Tridiagonal pairs of q -Racah type and the μ -conjecture
aa r X i v : . [ m a t h . R A ] A ug Tridiagonal pairs of q -Racah typeand the µ -conjecture Kazumasa Nomura and Paul Terwilliger
Abstract
Let K denote a field and let V denote a vector space over K with finite positivedimension. We consider a pair of linear transformations A : V → V and A ∗ : V → V that satisfy the following conditions: (i) each of A, A ∗ is diagonalizable; (ii) thereexists an ordering { V i } di =0 of the eigenspaces of A such that A ∗ V i ⊆ V i − + V i + V i +1 for0 ≤ i ≤ d , where V − = 0 and V d +1 = 0; (iii) there exists an ordering { V ∗ i } δi =0 of theeigenspaces of A ∗ such that AV ∗ i ⊆ V ∗ i − + V ∗ i + V ∗ i +1 for 0 ≤ i ≤ δ , where V ∗− = 0 and V ∗ δ +1 = 0; (iv) there is no subspace W of V such that AW ⊆ W , A ∗ W ⊆ W , W = 0, W = V . We call such a pair a tridiagonal pair on V . It is known that d = δ and for0 ≤ i ≤ d the dimensions of V i , V d − i , V ∗ i , V ∗ d − i coincide. We say the pair A, A ∗ is sharp whenever dim V = 1. It is known that if K is algebraically closed then A, A ∗ is sharp.A conjectured classification of the sharp tridiagonal pairs was recently introduced byT. Ito and the second author. Shortly afterwards we introduced a conjecture, calledthe µ -conjecture , which implies the classification conjecture. In this paper we showthat the µ -conjecture holds in a special case called q -Racah. Keywords . Tridiagonal pair, Leonard pair, q -Racah polynomial. . Primary: 15A21. Secondary: 05E30,05E35, 17Bxx. Throughout the paper K denotes a field.We begin by recalling the notion of a tridiagonal pair. We will use the following terms.Let V denote a vector space over K with finite positive dimension. For a linear transfor-mation A : V → V and a subspace W ⊆ V , we call W an eigenspace of A whenever W = 0and there exists θ ∈ K 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 theeigenspaces of A . efinition 1.1 [1, Definition 1.1] Let V denote a vector space over K with finite positivedimension. By a tridiagonal pair on V we mean an ordered pair of linear transformations A : V → V and A ∗ : V → V that satisfy the following four conditions.(i) Each of A, A ∗ is diagonalizable.(ii) There exists an ordering { V i } di =0 of the eigenspaces of A such that A ∗ V i ⊆ V i − + V i + V i +1 (0 ≤ i ≤ d ) , (1)where V − = 0 and V d +1 = 0.(iii) There exists an ordering { V ∗ i } δi =0 of the eigenspaces of A ∗ such that AV ∗ i ⊆ V ∗ i − + V ∗ i + V ∗ i +1 (0 ≤ i ≤ δ ) , (2)where V ∗− = 0 and V ∗ δ +1 = 0.(iv) There does not exist a subspace W of V such that AW ⊆ W , A ∗ W ⊆ W , W = 0, W = V .We say that the pair A, A ∗ is over K . Note 1.2
According to a common notational convention A ∗ denotes the conjugate-transposeof A . We are not using this convention. In a tridiagonal pair A, A ∗ the linear transforma-tions A and A ∗ are arbitrary subject to (i)–(iv) above.We refer the reader to [1–12, 14–16] for background on tridiagonal pairs.In order to motivate our results we recall some facts about tridiagonal pairs. Let A, A ∗ denote a tridiagonal pair on V , as in Definition 1.1. By [1, Lemma 4.5] the integers d and δ from (ii), (iii) are equal; we call this common value the diameter of the pair. Anordering 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 i } di =0 denote astandard ordering of the eigenspaces of A . By [1, Lemma 2.4], the ordering { V d − i } di =0 is alsostandard and no further ordering is standard. A similar result holds for the eigenspaces of A ∗ . Let { V i } di =0 (resp. { V ∗ i } di =0 ) denote a standard ordering of the eigenspaces of A (resp. A ∗ ). By [1, Corollary 5.7], for 0 ≤ i ≤ d the spaces V i , V ∗ i have the same dimension; wedenote this common dimension by ρ i . By [1, Corollaries 5.7, 6.6] the sequence { ρ i } di =0 issymmetric and unimodal; that is ρ i = ρ d − i for 0 ≤ i ≤ d and ρ i − ≤ ρ i for 1 ≤ i ≤ d/ { ρ i } di =0 the shape of A, A ∗ . We say A, A ∗ is sharp whenever ρ = 1.If K is algebraically closed then A, A ∗ is sharp [11, Theorem 1.3].We now summarize the present paper. A conjectured classification of the sharp tridiag-onal pairs was introduced in [7, Conjecture 14.6] and studied carefully in [9–11]; see Con-jecture 3.1 below. Shortly afterwards we introduced a conjecture, called the µ -conjecture ,which implies the classification conjecture. The µ -conjecture is roughly described as follows.We start with a sequence p = ( { θ i } di =0 ; { θ ∗ i } di =0 ) of scalars taken from K that satisfy theknown constraints on the eigenvalues of a tridiagonal pair over K of diameter d ; these areconditions (i), (ii) of Conjecture 3.1. Following [11, Definition 2.4] we associate with p an2ssociative K -algebra T defined by generators and relations; see Definition 4.1 below. Weare interested in the K -algebra e ∗ T e ∗ where e ∗ is a certain idempotent element of T . Let { x i } di =1 denote mutually commuting indeterminates. Let K [ x , . . . , x d ] denote the K -algebraconsisting of the polynomials in { x i } di =1 that have all coefficients in K . In [12, Corollary6.3] we displayed a surjective K -algebra homomorphism µ : K [ x , . . . , x d ] → e ∗ T e ∗ . The µ -conjecture [12, Conjecture 6.4] asserts that µ is an isomorphism. We have shown thatthe µ -conjecture implies the classification conjecture [12, Theorem 10.1] and that the µ -conjecture holds for d ≤ µ -conjecture is true. There is a general class of parameters p said to have q -Racah type [8, Definition 3.1]. In [8, Theorem 3.3] T. Ito and the secondauthor verified the classification conjecture for the case in which p has q -Racah type and K is algebraically closed; see Proposition 3.2 below. Making heavy use of this result, weverify the µ -conjecture for the case in which p has q -Racah type, with no restriction on K .Our main result is Theorem 5.3. On our way to Theorem 5.3 we obtain two related resultsTheorem 5.1 and 5.2, which might be of independent interest. When working with a tridiagonal pair, it is often convenient to consider a closely relatedobject called a tridiagonal system. To define a tridiagonal system, we recall a few conceptsfrom linear algebra. Let V denote a vector space over K with finite positive dimension. LetEnd K ( V ) denote the K -algebra of all linear transformations from V to V . Let A denote adiagonalizable element of End K ( V ). Let { V i } di =0 denote an ordering of the eigenspaces of A and let { θ i } di =0 denote the corresponding ordering of the eigenvalues of A . For 0 ≤ i ≤ d define E i ∈ End K ( V ) such that ( E i − I ) V i = 0 and E i V j = 0 for j = i (0 ≤ j ≤ d ). Here I denotes the identity of End K ( V ). We call E i the primitive idempotent of A correspondingto V i (or θ i ). Observe that (i) I = P di =0 E i ; (ii) E i E j = δ i,j E i (0 ≤ i, j ≤ d ); (iii) V i = E i V (0 ≤ i ≤ d ); (iv) A = P di =0 θ i E i . Moreover E i = Y ≤ j ≤ d j = i A − θ j Iθ i − θ j . (3)Now let A, A ∗ denote a tridiagonal pair on V . An ordering of the primitive idempotentsor eigenvalues of A (resp. A ∗ ) is said to be standard whenever the corresponding orderingof the eigenspaces of A (resp. A ∗ ) is standard.3 efinition 2.1 [1, Definition 2.1] Let V denote a vector space over K with finite positivedimension. By a tridiagonal system on V we mean a sequenceΦ = ( A ; { E i } di =0 ; A ∗ ; { E ∗ i } di =0 )that satisfies (i)–(iii) below.(i) A, A ∗ is a tridiagonal pair on V .(ii) { E i } di =0 is a standard ordering of the primitive idempotents of A .(iii) { E ∗ i } di =0 is a standard ordering of the primitive idempotents of A ∗ .We say Φ is over K . We call V the vector space underlying Φ.The notion of isomorphism for tridiagonal systems is defined in [9, Definition 3.1]. Thefollowing result is immediate from lines (1), (2) and Definition 2.1.
