrix representing
is diagonal, and the matrix representing
is irreducible tridiagonal.

$A$$A^{*}$(ii) There exists a basis for with respect to which the matrix representing
is diagonal, and the matrix representing is irreducible tridiagonal.
$V$$A^{*}$$A$

We call such a pair a Leonard pair on . We present a classification of Leonard
pairs. We obtainnard pairs from irreducible representations of the quantum
Lie algebra. We show any Leonard pair satisfy two polynomial relations
called the Askey-Wilson relations. We obtain Leonard pairs from five families ofclassical posets.
$V$$\mathrm{L}\mathrm{e}o$$U_{q}(sl_{2})$1Introduction

Throughout this talk,$\mathcal{F}$will denote an arbitrary field.

Definition 1.1 Let denote a finite dimensional vector space over . By a Leonard
pair on , we mean an ordered pairwhere andare -linear transformations from to satisfying (i), (ii) below.
$V$$\mathcal{F}$$V$$(A, A^{*})_{f}$

$V$

$A$

$A^{*}$$\mathcal{F}$$V$(i) There exists a basis forwith respect to which the matrix representing$diagonal_{f}$ and the matrix represeniing
is irreducible iridiagonal.
$V$$A^{*}$is$A$(ii) There exists a basis for with respect to which the matrix representing
and the matrix representing
is irre. ducible tridiagonal.
$V$$onal_{j}$*Mathematics Department

Email:terwilli@math.wisc.edu$A$is diag-$A^{*}$

University of Wisconsin

480 Lincoln Drive

Madison, WI 53706

68(A tridiagonal matrix is said to be irreducible whenever all entries immediately aboveand below the main diagonal are nonzero).
Here is an example of a Leonard pair. Set $V=F^{4}$ (column vectors), set$A^{*}=$ ,$A=(_{0}^{0}1030020203000^{\backslash },1’$

as linear transformations on . We assume the characteristic of $F$
andis not 2 or 3, to insure is irreducible. Then $(A, A^{*})$ is a Leonard pair on $V$ . Indeed,
condition (i) of Definition 1.1 is satisfied by the basis for $V$ consisting of the columns of
the 4 by 4 identity matrix. To verify condition (ii), we display an invertible matrix $P$
such that $P^{-1}AP$ is diagonal, and such that $P^{-1}A^{*}P$ is irreducible tridiagonal. Putand view$A$$V$$A^{*}$$A$$P=$
By matrix multiplication $P^{2}=8I$ , so$P^{-1}$.

exists. Also by matrix multiplication,$\mathrm{A}P=P\mathrm{A}^{*}$.

, and is therefor diagonal. By the above line, and since
Apparently $P^{-1}AP$ equals
is a scalar multiple of , we find $P^{-1}A^{*}P$ equals , and is therefor irreducible
tridiagonal. Now condition (ii) of Definition 1.1 is satisfied by the basis for consistingof the columns of .
(and ) are 3, $-1,$ $-3$ ,Referring to the above example, apparently the eigenvalues ofand we observe these are distinct. This will always be the case. In fact, it is an easy
exercise to show the following.
$A^{*}$$A$$P$$P^{-1}$

$V$

$P$

$A$

$A^{*}$$1,$

Lemma 1.2 With reference to Definition 1.1, let $(A, A^{*})$ denote a Leonard pair on .
Then the eigenvatues of are $distinct_{f}$ and contained in F. $Moreover_{2}$ the eigenvaluesof are distinct, and contained in $F$ .
$V$

$A$

$A^{*}$

When studying Leonard pairs, it is often convenient to consider a related and somewhat
more abstract object, which we call a Leonard system. To define this, we need a few
denote the F-algebra
denote a nonnegative integer, and let Mat
terms. Let. We view the rows andconsisting of all $d+1$ by $d+1$ matrices with entries in$A$. For the rest of this talk,will denote an F-algebracolumns as indexed by $0,1,$. An element $A\in A$ will be called multiplicity-free wheneverisomorphic toit has $d+1$ distinct eigenvalues, all of which are in . Assume A is multiplicity free,,and let denote the subalgebra of $A$ generated by . Then has a basis
such that$d$$d+1(\mathcal{F})$$\mathcal{F}$$d$$\ldots,$$\mathrm{M}\mathrm{a}\mathrm{t}_{d+1}(F)$$\mathcal{F}$$A$$\prime D$$(0\leq i,j\leq d)$$E_{i}E_{j}=\delta_{ij}E_{i}$$\sum_{i=0}^{d}E_{i}=I$$D$.

