A classification of sharp tridiagonal pairs

0.41 MB·English

by  Tatsuro Ito

A classification of sharp
tridiagonal pairs
Tatsuro Ito∗, Kazumasa Nomura, and Paul Terwilliger
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1
0
2 Abstract
n Let F denote a field and let V denote a vector space over F with finite positive dimen-
a sion. We consider a pair of linear transformations A : V → V and A∗ : V → V that
J
satisfy the following conditions: (i) each of A,A∗ is diagonalizable; (ii) there exists an
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1 ordering{Vi}di=0 oftheeigenspaces ofAsuchthatA∗Vi ⊆ Vi−1+Vi+Vi+1 for0 ≤ i ≤ d,
] where V−1 = 0 and Vd+1 = 0; (iii) there exists an ordering {Vi∗}δi=0 of the eigenspaces
A of A∗ such that AV∗ ⊆ V∗ +V∗ +V∗ for 0 ≤ i ≤ δ, where V∗ = 0 and V∗ = 0;
i i−1 i i+1 −1 δ+1
R (iv) there is no subspace W of V such that AW ⊆ W, A∗W ⊆ W, W 6= 0, W 6= V.
. We call such a pair a tridiagonal pair on V. It is known that d = δ and for 0 ≤ i ≤ d
h
at dthime dVim=en1si.onIts iosfkVni,oVwdn−it,hVai∗t,iVfd∗F−iiscoailngceibdrea.icTahlleypcaloirseAd,tAh∗enisAc,aAlle∗disshsahraprpw.hIennetvheisr
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paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we
[
classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We

1
obtain these classifications by proving the µ-conjecture.
v
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1 Keywords. Tridiagonal pair, Leonard pair, q-Racah polynomials.
8 2000 Mathematics Subject Classification. Primary: 15A21. Secondary: 05E30,
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05E35.
.
1
0
0
1 Tridiagonal pairs
1
:
v
i Throughout this paper F denotes a field and F denotes the algebraic closure of F. An algebra
X
is meant to be associative and have a 1.
r
a
We begin by recalling the notion of a tridiagonal pair. We will use the following terms. Let
V denote a vector space over F with finite positive dimension. For a linear transformation
A : V → V and a subspace W ⊆ V, we call W an eigenspace of A whenever W 6= 0 and
there exists θ ∈ F such that W = {v ∈ V | Av = θv}; in this case θ is the eigenvalue
of A associated with W. We say that A is diagonalizable whenever V is spanned by the
eigenspaces of A.
Definition 1.1 [27, Definition 1.1] Let V denote a vector space over F with finite positive
dimension. By a tridiagonal pair (or TD pair) on V we mean an ordered pair of linear
transformations A : V → V and A∗ : V → V that satisfy the following four conditions.
∗
Supported in part by JSPS grant 18340022.
1
(i) Each of A,A∗ is diagonalizable.
(ii) There exists an ordering {V }d of the eigenspaces of A such that
i i=0
A∗V ⊆ V +V +V 0 ≤ i ≤ d, (1)
i i−1 i i+1
where V = 0 and V = 0.
−1 d+1
(iii) There exists an ordering {V∗}δ of the eigenspaces of A∗ such that
i i=0
AV∗ ⊆ V∗ +V∗ +V∗ 0 ≤ i ≤ δ, (2)
i i−1 i i+1
where V∗ = 0 and V∗ = 0.
−1 δ+1
(iv) There does not exist a subspace W of V such that AW ⊆ W, A∗W ⊆ W, W 6= 0,
W 6= V.
We say the pair A,A∗ is over F. We call V the underlying vector space.
Note 1.2 AccordingtoacommonnotationalconventionA∗ denotestheconjugate-transpose
of A. We are not using this convention. In a TD pair A,A∗ the linear transformations A
and A∗ are arbitrary subject to (i)–(iv) above.
We now give some background on TD pairs; for more information we refer the reader to
the survey [77]. The concept of a TD pair originated in algebraic graph theory, or more
precisely, the theory of Q-polynomial distance-regular graphs. The concept is implicit in [7,
p. 263], [43] and more explicit in [69, Theorem 2.1]. A systematic study began in [27].