Lemma 2.2 [10, Lemma 2.5]
Let ( A ; { E i } di =0 ; A ∗ ; { E ∗ i } di =0 ) denote a tridiagonal system.Then for ≤ i, j, k ≤ d the following (i) , (ii) hold. (i) E i A ∗ k E j = 0 if k < | i − j | . (ii) E ∗ i A k E ∗ j = 0 if k < | i − j | . Definition 2.3
Let Φ = ( A ; { E i } di =0 ; A ∗ ; { E ∗ i } di =0 ) denote a tridiagonal system on V . For0 ≤ i ≤ d let θ i (resp. θ ∗ i ) denote the eigenvalue of A (resp. A ∗ ) associated with theeigenspace E i V (resp. E ∗ i V ). We call { θ i } di =0 (resp. { θ ∗ i } di =0 ) the eigenvalue sequence (resp. dual eigenvalue sequence ) of Φ. We observe that { θ i } di =0 (resp. { θ ∗ i } di =0 ) are mutuallydistinct and contained in K . We say Φ is sharp whenever the tridiagonal pair A, A ∗ is sharp.Let Φ denote a tridiagonal system over K with eigenvalue sequence { θ i } di =0 and dualeigenvalue sequence { θ ∗ i } di =0 . By [1, Theorem 11.1] the expressions θ i − − θ i +1 θ i − − θ i , θ ∗ i − − θ ∗ i +1 θ ∗ i − − θ ∗ i (4)are equal and independent of i for 2 ≤ i ≤ d −
1. For this constraint the “most general”solution is θ i = a + bq i − d + cq d − i (0 ≤ i ≤ d ) , (5) θ ∗ i = a ∗ + b ∗ q i − d + c ∗ q d − i (0 ≤ i ≤ d ) , (6) q, a, b, c, a ∗ , b ∗ , c ∗ ∈ K , (7) q = 0 , q = 1 , q = − , bb ∗ cc ∗ = 0 , (8)where K denotes the algebraic closure of K . For this solution q + q − + 1 is the commonvalue of (4). The tridiagonal system Φ is said to have q-Racah type whenever (5)–(8) hold.The following definition is more general. 4 efinition 2.4 Let d denote a nonnegative integer and let ( { θ i } di =0 ; { θ ∗ i } di =0 ) denote asequence of scalars taken from K . We call this sequence q -Racah whenever the following(i), (ii) hold.(i) θ i = θ j , θ ∗ i = θ ∗ j if i = j (0 ≤ i, j ≤ d ).(ii) There exist q, a, b, c, a ∗ , b ∗ , c ∗ that satisfy (5)–(8).We will return to the subject of q -Racah a bit later. We now recall the split sequenceof a sharp tridiagonal system. We will use the following notation. Definition 2.5
Let λ denote an indeterminate and let K [ λ ] denote the K -algebra consistingof the polynomials in λ that have all coefficients in K . Let d denote a nonnegative integerand let ( { θ i } di =0 ; { θ ∗ i } di =0 ) denote a sequence of scalars taken from K . Then for 0 ≤ i ≤ d we define the following polynomials in K [ λ ]: τ i = ( λ − θ )( λ − θ ) · · · ( λ − θ i − ) ,η i = ( λ − θ d )( λ − θ d − ) · · · ( λ − θ d − i +1 ) ,τ ∗ i = ( λ − θ ∗ )( λ − θ ∗ ) · · · ( λ − θ ∗ i − ) ,η ∗ i = ( λ − θ ∗ d )( λ − θ ∗ d − ) · · · ( λ − θ ∗ d − i +1 ) . Note that each of τ i , η i , τ ∗ i , η ∗ i is monic with degree i . Definition 2.6 [12, Definition 2.5] Let ( A ; { E i } di =0 ; A ∗ ; { E ∗ i } di =0 ) denote a sharp tridiago-nal system over K , with eigenvalue sequence { θ i } di =0 and dual eigenvalue sequence { θ ∗ i } di =0 .By [11, Lemma 5.4], for 0 ≤ i ≤ d there exists a unique ζ i ∈ K such that E ∗ τ i ( A ) E ∗ = ζ i E ∗ ( θ ∗ − θ ∗ )( θ ∗ − θ ∗ ) · · · ( θ ∗ − θ ∗ i ) . Note that ζ = 1. We call { ζ i } di =0 the split sequence of the tridiagonal system. Definition 2.7 [9, Definition 6.2] Let Φ denote a sharp tridiagonal system. By the pa-rameter array of Φ we mean the sequence ( { θ i } di =0 ; { θ ∗ i } di =0 ; { ζ i } di =0 ) where { θ i } di =0 (resp. { θ ∗ i } di =0 ) is the eigenvalue sequence (resp. dual eigenvalue sequence) of Φ and { ζ i } di =0 is thesplit sequence of Φ. To motivate our results we recall a conjectured classification of the tridiagonal systemsdue to T. Ito and the second author. 5 onjecture 3.1 [7, Conjecture 14.6]
