Two permanents in the universal enveloping algebras of the symplectic Lie algebras
aa r X i v : . [ m a t h . R T ] F e b TWO PERMANENTSIN THE UNIVERSAL ENVELOPING ALGEBRASOF THE SYMPLECTIC LIE ALGEBRAS
Minoru ITOH
Department of Mathematics and Computer Science, Faculty of Science,Kagoshima University, Kagoshima 890-0065, [email protected]
Abstract.
This paper presents new generators for the center of the universal envelopingalgebra of the symplectic Lie algebra. These generators are expressed in terms of thecolumn-permanent, and it is easy to calculate their eigenvalues on irreducible represen-tations. We can regard these generators as the counterpart of central elements of theuniversal enveloping algebra of the orthogonal Lie algebra given in terms of the column-determinant by A. Wachi. The earliest prototype of all these central elements is the Capellideterminants in the universal enveloping algebra of the general linear Lie algebra.
Introduction.
In this paper we give new generators for the center of the universalenveloping algebra of the symplectic Lie algebra sp N . These generators D k ( u ) areexpressed in terms of the “column-permanent,” and similar to the Capelli determinants,i.e., well-known central elements of the universal enveloping algebra of the general linearLie algebra gl N . As the key of the Capelli identity, these Capelli determinants are usedto analyze the representations of gl N acting via the polarization operators (see [Ca1],[Ca2], [HU]). One of the remarkable properties of the Capelli determinants is that wecan easily calculate their eigenvalues on irreducible representations. It is also easy tocalculate the eigenvalues of our central elements D k ( u ).On the other hand, it is not so obvious that D k ( u ) is actually central in the universalenveloping algebra. This fact can be proved as follows. In addition to D k ( u ), we consideranother central element D ′ k ( u ) expressed in terms of the “symmetrized permanent.” Wecan easily check that this D ′ k ( u ) is central, but its eigenvalue is difficult to calculate. Inspite of this difference, these D k ( u ) and D ′ k ( u ) are actually equal. We will prove this Mathematics Subject Classification.
Primary 17B35; Secondary 15A15.
Key words and phrases. symplectic Lie algebras, central elements of universal enveloping algebras,Capelli identity. Typeset by
AMS -TEX M. ITOH coincidence to show the centrality of D k ( u ). Then, at the same time, we can also seethe eigenvalue of D ′ k ( u ).More directly, our central elements are regarded as the counterpart of the centralelements of the universal enveloping algebra of the orthogonal Lie algebra o N recentlygiven by A. Wachi [W] in terms of the “column-determinant.” The discussion between D ( u ) and D ′ ( u ) above can be applied to Wachi’s elements ([I4]).Let us explain the main result precisely. Let J ∈ Mat N ( C ) be a non-degeneratealternating matrix of size N . We can realize the symplectic Lie group as the isometrygroup with respect to the bilinear form determined by J : Sp ( J ) = { g ∈ GL N | t gJ g = J } . The corresponding Lie algebra is expressed as sp ( J ) = { Z ∈ gl N | t ZJ + J Z = 0 } . As generators of this sp ( J ), we can take F sp ( J ) ij = E ij − J − E ji J , where E ij is thestandard basis of gl N . We introduce the N × N matrix F sp ( J ) whose ( i, j )th entryis this generator: F sp ( J ) = ( F sp ( J ) ij ) ≤ i,j ≤ N . We regard this matrix as an element ofMat N ( U ( sp ( J ))).In the representation theory, the case J = J = . . . − . . . − is important. Indeed, we can take a triangular decomposition of sp ( J ) simply as follows: sp ( J ) = n − ⊕ h ⊕ n + . Here n − , h , and n + are the subalgebra of sp ( J ) spanned by the elements F sp ( J ) ij suchthat i > j , i = j , and i < j respectively. Namely, the entries in the lower triangularpart, in the diagonal part, and in the upper triangular part of the matrix F sp ( J ) belongto n − , h , and n + respectively. We call this sp ( J ) be the “split realization” of thesymplectic Lie algebras.The following is the main theorem of this paper: Theorem A.