Some notable papers on the topic are [6,13,28–31,35–37,71]. There are connections to
representation theory [2,9,21,26,29,31,39,41,42,64,65,75], partially ordered sets [73], the
bispectral problem [5,6,22–24,82], statistical mechanical models [8–14,17–20,62], and other
areas of physics [61,81,83].
LetA,A∗ denoteaTDpaironV,asinDefinition1.1. By[27,Lemma4.5]theintegersdandδ
from(ii), (iii) areequal; we call thiscommon valuethe diameterof thepair. By [27, Theorem
10.1] the pair A,A∗ satisfy two polynomial equations called the tridiagonal relations; these
generalize the q-Serre relations [72, Example 3.6] and the Dolan-Grady relations [72, Exam-
ple 3.2]. See [13,24,35,37,45,70,72,75,79,80] for results on the tridiagonal relations. An
ordering of the eigenspaces of A (resp. A∗) is said to be standard whenever it satisfies (1)
(resp. (2)). We comment on the uniqueness of the standard ordering. Let {V }d denote
i i=0
a standard ordering of the eigenspaces of A. By [27, Lemma 2.4], the ordering {V }d is
d−i i=0
also standard and no further ordering is standard. A similar result holds for the eigenspaces
of A∗. Let {V }d (resp. {V∗}d ) denote a standard ordering of the eigenspaces of A (resp.
i i=0 i i=0
A∗). For 0 ≤ i ≤ d let θ (resp. θ∗) denote the eigenvalue of A (resp. A∗) associated with V
i i i
(resp. V∗). By [27, Theorem 11.1] the expressions
i
θ −θ θ∗ −θ∗
i−2 i+1, i−2 i+1
θ −θ θ∗ −θ∗
i−1 i i−1 i
2
are equal and independent of i for 2 ≤ i ≤ d − 1. We call the sequence {θ }d (resp.
i i=0
{θ∗}d ) the eigenvalue sequence (resp. dual eigenvalue sequence) for the given standard
i i=0
orderings. See [27,36,63,71,72] for results on the eigenvalues and dual eigenvalues. By [27,
Corollary 5.7], for 0 ≤ i ≤ d the spaces V , V∗ have the same dimension; we denote this
i i
common dimension by ρ . By [27, Corollaries 5.7, 6.6] the sequence {ρ }d is symmetric and
i i i=0
unimodal; that is ρ = ρ for 0 ≤ i ≤ d and ρ ≤ ρ for 1 ≤ i ≤ d/2. By [59, Theorem 1.3]
i d−i i−1 i
we have ρ ≤ ρ d for 0 ≤ i ≤ d. We call the sequence {ρ }d the shape of A,A∗.
i 0(cid:0)i(cid:1) i i=0
See [28,38,46,47,55,59] for results on the shape. The TD pair A,A∗ is called sharp whenever
ρ = 1. By [57, Theorem 1.3], if F is algebraically closed then A,A∗ is sharp. In any case
0
A,A∗ can be “sharpened” by replacing F with a certain field extension K of F that has index
[K : F] = ρ [38, Theorem 4.12]. Suppose that A,A∗ is sharp. Then by [57, Theorem 1.4],
0
there exists a nondegenerate symmetric bilinear form h, i on V such that hAu,vi = hu,Avi
and hA∗u,vi = hu,A∗vi for all u,v ∈ V. See [1,55,67,68] for results on the bilinear form.