Let d denote a nonnegative integer and let ( { θ i } di =0 ; { θ ∗ i } di =0 ; { ζ i } di =0 ) (9) denote a sequence of scalars taken from K . Then there exists a sharp tridiagonal system Φ over K with parameter array (9) if and only if (i)–(iii) hold below. (i) θ i = θ j , θ ∗ i = θ ∗ j if i = j (0 ≤ i, j ≤ d ) . (ii) The expressions θ i − − θ i +1 θ i − − θ i , θ ∗ i − − θ ∗ i +1 θ ∗ i − − θ ∗ i (10) are equal and independent of i for ≤ i ≤ d − . (iii) ζ = 1 , ζ d = 0 and = d X i =0 η d − i ( θ ) η ∗ d − i ( θ ∗ ) ζ i . (11) Suppose (i)–(iii) hold. Then Φ is unique up to isomorphism of tridiagonal systems. In [9, Section 8] we proved the “only if” direction of Conjecture 3.1. In [11, Theorem 1.6]we proved the last assertion of Conjecture 3.1. Concerning the “if” direction of Conjecture3.1, we proved this for d ≤ Proposition 3.2 [8, Theorem 3.3]
Assume K is algebraically closed. Let d denote anonnegative integer and let ( { θ i } di =0 ; { θ ∗ i } di =0 ) denote a sequence of scalars taken from K that is q -Racah in the sense of Definition . Let { ζ i } di =0 denote a sequence of scalars takenfrom K that satisfies condition (iii) of Conjecture . Then there exists a sharp tridiagonalsystem over K that has parameter array ( { θ i } di =0 ; { θ ∗ i } di =0 ; { ζ i } di =0 ) . µ -conjecture In this section we recall the µ -conjecture. It has to do with the following algebra.6 efinition 4.1 [11, Definition 2.4] Let d denote a nonnegative integer, and let p =( { θ i } di =0 ; { θ ∗ i } di =0 ) denote a sequence of scalars taken from K that satisfies conditions (i),(ii) of Conjecture 3.1. Let T = T ( p, K ) denote the associative K -algebra with 1, defined bygenerators a , { e i } di =0 , a ∗ , { e ∗ i } di =0 and relations e i e j = δ i,j e i , e ∗ i e ∗ j = δ i,j e ∗ i (0 ≤ i, j ≤ d ) , (12)1 = d X i =0 e i , d X i =0 e ∗ i , (13) a = d X i =0 θ i e i , a ∗ = d X i =0 θ ∗ i e ∗ i , (14) e ∗ i a k e ∗ j = 0 if k < | i − j | (0 ≤ i, j, k ≤ d ) , (15) e i a ∗ k e j = 0 if k < | i − j | (0 ≤ i, j, k ≤ d ) . (16)The algebra T is related to tridiagonal systems as follows. Lemma 4.2 [11, Lemma 2.5]
Let V denote a vector space over K with finite positive di-mension. Let ( A ; { E i } di =0 ; A ∗ ; { E ∗ i } di =0 ) denote a tridiagonal system on V with eigenvaluesequence { θ i } di =0 and dual eigenvalue sequence { θ ∗ i } di =0 . For the sequence p = ( { θ i } di =0 ; { θ ∗ i } di =0 ) let T = T ( p, K ) denote the algebra from Definition . Then there exists a unique T -modulestructure on V such that a , a ∗ , e i , e ∗ i acts on V as A , A ∗ , E i , E ∗ i , respectively. Moreoverthis T -module is irreducible. For the rest of this section, let d denote a nonnegative integer and let p = ( { θ i } di =0 ; { θ ∗ i } di =0 )denote a sequence of scalars taken from K that satisfies conditions (i), (ii) of Conjecture3.1. Let T = T ( p, K ) denote the corresponding algebra from Definition 4.1. Observe that e ∗ T e ∗ is a K -algebra with multiplicative identity e ∗ . Lemma 4.3 [11, Theorem 2.6]
The algebra e ∗ T e ∗ is commutative and generated by e ∗ τ i ( a ) e ∗ (1 ≤ i ≤ d ) . Definition 4.4