$E_{0},$ $E_{1},$,$\ldots$$E_{d}$

69

The elements
idempotents of$E_{0},$ $E_{1},$$A$$\ldots,$.

$E_{d}$

are unique up to permutation, and are called the primitive

Definition 1.3 Let denote a nonnegative integer, let $F$ denote a field, and let $A$
denote an -algebra isomorphic to $Mat_{d+1}(F)$ . By a Leonard System in $A$, we mean asequence$d$$\mathcal{F}$(1)$\Phi=(A;E_{0}, E_{1}, \ldots, E_{d};A^{*};E_{0}^{*}, E_{1}^{*}, \ldots, E_{d}^{*})$

that satisfies $(i)-(v)$ below.

(ii)
(iii)

are both multiplicity-free elements in$E_{0},$$E_{1},$$\ldots)E_{d}$is an ordering of the primitive idempotents of$A$$E_{0}^{*},$$E_{1}^{*},$,

is an ordering of the primitive idempotents of$A^{*}$$\ldots$$E_{d}^{*}$,

if $|i-j|>1_{j}$, if $|i-j|=1$,
if $|i-j|>1_{j}$, if $|i-j|=1$
as the diameter of , and say(iv) $E_{i}A^{*}E_{j}=\{$(v).

$A^{*}$(i) $A,$

$0$

$A$

$0$

$(0\leq i,j\leq d)$$\neq 0$We refer to$d$$\Phi$$\Phi$

is over$\mathcal{F}$.

.

$(0\leq i,j\leq d)$$\neq 0$$E_{i}^{*}AE_{j}^{*}=\{$

.

.

.

To see the connection between Leonard pairs and Leonard systems, observe conditions(ii), (iv) above assert that with respect to an appropriate basis consisting of eigenvectors
for , the matrix representing
is irreducible tridiagonal. Similarily, conditions (iii),(v) assert that with respect to an appropriate basis consisting of eigenvectors for , thematrix representing is irreducible tridiagonal.
$A$$A^{*}$$A^{*}$$A$

Definition 1.4 Let the Leonard system
be as in (1). We(resp.
) denoie$0\leqi\leq
d$ .
the eigenvalue of(resp.
) associated with(resp.
), forWe
callthe eigenvalue sequence of . We callthe dual eigenvalue
sequence of .
$\Phi$$A$$\theta_{0},$$\theta_{1},$$lei\theta_{i}$$A^{*}$$E_{i}$$\Phi$$\theta_{d}$$\ldots,$$\theta_{i}^{*}$$E_{i}^{*}$$\theta_{0}^{*},$$\theta_{1}^{*},$$\theta_{d}^{*}$$\ldots,$$\Phi$

Given a Leonard system$\Phi=(A;E_{0}, E_{1}, \ldots, E_{d};A^{*}; E_{0)}^{*}E_{1}^{*}, \ldots, E_{d}^{*})$,

we can get more Leonard systems. For example$\Phi^{*}$$\Phi^{\downarrow}$$\Phi^{\Downarrow}$

are Leonard systems.

.

,

$:=$

$(A^{*};E_{0}^{*}, E_{1}^{*}, \ldots, E_{d}^{*};A;E_{0}, E_{1}, \ldots, E_{d})$$:=$$(A;E_{0}, E_{1}, \ldots, E_{d;}A^{*};E_{d}^{*}, E_{d-1}^{*}, \ldots, E_{0}^{*})$$:=$$(A;E_{d}, E_{d-1}, \ldots, E_{0};A^{*};E_{0}^{*}, E_{1}^{*}, \ldots, E_{d}^{*})$$\mathrm{V}\mathrm{i}\mathrm{e}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}*,$$\downarrow,$$\Downarrow \mathrm{a}\mathrm{s}$permutations on the set of$\downarrow*--*\Downarrow,\downarrow\Downarrow=\Downarrow\downarrow*^{2}=\downarrow^{2}=\Downarrow^{2}=1,$.