The following special cases of TD pairs have been studied extensively. In [78] the TD pairs
of shape (1,2,1) are classified and described in detail. A TD pair of shape (1,1,...,1) is
called a Leonard pair [71, Definition 1.1], and these are classified in [71, Theorem 1.9]. This
classification yields a correspondence between the Leonard pairs and a family of orthogonal
polynomials consisting of the q-Racah polynomials and their relatives [4,74,76]. This family
coincides with the terminating branch of the Askey scheme [40]. See [15,16,44,48–54,77]
and the references therein for results on Leonard pairs. Our TD pair A,A∗ is said to have
Krawtchouk type (resp. q-geometric type) whenever {d − 2i}d (resp. {qd−2i}d ) is both
i=0 i=0
an eigenvalue sequence and dual eigenvalue sequence for the pair. In [26, Theorems 1.7,
F F
1.8] Hartwig classified the TD pairs over that have Krawtchouk type, provided that
is algebraically closed with characteristic zero. By [26, Remark 1.9] these TD pairs are in
bijection with the finite-dimensional irreducible modules for the three-point loop algebra
sl ⊗F[t,t−1,(t−1)−1]. See [25,26,32,33,36] for results on TD pairs of Krawtchouk type. In
2
[30,Theorems 1.6,1.7] we classified theTD pairsover F thathave q-geometrictype, provided
that F is algebraically closed and q is not a root of unity. By [31, Theorems 10.3, 10.4]
these TD pairs are in bijection with the type 1, finite-dimensional, irreducible modules for
the F-algebra ⊠ ; this is a q-deformation of sl ⊗ F[t,t−1,(t − 1)−1] as explained in [31].
q 2
See [2,3,28–31,34,36] for results on q-geometric TD pairs. There is a general family of
TD pairs said to have q-Racah type; these have an eigenvalue sequence and dual eigenvalue
sequence of the form (11)–(15) below. The Leonard pairs of q-Racah type correspond to the
q-Racah polynomials [76, Example 5.3]. In [37, Theorem 3.3] we classified the TD pairs over
F that have q-Racah type, provided that F is algebraically closed. See [35–37] for results on
TD pairs of q-Racah type.
Turning to the present paper, in our main result we classify up to isomorphism the sharp
TD pairs. Here is a summary of the argument. In [33, Conjecture 14.6] we conjectured
how a classification of all the sharp TD pairs would look; this is the classification conjec-
ture. Shortly afterwards we introduced a conjecture, called the µ-conjecture, which implies
the classification conjecture. The µ-conjecture is roughly described as follows. Start with a
sequence p = ({θ }d ;{θ∗}d ) of scalars taken from F that satisfy the known constraints
i i=0 i i=0
on the eigenvalues of a TD pair over F of diameter d; these are conditions (i), (ii) in The-
orem 3.1 below. Following [57, Definition 2.4] we associate with p an F-algebra T defined
3
by generators and relations; see Definition 3.4 for the precise definition. We are interested
in the F-algebra e∗Te∗ where e∗ is a certain idempotent element of T. Let {x }d denote
0 0 0 i i=1
mutually commuting indeterminates. Let F[x ,...,x ] denote the F-algebra consisting of
1 d
the polynomials in {x }d that have all coefficients in F. In [58, Corollary 6.3] we displayed
i i=1
a surjective F-algebra homomorphism µ : F[x ,...,x ] → e∗Te∗. The µ-conjecture [58, Con-
1 d 0 0
jecture 6.4] asserts that µ is an isomorphism. By [58, Theorem 10.1] the µ-conjecture implies
the classification conjecture. In [58, Theorem 12.1] we showed that the µ-conjecture holds
for d ≤ 5. In [60, Theorem 5.3] we showed that the µ-conjecture holds for the case in which
p has q-Racah type. In the present paper we combine this fact with some algebraic geometry
to prove the µ-conjecture in general. The µ-conjecture (now a theorem) is given in Theorem
3.9. Theorem 3.9 implies the classification conjecture, and this yields our classification of
the sharp TD pairs. The classification is given in Theorem 3.1. As a corollary, we classify
up to isomorphism the TD pairs over an algebraically closed field. This result can be found
in Corollary 18.1.
Section 3 contains the precise statements of our main results. In Section 2 we review the
concepts needed to make these statements.
2 Tridiagonal systems
When working with a TD pair, it is often convenient to consider a closely related object
called a TD system. To define a TD system, we recall a few concepts from linear algebra.