Let { x i } di =1 denote mutually commuting indeterminates. We denote by K [ x , . . . , x d ] the K -algebra consisting of the polynomials in { x i } di =1 that have all coefficientsin K .From Lemma 4.3 we immediately obtain the following. Lemma 4.5 [12, Corollary 6.3]
There exists a surjective K -algebra homomorphism µ : K [ x , . . . , x d ] → e ∗ T e ∗ that sends x i e ∗ τ i ( a ) e ∗ for ≤ i ≤ d . The following conjecture was introduced in [12, Conjecture 6.4].7 onjecture 4.6 ( µ -conjecture) The map µ from Lemma is an isomorphism. We have shown that the µ -conjecture implies Conjecture 3.1 [12, Theorem 10.1], andthat the µ -conjecture holds for d ≤ p is q -Racah. In our proof we make heavy use of Proposition 3.2.Our main result is Theorem 5.3. In this section we obtain our main result, which is Theorem 5.3. On our way to this re-sult we obtain two other results Theorem 5.1 and 5.2, which may be of independent interest.Throughout this section, let d denote a nonnegative integer and let p = ( { θ i } di =0 ; { θ ∗ i } di =0 )denote a sequence of scalars taken from K that satisfies conditions (i), (ii) of Conjecture3.1. Theorem 5.1
Assume the field K is infinite and let T = T ( p, K ) denote the K -algebrafrom Definition . Assume that, for every sequence { ζ i } di =0 of scalars taken from K thatsatisfies condition (iii) of Conjecture , there exists a sharp tridiagonal system over K thathas parameter array ( { θ i } di =0 ; { θ ∗ i } di =0 ; { ζ i } di =0 ) . Then the map µ : K [ x , . . . , x d ] → e ∗ T e ∗ from Lemma is an isomorphism. Proof.
We assume d ≥
1; otherwise the result is obvious. The map µ is surjective byLemma 4.5, so it suffices to show that µ is injective. We pick any f ∈ K [ x , . . . , x d ] suchthat µ ( f ) = 0, and show f = 0. Instead of working directly with f , it will be convenientto work with the product ψ = f gh , where g = η ∗ d ( θ ∗ ) x d and h is η ∗ d ( θ ∗ ) times η d ( θ ) + d X i =1 η d − i ( θ ) x i . By construction η ∗ d ( θ ∗ ) = 0 so each of g, h is nonzero. To show that f = 0, we show ψ = 0 and invoke the fact that K [ x , . . . , x d ] is a domain [13, page 129]. We now show that ψ = 0. Since the field K is infinite it suffices to show that ψ ( ξ , . . . , ξ d ) = 0 for all d -tuples( ξ , . . . , ξ d ) of scalars taken from K [13, Proposition 6.89]. Let ( ξ , . . . , ξ d ) be given. Define ζ i = ξ i ( θ ∗ − θ ∗ )( θ ∗ − θ ∗ ) · · · ( θ ∗ − θ ∗ i ) (1 ≤ i ≤ d ) (17)and ζ = 1. Observe g ( ξ , . . . , ξ d ) = ζ d ,h ( ξ , . . . , ξ d ) = d X i =0 η d − i ( θ ) η ∗ d − i ( θ ∗ ) ζ i . First assume { ζ i } di =0 does not satisfy condition (iii) of Conjecture 3.1. Then either ζ d = 0, inwhich case g ( ξ , . . . , ξ d ) = 0, or 0 = P di =0 η d − i ( θ ) η ∗ d − i ( θ ∗ ) ζ i , in which case h ( ξ , . . . , ξ d ) =8. Either way ψ ( ξ , . . . , ξ d ) = 0. Next assume { ζ i } di =0 does satisfy condition (iii) of Conjec-ture 3.1. By the assumption of the present theorem, there exists a sharp tridiagonal systemΦ = ( A ; { E i } di =0 ; A ∗ ; { E ∗ i } di =0 ) over K that has parameter array ( { θ i } di =0 ; { θ ∗ i } di =0 ; { ζ i } di =0 ).Let V denote the vector space underlying Φ. By Definition 2.6 the following holds on V : E ∗ τ i ( A ) E ∗ = ζ i E ∗ ( θ ∗ − θ ∗ )( θ ∗ − θ ∗ ) · · · ( θ ∗ − θ ∗ i ) (1 ≤ i ≤ d ) . (18)Consider the T -module structure on V from Lemma 4.2. Using (17), (18) we find