$\mathrm{a}.11$,

Leonard systems,

70

to the above relations is the dihedral
The group generated bygroup
is the grof symmetries of a square, and has 8 elements.
. We recall
induce an action of
on the set of all Leonard systems. We say two
action.
Leonard systems are relatives whenever they are in the same orbit of this$\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{b}\mathrm{o}\mathrm{l}\mathrm{s}*,$$D_{4}$$D_{4}$$\mathrm{A}\mathrm{p}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{l}\mathrm{y}*,$$\downarrow,$$\downarrow,$$\Downarrow \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}$$o\mathrm{u}\mathrm{p}$$D_{4}$$\Downarrow$$D_{4}$

In view of our above comments, when we discuss Leonard systems, we are often notinterested in the orderings of the primitive idempotents involved; we just care howandinteract. This brings us back to the notion of a Leonard pair.

$A$$A^{*}$

Definition 1.5 Let denote a nonnegative $integer_{J}$ let $F$ denote a field, and let $A$. By a Leonard pair in $A$ , we mean an
denote an $F$ -algebra isomorphic toordered pair $(A, A^{*})$ such that$d$$Mat_{d+1}(\mathcal{F})$(i) $A,$$A^{*}$

are both multiplicity free elements of(ii) There exists an ordering
there exists an orderingthat$E_{0},$$E_{0}^{*},$$E_{1},$$E_{1}^{*},$$\ldots,$$\ldots,$$E_{d}$$E_{d}^{*}$$A_{l}$

andof the primitive idempotents of andof the primitive idempotents of , such
$A_{j}$$A^{*}$$(A;E_{0}, E_{1}, \ldots, E_{d;}A^{*};E_{0}^{*}, E_{1}^{*}, \ldots, E_{d}^{*})$

is a Leonard System.

A classification of Leonard systems

2

When studying a Leonard system , it is often useful to examine a second Leonard
but in a particularily nice form. We present such asystem that is isomorphic to‘canonical form’. To describe it, we use the following notation. Let$\Phi$$\Phi$$\Phi=(A;E_{0}, E_{1}, \ldots, E_{d;}A^{*};E_{0}^{*}, E_{1}^{*}, \ldots, E_{d}^{*})$

denote a Leonard system in $A$ , and let
Then we write$\Phi^{\sigma}:=$and observe$\Phi^{\sigma}$$\sigma$: $Aarrow A’$ denote an isomorphism of -algebras.
$\mathcal{F}$$(A^{\sigma};E_{0}^{\sigma}, E_{1}^{\sigma}, \ldots, E_{d}^{\sigma};A^{*\sigma};E_{0}^{*\sigma}, E_{1}^{*\sigma}, \ldots , E_{d}^{*\sigma})$,

is a Leonard system in $A’$ .

Let us say a matrix$X\in \mathrm{M}\mathrm{a}\mathrm{t}_{d+1}(F)$$X_{ij}\neq 0$$arrow$

is lower$di$-diagonal whenever$i-j\in\{0,1\}$$(0\leq i,j\leq d)$.

That is, $X$ is lower -diagonal whenever each nonzero entry lies either on or immediately
below the main diagonal. We say $X$ is upper -diagonal whenever the transposeislower di-diagonal.
$\mathrm{d}\mathrm{i}$

$di$

Let

$\Phi$

denote the Leonard system in (1). We say
hold below.

$X^{t}$$\Phi$

is in split canonical form whenever$(\mathrm{i})-(\mathrm{i}\mathrm{i}\mathrm{i})$(i) $A=$ Mat(ii)$d+1(\mathcal{F})$.

is lower -diagonal, with $A_{i,i-1}=1$ for $1\leq i\leq d$ , andwhere denotes the eigenvalue of associated with .
$A$$\mathrm{d}\mathrm{i}$$\theta_{i}$$A$$E_{i}$$A_{ii}=\theta_{i}$for $0\leq i\leq d$,

71

is upper -diagonal, withassociated witheigenvalue of(iii)$\mathrm{A}^{*}$$\mathrm{d}\mathrm{i}$$A_{ii}^{*}=\theta_{i}^{*}$$A^{*}$$E_{i}^{*}$.

for $0\leq i\leq d$ , where

We show there there exists a unique isomorphism of $F$-algebras
is in split canonical form. Apparently
such that

$\wp$

:

$\theta_{i}^{*}$

denotes the$Aarrow \mathrm{M}\mathrm{a}\mathrm{t}_{d+1}(F)$$\Phi^{\varphi}$$\theta_{1}1$$A^{*\mathrm{c})}=$ ,,$\theta_{2}$$A^{\mathrm{Q}}=(_{0}^{\theta_{0}}1$$\theta_{d}0\backslash ,$1
whereof . Let
we have$\varphi_{1},$$\Phi$$\varphi_{2},$$\phi_{1},$,$\ldots$$\varphi_{d}$$\phi_{2},$are appropriate scalars in$\phi_{d}$$\ldots,$$\phi_{1},$$\phi_{2},$$\phi_{d}$$\ldots,$. We calldenote the -sequence for$\Phi^{\Downarrow}$$\varphi$$A^{\theta}=$ ,We call$\mathcal{F}$the -sequence of$\phi$,the. Then abbreviating$\varphi_{1},$$\varphi_{2},$$\varphi_{d}$$\ldots$$\varphi$-sequence$\theta:=\wp(\Phi^{\Downarrow})$,$A^{*\theta}=$ .
$\Phi$.

We obtain the following classification of Leonard systems.
Theorem 2.1 [7] Let$d$denote a nonnegative integer, let$\theta_{0},$$\theta_{d;}$$\theta_{1},$$\theta_{0}^{*},$$\theta_{1}^{*},$$\varphi_{1},$$\varphi_{2},$$\ldots,$$\theta_{d}^{*}$$\ldots,$$\ldots,$$\varphi_{d}$;$\phi_{1},$$\mathcal{F}$$\phi_{2},$denote a field, and let;$\phi_{d}$$\ldots,$over $F$ with eigenvalue
denote scalars in . Then there exists a Leonard System,,-sequence, , dual eigenvalue sequence
sequence, if and only if $(i)-(v)$ hold below.
and -sequence$\Phi$$\mathcal{F}$$\theta_{0},$$\theta_{1},$,(i)$\varphi_{i}\neq 0$(ii)$\theta_{i}\neq\theta_{j}$

(iii)
(iv)

$\theta_{d}$$\theta_{0}^{*},$$\phi_{2)}\ldots$$\theta_{i}^{*}\neq\theta_{j}^{*}$$\ldots$$\theta_{df}^{*}\varphi$$\varphi_{1},$$(1\leq i\leq d)$if$i\neq j$,$\varphi_{i}=\phi_{1}\sum_{h=0}^{i-1}\frac{\theta_{h}-\theta_{d-h}}{\theta_{0}-\theta_{d}}+(\theta_{i}^{*}-\theta_{0}^{*})(\theta_{i-1}-\theta_{d})$$\phi_{i}=\varphi_{1}\sum_{h=0}^{\dot{\mathrm{t}}-1}\frac{\theta_{h}-\theta_{d-h}}{\theta_{0}-\theta_{d}}+(\theta_{i}^{*}-\theta_{0}^{*})(\theta_{d-i+1}-\theta_{0})$$\varphi_{d}$$\varphi_{2}.’\ldots,$$\phi_{d}$$\phi_{i}\neq 0$,$\theta_{1}^{*},$$\ldots$$\phi_{1},$$\phi$,$(0\leq i,j\leq d)_{f}$$(1\leq i\leq d)$,$(1\leq i\leq d)$,

72(v) The expressions$\frac{\theta_{i-2}-\theta_{i+1}}{\theta_{i-1}-\theta_{i}}$,(2)$\frac{\theta_{i-2}^{*}-\theta_{i+1}^{*}}{\theta_{i-1}^{*}-\theta_{i}^{*}}$

are equal and independent of , for $2\leq i\leq d-1$ .
$i$$Moreover_{J}$

if $(i)-(v)$ hold above then$\Phi$

is unique up$io$isomorphism of Leonard Systems.

From the above theorem, we routinely obtain the following corollary.
Corollary 2.2 [7] Let denote a nonnegative integer, and let $F$ denote a field. Let
denote any matrices inandof the form$d$$A^{*}$$A$$Mat_{d+1}(\mathcal{F})$$A=$$A^{*}=$ .

,Then the following are equivalent.
(i) $(A, A^{*})$ is a Leonard pair.