Let V denote a vector space over F with finite positive dimension. Let End(V) denote the
F-algebra of all linear transformations from V to V. Let A denote a diagonalizable element
of End(V). Let {V }d denote an ordering of the eigenspaces of A and let {θ }d denote
i i=0 i i=0
the corresponding ordering of the eigenvalues of A. For 0 ≤ i ≤ d define E ∈ End(V) such
i
that (E − I)V = 0 and E V = 0 for j 6= i (0 ≤ j ≤ d). Here I denotes the identity of
i i i j
End(V). We call E the primitive idempotent of A corresponding to V (or θ ). Observe
i i i
d
that (i) I = E ; (ii) E E = δ E (0 ≤ i,j ≤ d); (iii) V = E V (0 ≤ i ≤ d); (iv)
Pi=0 i i j i,j i i i
d
A = θ E . Here δ denotes the Kronecker delta. Note that
Pi=0 i i i,j
A−θ I
j
Ei = Y 0 ≤ i ≤ d. (3)
θ −θ
i j
0≤j≤d
j6=i
Observe that each of {Ai}d , {E }d is a basis for the F-subalgebra of End(V) generated
i=0 i i=0
by A. Moreover d (A−θ I) = 0. Now let A,A∗ denote a TD pair on V. An ordering of
Qi=0 i
the primitive idempotents of A (resp. A∗) is said to be standard whenever the corresponding
ordering of the eigenspaces of A (resp. A∗) is standard.
Definition 2.1 [27, Definition 2.1] Let V denote a vector space over F with finite positive
dimension. By a tridiagonal system (or TD system) on V we mean a sequence
Φ = (A;{E }d ;A∗;{E∗}d )
i i=0 i i=0
that satisfies (i)–(iii) below.
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(i) A,A∗ is a TD pair on V.
(ii) {E }d is a standard ordering of the primitive idempotents of A.
i i=0
(iii) {E∗}d is a standard ordering of the primitive idempotents of A∗.
i i=0
We say that Φ is over F. We call V the underlying vector space.
The following result is immediate from lines (1), (2) and Definition 2.1.
Lemma 2.2 Let (A;{E }d ;A∗;{E∗}d ) denote a TD system. Then the following hold for
i i=0 i i=0
0 ≤ i,j,k ≤ d.
(i) E∗AkE∗ = 0 if k < |i−j|;
i j
(ii) E A∗kE = 0 if k < |i−j|.
i j
The notion of isomorphism for TD systems is defined in [55, Section 3].
Definition 2.3 Let Φ = (A;{E }d ;A∗; {E∗}d ) denote a TD system on V. For 0 ≤ i ≤ d
i i=0 i i=0
let θ (resp. θ∗) denote the eigenvalue of A (resp. A∗) associated with the eigenspace E V
i i i
(resp. E∗V). We call {θ }d (resp. {θ∗}d ) the eigenvalue sequence (resp. dual eigenvalue
i i i=0 i i=0
sequence) of Φ. Observe that {θ }d (resp. {θ∗}d ) are mutually distinct and contained in
i i=0 i i=0
F. We call Φ sharp whenever the TD pair A,A∗ is sharp.
The following notation will be useful.
Definition 2.4 Let x denote an indeterminate and let F[x] denote the F-algebra consisting
of the polynomials in x that have all coefficients in F. Let {θ }d and {θ∗}d denote scalars
i i=0 i i=0
in F. For 0 ≤ i ≤ d define the following polynomials in F[x]:
τ = (x−θ )(x−θ )···(x−θ ),
i 0 1 i−1
η = (x−θ )(x−θ )···(x−θ ),
i d d−1 d−i+1
τ∗ = (x−θ∗)(x−θ∗)···(x−θ∗ ),
i 0 1 i−1
η∗ = (x−θ∗)(x−θ∗ )···(x−θ∗ ).
i d d−1 d−i+1
Note that each of τ , η , τ∗, η∗ is monic with degree i.
i i i i
We now recall the split sequence of a sharp TD system. This sequence was originally defined
in [33, Section 5] using the split decomposition [27, Section 4], but in [58] an alternate
definition was introduced that is more convenient to our purpose.
Definition 2.5 [58, Definition 2.5] Let (A;{E }d ;A∗;{E∗}d ) denote a sharp TD system
i i=0 i i=0
over F, with eigenvalue sequence {θ }d and dual eigenvalue sequence {θ∗}d . By [57,
i i=0 i i=0
Lemma 5.4], for 0 ≤ i ≤ d there exists a unique ζ ∈ F such that
i
ζ E∗
E∗τ (A)E∗ = i 0 .