that thefollowing holds on V : e ∗ τ i ( a ) e ∗ = ξ i e ∗ (1 ≤ i ≤ d ) . (19)Pick an integer i (1 ≤ i ≤ d ). By (19) the element e ∗ τ i ( a ) e ∗ acts on e ∗ V as ξ i times theidentity map. Recall that µ sends x i to e ∗ τ i ( a ) e ∗ , so µ ( x i ) acts on e ∗ V as ξ i times theidentity map. By these comments µ ( f ) acts on e ∗ V as f ( ξ , . . . , ξ d ) times the identity map.But µ ( f ) = 0 and e ∗ V = 0 so f ( ξ , . . . , ξ d ) = 0, and therefore ψ ( ξ , . . . , ξ d ) = 0. We haveshown ψ ( ξ , . . . , ξ d ) = 0 for all d -tuples ( ξ , . . . , ξ d ) of scalars taken from K , and therefore ψ = 0. The result follows. (cid:3) Theorem 5.2
Let F denote a field extension of K . Let the algebras T K = T ( p, K ) and T F = T ( p, F ) be as in Definition . Let µ K : K [ x , . . . , x d ] → e ∗ T K e ∗ ,µ F : F [ x , . . . , x d ] → e ∗ T F e ∗ denote the maps from Lemma and assume that µ F is an isomorphism. Then µ K is anisomorphism. Proof.
The map µ K is surjective by Lemma 4.5, so it suffices to show that µ K is injective.Since K is a subfield of F we may view any F -algebra as a K -algebra. The inclusion map K → F induces an injective K -algebra homomorphism ι : K [ x , . . . , x d ] → F [ x , . . . , x d ]. Inthe K -algebra T F the defining generators a , { e i } di =0 , a ∗ , { e ∗ i } di =0 satisfy the defining relationsfor T K . Therefore there exists a K -algebra homomorphism N : T K → T F that sends eachof the T K generators a , { e i } di =0 , a ∗ , { e ∗ i } di =0 to the corresponding generator in T F . Therestriction of N to e ∗ T K e ∗ is a K -algebra homomorphism e ∗ T K e ∗ → e ∗ T F e ∗ ; we denote thishomomorphism by ν . By construction the following diagram commutes: K [ x , . . . , x d ] ι −−−−→ F [ x , . . . , x d ] µ K y y µ F e ∗ T K e ∗ ν −−−−→ e ∗ T F e ∗ The maps ι and µ F are injective so their composition µ F ◦ ι is injective. But µ F ◦ ι = ν ◦ µ K so ν ◦ µ K is injective. Therefore µ K is injective and hence an isomorphism. (cid:3) The following is our main result. 9 heorem 5.3
Let K denote a field and let d denote a nonnegative integer. Let p =( { θ i } di =0 ; { θ ∗ i } di =0 ) denote a sequence of scalars taken from K that is q -Racah in the senseof Definition . Let the K -algebra T = T ( p, K ) be as in Definition . Then the corre-sponding map µ : K [ x , . . . , x d ] → e ∗ T e ∗ from Lemma is an isomorphism. Proof.
Abbreviate F = K for the algebraic closure of K , and note that F is infinite. Let T F = T ( p, F ) denote the F -algebra from Definition 4.1 and let µ F : F [ x , . . . , x d ] → e ∗ T F e ∗ be the corresponding map from Lemma 4.5. By Proposition 3.2, for every sequence { ζ i } di =0 of scalars taken from F that satisfies condition (iii) of Conjecture 3.1, there exists a sharptridiagonal system over F that has parameter array ( { θ i } di =0 ; { θ ∗ i } di =0 ; { ζ i } di =0 ). By this andTheorem 5.1 the map µ F is an isomorphism. Now µ is an isomorphism by Theorem 5.2. (cid:3) We finish with a comment.
Lemma 5.4
Proposition remains true if we drop the assumption that K is algebraicallyclosed. Proof.
Immediate from Theorem 5.3 and [12, Theorem 10.1]. (cid:3)
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Paul TerwilligerDepartment of MathematicsUniversity of Wisconsin480 Lincoln DriveMadison, WI 53706-1388 USAemail: [email protected]@math.wisc.edu