(ii) There exist scalars$\phi_{1},$$\phi_{2},$$\ldots$,$\phi_{d}$in$\mathcal{F}$such that conditions $(i)-(v)$ hold in Theorem

2.1.

3

The quantum Lie algebra$U_{q}(sl_{2})$In this section, we obtain Leonard pairs from irreducible representations of the quantum. Throughout this section, we assume our ground field is algeLie algebra
braically closed with characteristic zero. We let denote a nonzero element in $F$ , andassume is not a root of 1.
$\mathcal{F}$$U_{q}(sl_{2})$

$q$

$q$

is the associative $F$-algebra with 1 generated by symbols
Recallject to the relations$U_{q}(sl_{2})$$f,$ $k,$ $k^{-1}$

sub(3)$kk^{-1}=k^{-1}k=1$ ,$ke=q^{2}ek$ ,$e,$$kf=q^{-2}fk$ ,

(4)

.

(5)

$ef– fe= \frac{k-k^{-1}}{q-q^{-1}}$

73

Let

$d$

denote a nonnegative integer, and put$E=$ $F=$
,

,

where
$[i]= \frac{q^{i}-q^{-i}}{q-q^{-1}}$$(\forall i\in \mathbb{Z})$.

Also put$K=$ diag $(q^{d}, q^{d-2}, q^{d-4}, \ldots, q^{-d})$.

Then $K$ is invertible, and $E,$ $F,$ $K$ satisfy the equations (4), (5), so they support a
representation of. It can be shown the representation is irreducible.
$U_{q}(sl_{2})$Let$\alpha$and$\alpha^{*}$

denote nonzero elements in $F$ such that$A=$
$A^{*}$$=$$\alpha\alpha^{*}$

is not a power of , and put$q$$\alpha F+\underline{K}$$q-q^{-1}$’$\alpha^{*}E+\frac{K^{-1}}{q-q^{-1}}$

We claim $(A, A^{*})$ is a Leonard pair. To seeMatsatisfying

, let

.
denote the automorphism of$\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{s}^{r}$$\sigma$$d+1(\mathcal{F})$$X^{\sigma}=D^{-1}XD$

where $D$ is the diagonal matrix in$(\forall X\in \mathrm{M}\mathrm{a}\mathrm{t}_{d+1}(\mathcal{F}))$$\mathrm{M}\mathrm{a}\mathrm{t}_{d+1}(\mathcal{F})$$D_{ii}=[1][2]\cdots[i]\alpha^{i}$,with entries$(0\leq i\leq d)$

.

Then

$\theta_{1}1$$\theta_{2}$$A^{\sigma}=(\theta_{0}01$,$\theta_{d}0/\backslash$$A^{*\sigma}=$ ,

1

where
$\theta_{i}=\frac{q^{d-2i}}{q-q^{-1}}$,$\theta_{i}^{*}=\frac{q^{2i-d}}{q-q^{-1}}$$(0\leq i\leq d)$,$\varphi_{i}=[i][d-i+1]\alpha\alpha^{*}$,$(0\leq i\leq d)$(6),$(1\leq i\leq d)$

(7)

.

(8)

74

Set
$(1\leq i\leq d)$$\phi_{i}=\varphi_{1}\sum_{h=0}^{i-1}\frac{\theta_{h}-\theta_{d-h}}{\theta_{0}-\theta_{d}}+(\theta_{i}^{*}-\theta_{0}^{*})(\theta_{d-i+1}-\theta_{0})$.

Evaluating this using (6) $-(8)$ , we obtain$(1\leq i\leq d)$$\phi_{i}=[i][d-i+1](\alpha\alpha^{*}-q^{2i-d-1})$

satisfy the
One readily checks the above scalars
is a Leonard pair by Corollary 2.2. ApplyingsoLeonard pair. For this example, it turns out$\theta_{i},$$\theta_{i}^{*},$$\varphi_{i},$$\phi_{i}$.

of Theorem 2.1,
, we find $(A, A^{*})$ is a$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}-\mathrm{l}$$(\mathrm{A}^{\sigma}, A^{*\sigma})$$\sigma$,$A^{2}A^{*}-(q^{2}+q^{-2})AA^{*}A+A^{*}A^{2}$$=\omega A+\eta I$$A^{*2}A-(q^{2}+q^{-2})A^{*}AA^{*}+AA^{*2}$$=\omega A^{*}+\eta^{*}I$

,

where
$\eta=\alpha\alpha^{*}\frac{q+q^{-1}}{q-q^{-1}}$,$\eta^{*}=\alpha\alpha^{*}\frac{q+q^{-1}}{q-q^{-1}}$$\omega=-1-\alpha\alpha^{*}(q^{-d-1}+q^{d+1})$

,

.