0 i 0 (θ∗ −θ∗)(θ∗ −θ∗)···(θ∗ −θ∗)
0 1 0 2 0 i
Note that ζ = 1. We call {ζ }d the split sequence of the TD system.
0 i i=0
5
Definition 2.6 Let Φ denote a sharp TD system. By the parameter array of Φ we mean the
sequence ({θ }d ;{θ∗}d ;{ζ }d ) where {θ }d (resp. {θ∗}d ) is the eigenvalue sequence
i i=0 i i=0 i i=0 i i=0 i i=0
(resp. dual eigenvalue sequence) of Φ and {ζ }d is the split sequence of Φ.
i i=0
The following result shows the significance of the parameter array.
Proposition 2.7 [35], [57, Theorem 1.6] Two sharp TD systems over F are isomorphic if
and only if they have the same parameter array.
3 Statement of results
In this section we state our main results. The first result below resolves [33, Conjecture 14.6].
Theorem 3.1 Let d denote a nonnegative integer and let
({θ }d ;{θ∗}d ;{ζ }d ) (4)
i i=0 i i=0 i i=0
F F
denote a sequence of scalars taken from . Then there exists a sharp TD system Φ over
with parameter array (4) if and only if (i)–(iii) hold below.
(i) θ 6= θ , θ∗ 6= θ∗ if i 6= j (0 ≤ i,j ≤ d).
i j i j
(ii) The expressions
θ −θ θ∗ −θ∗
i−2 i+1, i−2 i+1 (5)
θ −θ θ∗ −θ∗
i−1 i i−1 i
are equal and independent of i for 2 ≤ i ≤ d−1.
(iii) ζ = 1, ζ 6= 0, and
0 d
d
0 6= Xηd−i(θ0)ηd∗−i(θ0∗)ζi.
i=0
Suppose (i)–(iii) hold. Then Φ is unique up to isomorphism of TD systems.
In [58, Conjecture 6.4] we stated a conjecture called the µ-conjecture, and we proved that the
µ-conjecture implies Theorem 3.1. To obtain Theorem 3.1 we will prove the µ-conjecture.
We now explain this conjecture.
Definition 3.2 Let d denote a nonnegative integer and let ({θ }d ;{θ∗}d ) denote a se-
i i=0 i i=0
F
quenceofscalarstakenfrom . Thissequenceiscalledfeasiblewhenever itsatisfiesconditions
(i), (ii) of Theorem 3.1.
Definition 3.3 For all integers d ≥ 0 let Feas(d,F) denote the set of all feasible sequences
({θ }d ;{θ∗}d ) of scalars taken from F.
i i=0 i i=0
6
Definition 3.4 [57,Definition2.4] Fixanintegerd ≥ 0andasequencep = ({θ }d ;{θ∗}d )
i i=0 i i=0
in Feas(d,F). Let T = T(p,F) denote the F-algebra defined by generators a, {e }d , a∗,
i i=0
{e∗}d and relations
i i=0
e e = δ e , e∗e∗ = δ e∗ 0 ≤ i,j ≤ d, (6)
i j i,j i i j i,j i
d d
1 = Xei, 1 = Xe∗i, (7)
i=0 i=0
d d
a = Xθiei, a∗ = Xθi∗e∗i, (8)
i=0 i=0
e∗ake∗ = 0 if k < |i−j| 0 ≤ i,j,k ≤ d, (9)
i j
e a∗ke = 0 if k < |i−j| 0 ≤ i,j,k ≤ d. (10)
i j
Lemma 3.5 [58, Lemma 4.2] In the algebra T from Definition 3.4, the elements {e }d
i i=0
are linearly independent and the elements {e∗}d are linearly independent.
i i=0
The algebra T is related to TD systems as follows.
Lemma 3.6 [57, Lemma 2.5] Let V denote a vector space over F with finite positive di-
mension. Let (A;{E }d ;A∗;{E∗}d ) denote a TD system on V with eigenvalue sequence
i i=0 i i=0
{θ }d and dual eigenvalue sequence {θ∗}d . Let T denote the F-algebra from Definition
i i=0 i i=0
3.4 corresponding to ({θ }d ;{θ∗}d ). Then there exists a unique T-module structure on V
i i=0 i i=0
such that a, e , a∗, e∗ acts as A, E , A∗, E∗ respectively. This T-module is irreducible.
i i i i
Fixanintegerd ≥ 0andasequencep ∈ Feas(d,F). LetT = T(p,F)denotethecorresponding
algebra from Definition 3.4. Observe that e∗Te∗ is an F-algebra with multiplicative identity
0 0
e∗.
0
Lemma 3.7 [57, Theorem 2.6] With the above notation, the algebra e∗Te∗ is commutative
0 0
and generated by
e∗τ (a)e∗ 1 ≤ i ≤ d.
0 i 0
Corollary 3.8 [58, Corollary 6.3] With the above notation, there exists a surjective F-
algebra homomorphism µ : F[x ,...,x ] → e∗Te∗ that sends x 7→ e∗τ (a)e∗ for 1 ≤ i ≤ d.
1 d 0 0 i 0 i 0
In [58, Conjecture 6.4] we conjectured that the map µ from Corollary 3.8 is an isomorphism.
This is the µ-conjecture. The following result resolves the µ-conjecture.
Theorem 3.9 Fix an integer d ≥ 0 and a sequence p ∈ Feas(d,F). Let T = T(p,F) denote
the corresponding algebra from Definition 3.4. Then the map µ : F[x ,...,x ] → e∗Te∗ from
1 d 0 0
Corollary 3.8 is an isomorphism.
We will prove Theorem 3.1 and Theorem 3.9 in Section 17.
7
4 The q-Racah case
Our proof of Theorem 3.9 will use the fact that the theorem is known to be true in a special
case called q-Racah [60, Theorem 5.3]. In this section we describe the q-Racah case. We
start with some comments about the feasible sequences from Definition 3.2.
Lemma 4.1 Assume F is infinite. Then forall integersd ≥ 0 the setFeas(d,F)is nonempty.
Proof: Consider thepolynomial d (xn−1)inF[x]. SinceFisinfinitethereexistsanonzero
Qn=1
ϑ ∈ F that is not a root of this polynomial. Define θ = ϑi and θ∗ = ϑi for 0 ≤ i ≤ d. Then
i i
({θ }d ;{θ∗}d ) satisfies the conditions (i), (ii) of Theorem 3.1 and is therefore feasible.
i i=0 i i=0
2
The result follows.
Fix an integer d ≥ 0 and a sequence ({θ }d ;{θ∗}d ) in Feas(d,F). This sequence must
i i=0 i i=0
satisfy condition (ii) in Theorem 3.1. For this constraint the “most general” solution is
θ = α+bq2i−d +cqd−2i 0 ≤ i ≤ d, (11)
i
θ∗ = α∗ +b∗q2i−d +c∗qd−2i 0 ≤ i ≤ d, (12)
i
q, α, b, c, α∗, b∗, c∗ ∈ F, (13)
q 6= 0, q2 6= 1, q2 6= −1. (14)
We have a few comments about this solution. For the moment define β = q2 + q−2, and
observethatβ+1isthecommonvalueof(5). Wehaveβ−2 = (q−q−1)2 andβ+2 = (q+q−1)2.
Therefore β 6= 2, β 6= −2 in view of (14). Using (11), (12) we obtain
(θ −θ )2 −β(θ −θ )(θ −θ )+(θ −θ )2
0 1 0 1 1 2 1 2
bc = ,
(β −2)2(β +2)
(θ∗ −θ∗)2 −β(θ∗ −θ∗)(θ∗ −θ∗)+(θ∗ −θ∗)2
b∗c∗ = 0 1 0 1 1 2 1 2
(β −2)2(β +2)
provided d ≥ 2. We will focus on the case
bb∗cc∗ 6= 0. (15)
Definition 4.2 [37,Definition3.1] Letddenoteanonnegativeintegerandlet({θ }d ;{θ∗}d )
i i=0 i i=0
denote a sequence of scalars taken from F. We call this sequence q-Racah whenever the fol-
lowing (i), (ii) hold:
(i) θ 6= θ , θ∗ 6= θ∗ if i 6= j (0 ≤ i,j ≤ d);
i j i j
(ii) there exist q,α,b,c,α∗,b∗,c∗ that satisfy (11)–(15).