(9)

We comment there is a second Leonard pair associated withabove, and put
$B$

$=$

$\alpha K-(1-q^{-2})KE$ ,$B^{*}$$=$$\alpha^{*}K^{-1}-(1-q^{-2})K^{-1}F$$U_{q}(sl_{2})$. Let$\alpha^{*}$$\alpha,$

be as

.

Then $(B, B^{*})$ is a Leonard pair. The proof is similar, and omitted.

4

The Askey-Wilson relations

satisfied
In the previous section, we obtained a Leonard pair whose elements
pair ofpair
satisfies
similaraevery
Leonard
It
turns
out
equations.
two polynomial
equations.
$\mathrm{A},$$A^{*}$

Theorem 4.1 [6] Let denote a nonnegative $integer_{f}$ let $F$ denote any field, and let $A$. Let $(A, A^{*})$ denote a Leonard pair in $A$ .
denote an $F$ -algebra isomorphic to$\eta^{*}from$ $F$ such that
Then there exists a sequence of scalars$d$$Mat_{d+1}(\mathcal{F})$$\beta,$$\gamma,$$\gamma^{*},$$\rho,$$\rho^{*},$$\omega,$$\eta,$$A^{2}A^{*}-\beta \mathrm{A}A^{*}A+A^{*}A^{2}-\gamma(AA^{*}+A^{*}A)-\rho A^{*}$$A^{*2}A-\beta A^{*}AA^{*}+AA^{*2}-\gamma^{*}(A^{*}A+AA^{*})-\rho^{*}A$

The sequence is unique if $d\geq 3$ .

$=\gamma^{*}A^{2}+\omega A+\eta I$,$=\gamma A^{*2}+\omega A^{*}+\eta^{*}I$.

75

The above equations are known as the Askey-Wilson relations [1], [2], [3], [4], [5], [9],[10], [11].

Concerning the converse to the above theorem, we have the following.
Theorem 4.2 [6] Let denote a nonnegative integer, let $F$ denote any field, and let
A denote an $F$ -algebra isomorphic to. Let $A,$ denote multiplicity free elements in $A$ , and assume the irreducible $A$ -module is irreducible as an $(A, A^{*})$ -module.
$\eta^{*}from$, and assume $A,$ satisfy the correspondPick any scalars
ing Askey-Wilson relations. Assume further that none of the following $(i)-(iii)$ occur:$d$$Mat_{d+1}(\mathcal{F})$$\beta,$(i)$q$$\gamma,$$\gamma^{*},$$\rho,$$\rho^{*},$$A^{*}$$A^{*}$$\mathcal{F}$$\omega,$$\eta,$

is a primitive $d+1^{st}$ root of 1, where $q+q^{-1}=\beta$ .

(ii)$\beta=2$(iii)$\beta=-2$

and$d+1=char(\mathcal{F})$.

and $d+1=2char(F)$ .

Then $(A, A^{*})$ is a Leonard pair in$A$.

Leonard pairs from the classical posets

5

There is a way to obtain Leonard pairs from the following classical posets: (i) the subset
lattice, (ii) the subspace lattice, (iii) the Hamming semi-lattice, (iv) the attenuatedspaces, (v) the classical polar spaces. For the definitions of these posets, see [8]. The
argument in each case is similar. To illustrate it, we will consider the attenuated spaces
in some detail.
Definition 5.1 Let $F$ denote any field, let denote a finite dimensional vector spaceover , and let and denote -linear iransformations from to V. We say $(A, A^{*})$
is a generalized Leonard pair on whenever there exists a decomposition$V$$\mathcal{F}$$A$$A^{*}$$V$$\mathcal{F}$$V$(direct sum),$V=V_{1}+V_{2}+\cdots+V_{n}$such that$AV_{i}\subseteq V_{i}$,$A^{*}V_{i}\subseteq V_{i}$,$(1 \leq i\leq n)$

and such that$(A|_{V;}, A^{*}|_{V}.\cdot)$is a Leonard Pair$(1 \leq i\leq n)$.