Definition 4.3 For all integers d ≥ 0 let Rac(d,F) denote the set of all q-Racah sequences
({θ }d ;{θ∗}d ) of scalars taken from F.
i i=0 i i=0
Observe that the set Rac(d,F) from Definition 4.3 is contained in the set Feas(d,F) from
Definition 3.3. In the next lemma we characterize Rac(d,F) as a subset of Feas(d,F). To
avoid trivialities we assume d ≥ 3.
8
Lemma 4.4 Fix an integer d ≥ 3 and a sequence ({θ }d ;{θ∗}d ) in Feas(d,F). Let β+1
i i=0 i i=0
denote the common value of (5). Then the sequence is in Rac(d,F) if and only if each of the
following hold:
(i) β2 6= 4;
(ii) (θ −θ )2 −β(θ −θ )(θ −θ )+(θ −θ )2 6= 0;
0 1 0 1 1 2 1 2
(iii) (θ∗ −θ∗)2 −β(θ∗ −θ∗)(θ∗ −θ∗)+(θ∗ −θ∗)2 6= 0.
0 1 0 1 1 2 1 2
2
Proof: Use the comments above Definition 4.2.
Proposition 4.5 Assume F is infinite and pick an integer d ≥ 3. Let h denote a polynomial
in 2d + 2 mutually commuting indeterminates that has all coefficients in F. Suppose that
h(p) = 0 for all p ∈ Rac(d,F). Then h(p) = 0 for all p ∈ Feas(d,F).
Proof: Let ♭,{t }2 ,{t∗}2 denote mutually commuting indeterminates. Consider the F-
i i=0 i i=0
algebra F[♭,t ,t ,t ,t∗,t∗,t∗] consisting of the polynomials in ♭,t ,t ,t ,t∗,t∗,t∗ that have all
0 1 2 0 1 2 0 1 2 0 1 2
coefficients in F. For 3 ≤ i ≤ d define t ,t∗ ∈ F[♭,t ,t ,t ,t∗,t∗,t∗] by
i i 0 1 2 0 1 2
0 = t −(♭+1)t +(♭+1)t −t ,
i i−1 i−2 i−3
0 = t∗ −(♭+1)t∗ +(♭+1)t∗ −t∗ .
i i−1 i−2 i−3
Define a polynomial f ∈ F[♭,t ,t ,t ,t∗,t∗,t∗] to be the composition
0 1 2 0 1 2
f = h(t ,t ,...,t ,t∗,t∗,...,t∗).
0 1 d 0 1 d
We mention one significance of f. Given a sequence p = ({θ }d ;{θ∗}d ) in Feas(d,F),
i i=0 i i=0
let β + 1 denote the common value of (5). Define the sequence s = (β,θ ,θ ,θ ,θ∗,θ∗,θ∗).
0 1 2 0 1 2
Observe that θ = t (s) and θ∗ = t∗(s) for 0 ≤ i ≤ d. Therefore
i i i i
f(s) = h(p). (16)
We show f = 0. Instead of working directly with f, it will be convenient to work with the
product Ψ = fξξ∗ωω∗(♭2 −4), where
ξ = Y (ti −tj), (17)
0≤i<j≤d
ξ∗ = Y (t∗ −t∗), (18)
i j
0≤i<j≤d
ω = (t −t )2 −♭(t −t )(t −t )+(t −t )2, (19)
0 1 0 1 1 2 1 2
ω∗ = (t∗ −t∗)2 −♭(t∗ −t∗)(t∗ −t∗)+(t∗ −t∗)2. (20)
0 1 0 1 1 2 1 2
Each of ξ,ξ∗ is nonzero by Lemma 4.1 and since F is infinite. Each of ω,ω∗,♭2 − 4 is
nonzero by construction. To show f = 0 we will show that Ψ = 0 and invoke the fact that
F[♭,t ,t ,t ,t∗,t∗,t∗] is a domain [66, p. 129]. We now show that Ψ = 0. Since F is infinite
0 1 2 0 1 2
9
it suffices to show that Ψ(s) = 0 for all sequences s = (β,θ ,θ ,θ ,θ∗,θ∗,θ∗) of scalars taken
0 1 2 0 1 2
from F [66, Proposition 6.89]. Let s be given. For 3 ≤ i ≤ d define θ = t (s), θ∗ = t∗(s) and
i i i i
put p = ({θ }d ;{θ∗}d ). We may assume {θ }d are mutually distinct; otherwise ξ(s) = 0
i i=0 i i=0 i i=0
so Ψ(s) = 0. We may assume {θ∗}d are mutually distinct; otherwise ξ∗(s) = 0 so Ψ(s) = 0.
i i=0
By construction p satisfies condition (ii) of Theorem 3.1, with β+1 the common value of (5).
Therefore p is feasible by Definition 3.2. For the moment assume that p ∈ Rac(d,F). Then
f(s) = 0 by (16) and since h(p) = 0. Therefore Ψ(s) = 0. Next assume that p 6∈ Rac(d,F).
Then the product ωω∗(♭2−4) vanishes at s in view of Lemma 4.4. The product ωω∗(♭2−4)
is a factor of Ψ so Ψ(s) = 0. By the above comments Ψ(s) = 0 for all sequences of scalars
s = (β,θ ,θ ,θ ,θ∗,θ∗,θ∗) taken from F. Therefore Ψ = 0 so f = 0. Now consider any
0 1 2 0 1 2
sequence p = ({θ }d ;{θ∗}d ) in Feas(d,F). Then h(p) = 0 by (16) and since f = 0. 2
i i=0 i i=0
ˇ
5 The algebra T
In order to prove Theorem 3.9 we will need some detailed results about the algebra T from
Definition 3.4. In order to obtain these results it is helpful to first consider the following
ˇ
algebra T.
Definition 5.1 Fix an integer d ≥ 0. Let Tˇ = Tˇ(d,F) denote the F-algebra defined by
generators {ǫ }d , {ǫ∗}d and relations
i i=0 i i=0
ǫ ǫ = δ ǫ , ǫ∗ǫ∗ = δ ǫ∗ 0 ≤ i,j ≤ d. (21)
i j i,j i i j i,j i
Definition 5.2 Referring to Definition 5.1, we call {ǫ }d and {ǫ∗}d the idempotent gen-
i i=0 i i=0
erators for Tˇ. We say that the {ǫ∗}d are starred and the {ǫ }d are nonstarred.
i i=0 i i=0
Definition 5.3 A pair of idempotent generators for Tˇ is called alternating whenever one of
them is starred and the other is nonstarred. For an integer n ≥ 0, by a word of length n in Tˇ
we mean a product g g ···g such that {g }n are idempotent generators for Tˇ and g ,g
1 2 n i i=1 i−1 i
are alternating for 2 ≤ i ≤ n. We interpret the word of length 0 to be the identity of Tˇ. We
call this word trivial.
Proposition 5.4 The F-vector space Tˇ has a basis consisting of its words.
Proof: Let S denote the set of words in Tˇ. By construction S spans Tˇ. We show that
S is linearly independent. To this end we introduce some indeterminates {f }d ; {f∗}d
i i=0 i i=0
called formal idempotents. We call the {f∗}d starred and the {f }d nonstarred. A pair
i i=0 i i=0
of formal idempotents is said to be alternating whenever one of them is starred and the
other is nonstarred. For an integer n ≥ 0, by a formal word of length n we mean a se-
quence (y ,y ,...,y ) such that {y }n are formal idempotents and y ,y are alternating
1 2 n i i=1 i−1 i
for 2 ≤ i ≤ n. The formal word of length 0 is called trivial and denoted by 1. Let S denote
the set of all formal words. Let V denote the vector space over F consisting of the F-linear
combinations of S that have finitely many nonzero coefficients. The set S is a basis for
V. For 0 ≤ i ≤ d we define linear transformations F : V → V and F∗ : V → V. To do
i i
10