The posets mentioned above all suppoTt generalized Leonard pairs. In each case, theunderlying vector space has the following form. Let $X$ denote a finite set. By $FX$ ,we mean the vector space over $F$ consisting of all formal sums
$V$$\sum_{x\in X}\alpha_{x}x$

,

76

where

$\alpha_{x}\in F$

for all $x\in X$ .

We will be discussing posets, so let us recall some terms. let denote a poset. For all$y\in P$ , we saycovers whenever $x<y$ , and there does not exist $z\in P$ such thatwith
$x<z<y$ . In this case, we write $x\prec y$ . Let denote the matrix in Mat
$P$

$x,$

$x$

$y$

$L$

$P(\oplus)$

entries

1, if $x\prec y$ ;, if $x\neq y$$L_{xy}=\{$$(\forall x, y\in P)$$0$

Viewingas a linear transformation on$L$$\mathbb{C}P$,.

$(\forall x\in P)$$Lx= \sum_{y\in P,y\prec x}y$

We call$L$the lowering matrix on$R_{xy}=\{$

$P$

. Let

$R$

denote the matrix in Mat

1, if $y\prec x$ ,, if $y\# x$$(\forall x, y\in P)$$0$

Viewing$R$

as a linear transformation on$\Phi 1P$$(\forall x\in P)$

the raising matrix on . Now assume$0\leq i\leq N$ , letdenote the diagonal matrix in
$P$

$R$

$F_{i}$$(F_{i})_{yy}=\{$

1,
,

$0$

We refer to$F_{i}$

as the$i^{\mathrm{t}\mathrm{h}}$$P(\oplus)$with entries.

,$Rx= \sum_{y\in P,x\prec y}y$

We call.

$P$

is ranked, with rank denoted $N$ . Forwith $yy$ entry$\mathrm{M}\mathrm{a}\mathrm{t}_{P}(\oplus)$

if rank $(y)=i$ ;if rank $(y)\neq i$projection matrix of$P$.

$(\forall y\in P)$. We observe$(0\leq i,j\leq N)$$F_{i}F_{j}=\delta_{ij}F_{i}$.

$F_{0}+F_{1}+\cdots+F_{N}=I$,.

Moreover,$(0\leq i\leq N)$$F_{i}V=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{x\in P|\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(x)=i\}$,where $V=\oplus P$ .
For each of the five families of classical posets we mentioned at the outset, we obtainof the form
generalized Leonard pairs on$V=\mathbb{C}P$

,

$A$

$=$

$\alpha R+\sum_{i=0}^{N}\theta_{i}F_{i}$$A^{*}$$=$$\alpha^{*}L+\sum_{i=0}^{N}\theta_{i}^{*}F_{i}$

(10)

,

(11)

77

where the$\alpha,$$\alpha^{*},$$\theta_{i},$$\theta_{i}^{*}$are complex scalars.

To illustrate, we now restrict our attention to the attenuated space poset. This$N$$M$
poset is defined as follows. Let
anddenote nonnegative integers, let $H$ denote a$\mathrm{A}_{q}(N, M)$

vector space of dimension $M+N$ over $GF(q))$ and fix a subspaceof dimension$M$ . Let$H$$x\caph=0$ . The
denote the poset consisting of all subspaces of such that$h\subseteq H$

$P$

$x$

partial order on $P$ iswhenever$x\leq y$

The poset$P$$(\forall x, y\in P)$$x\subseteq y$is ranked, withrank $(x)=\dim(x)$Apparently,.

$P$

has rank$NJ$$(\forall x\in P)$.

. For $0\leq i\leq N$ , each rank element of
$i$

$P$

covers exactly$\frac{q^{i}-1}{q-1}$

elements of$P$, and is covered by exactly$\frac{q^{N+M-i}-q^{M}}{q-1}$

elements in $P$ . Moreover, it is shown in [8] that(12)$\frac{q}{q+1}RL^{2}-LRL+\frac{1}{q+1}L^{2}R+f_{i}L$

vanishes on
and where$F_{i}V$, where

and

$R$

$L$

are the raising and lowering matrices, where $V=\oplus P$ ,$f_{i}=q^{N+M-i}$

.

(13)

Put
,

(14)

$A$

$=$

$R+ \sum_{i=0}^{N}\frac{q^{i}}{q-1}F_{i}$$A^{*}$$=$$\alpha^{*}L+\sum_{i=0}^{N}\frac{q^{-i}}{q-1}F_{i}$

,

(15)

$q^{-M-N}$ . We show $(A, A^{*})$
where is any scalarthat is not one of $q^{-M-1},$ $q^{-M-2},$
is a generalized Leonard pair on . Let denote the subalgebra of Mat
generated$L,$$R,$$R^{t}=L$by,. Observe, and each of,is symmetric, so
is closed under the conjugate-transpose map. It follows is semi-simple, so $V$ is adirect sum of irreducible -submodules. Let $W$ denote an irreducible -submodule of
V. The matrices andare contained in by (14), (15), so$\alpha^{*}$$\mathrm{i}\mathrm{n}\oplus$$\ldots,$$V$$F_{0},$ $F_{1},$$\ldots$$T$$P(\oplus)$$F_{N}$$F_{0},$ $F_{1},$$T$$\ldots$$F_{N}$

$T$

$T$

$A$

$T$

$T$

$A^{*}$,$AW\subseteq W$.

$\mathrm{A}^{*}W\subseteq W$

78

It remains to show that$(A|_{W}, A^{*}|_{W})$is a Leonard pair on $W$ . We do this as follows. Using (12), one can show there exists,
for $W$ such that
integers $r,p(0\leq r\leq p\leq N)$ and a basis$(r\leq i\leq p)$ ,(i) $w_{i}\in F_{i}V$$w_{r},$(ii) $Rw_{i}=w_{i+1}$$(r\leq i<p)$ ,$Rw_{p}=0$$(r<i\leq p)$ ,(iii) $Lw_{i}=x_{i}(7^{\cdot},p)w_{i-1}$$w_{r+1},$ $\ldots$$w_{p}$,$Lw_{r}=0$

,

where
(16)

$x_{i}(r,p)= \frac{q^{M+N-\mathrm{r}-p-i+1}(q^{i}-q^{f})(q^{p}-q^{i-1})}{(q-1)^{2}}$for $r<i\leq p$ . Let
respect to the basis$B$$w_{r},$(resp.

$B^{*}$) denote the matrix representing. Apparently$A$(resp.

$A^{*}$) with$w_{r+1},$ $\ldots,$ $w_{p}$$B=$$B^{*}=$,,

where $d=p-r$ , and where$\theta_{i}=\frac{q^{r+i}}{q-1}$,$(0\leq i\leq d)$$\theta_{i}^{*}=\frac{q^{-r-i}}{q-1}$$(1 \leq i\leq d)$$\varphi_{i}=\alpha^{*}x_{r+i}(r,p)$

(17)

,

.

(18)

Set
$\phi_{i}=\varphi_{1}\sum_{h=0}^{i-1}\frac{\theta_{h}-\theta_{d-h}}{\theta_{0}-\theta_{d}}+(\theta_{i}^{*}-\theta_{0}^{*})(\theta_{d-i+1}-\theta_{0})$$(1\leq i\leq d)$.

Evaluating this using (16), (17), and (18) we obtain$\phi_{i}=-\frac{(1-q^{i})(1-q^{d-i+1})(1-\alpha^{*}q^{M+N+i-r-d})}{(q-1)^{2}q^{i}}$One readily checks the above scalars
so $(B, B^{*})$ is a Leonard pair in$\theta_{i},$$\theta_{i}^{*},$$\mathrm{M}\mathrm{a}\mathrm{t}_{d+1}(\mathcal{F})$$(1\leq i\leq d)$.

satisfy the conditions of Theorem 2.1,
by Corollary 2.2. It follows$\varphi_{i},$$\phi_{i}$$(A|_{W}, \mathrm{A}^{*}|_{W})$is a Leonard pair on $W$ . We have now shown $(A, A^{*})$ is a generalized Leonard pair on
V. We remark that by (12), (14), (15), we have$[A, A^{2}A^{*}-(q+q^{-1})AA^{*}A+A^{*}A^{2}]=0$ ,$[A^{*}, A^{*2}A-(q+q^{-1})A^{*}AA^{*}+AA^{*2}]=0$

for this example.

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