The following element is central in U ( sp ( J )) for any u ∈ C : D k ( u ) = X ≤ α ≤···≤ α k ≤ N α ! per( e F sp ( J ) α + u α − α diag( k − , k − , . . . , − k )) . WO PERMANENTS FOR THE SYMPLECTIC LIE ALGEBRAS 3
The notation is as follows. First, the symbol “per” means the “column-permanent.”Namely, for an N × N matrix Z = ( Z ij ), we putper Z = X σ ∈ S N Z σ (1)1 Z σ (2)2 · · · Z σ ( N ) N , even if the entries Z ij are non-commutative. Secondly, e F sp ( J ) means the matrix e F sp ( J ) = F sp ( J ) − diag(0 , . . . , , , . . . , . Here the numbers of 0’s and 1’s are equal to N/
2. Thirdly, means the unit matrix.Moreover, for a matrix Z = ( Z ij ) and a non-decreasing sequence α = ( α , . . . , α k ),we denote the matrix ( Z α i α j ) ≤ i,j ≤ k by Z α . Finally, we put α ! = m ! · · · m N !, where m , . . . , m N are the multiplicities of α = ( α , . . . , α k ): α = ( α , . . . , α k ) = ( m z }| { , . . . , , m z }| { , . . . , , . . . , m N z }| { N, . . . , N ) . This central element D k ( u ) is remarkable, because we can easily calculate its eigen-value on irreducible representations of sp ( J ) (Theorem 4.4). However, the centralityof D k ( u ) (namely Theorem A) is not so obvious.To prove Theorem A, we consider another central element of U ( sp ( J )): D ′ k ( u ) = Per k ( F + u ; k − , k − , . . . , − k + 1 , k ” means the “symmetrized permanent.” Namely we putPer k ( Z ; a , . . . , a k ) = 1 k ! X ≤ α ≤···≤ α k ≤ N X σ,σ ′ ∈ S N α ! Z α σ (1) α σ ′ (1) ( a ) · · · Z α σ ( k ) α σ ′ ( k ) ( a k )with Z ij ( a ) = Z ij + δ ij a . From an invariance of this “Per k ” (Proposition 1.9), thecentrality of D ′ k ( u ) is almost obvious. However, its eigenvalue is difficult to calculate.These D k ( u ) and D ′ k ( u ) are actually equal: Theorem B.
We have D k ( u ) = D ′ k ( u ) . The centrality of D k ( u ) (namely Theorem A) and the eigenvalue of D ′ k ( u ) are bothimmediate from this Theorem B. Namely, proving Theorem B directly, we can settlethese two problems at the same time.Our elements D k ( u ) can be regarded as the counterpart of the central elementsof U ( o N ) recently given by A. Wachi [W]. Wachi’s elements are also expressed in twodifferent ways. The first expression C k ( u ) is given in terms of the “column-determinant,”and we can easily calculate its eigenvalue under this expression. On the other hand,the second expression C ′ k ( u ) is given in terms of the “symmetrized determinant,” andthe centrality is almost obvious under this expression. Proving the coincidence C k ( u ) = M. ITOH C ′ k ( u ) directly, we can settle the following two problems: (i) the centrality of C k ( u ), and(ii) the calculation of the eigenvalue of C ′ k ( u ). See Section 3 and [I4] for the details.We note some central elements related to these elements. First, Wachi’s element C k ( u ) is equal to the central elements of U ( o N ) given in terms of the Sklyanin determi-nant in [M] (see also [MNO]). This coincidence is seen by comparing their eigenvalues.Moreover, C k (0) and D k (0) are equal to the central elements defined by eigenvaluesin [MN]. For these elements, Capelli type identities are given.The symmetrized permanent was also introduced to give Capelli type identities forreductive dual pairs. See [I2] and [I3] for these Capelli type identities in terms of thesymmetrized determinant and the symmetrized permanent. These identities are closelyrelated to C k ( u ) and D k ( u ).The author is grateful to Professors Tˆoru Umeda and Akihito Wachi for the fruitfuldiscussions.
1. Capelli type elements for the general linear Lie algebras.
In this section,we recall the Capelli determinant, a famous central element of U ( gl N ) essentially givenin [Ca1]. We also recall its generalization in terms of minors given in [Ca2] and itsanalogue in terms of permanents due to M. Nazarov [N]. These are the prototypes ofthe main objects of this paper and Wachi’s elements. First let us recall the Capelli determinant. Let E ij be the standard basis of gl N ,and consider the matrix E = ( E ij ) ≤ i,j ≤ N in Mat N ( U ( gl N )). The following “Capellideterminant” in U ( gl N ) is well known as the key of the Capelli identity ([Ca1], [H],[U1]): C gl N ( u ) = det( E + u + diag( N − , N − , . . . , . Here the symbol “det” means the “column-determinant.” Namely, for N × N matrix Z = ( Z ij ), we put det Z = X σ ∈ S N sgn( σ ) Z σ (1)1 Z σ (2)2 · · · Z σ ( N ) N . Here each Z ij is an element of a (non-commutative) associative C -algebra A . This C gl N det ( u ) is known to be central: Theorem 1.1.
The element C gl N ( u ) is central in U ( gl N ) for any u ∈ C . The eigenvalue of this Capelli determinant on irreducible representations is easilycalculated:
Theorem 1.2.
For the irreducible representation π gl N λ of gl N determined by thepartition λ = ( λ , . . . , λ N ) , we have π gl N λ ( C gl N ( u )) = ( u + l ) · · · ( u + l N ) . Here we put l i = λ i + N − i . WO PERMANENTS FOR THE SYMPLECTIC LIE ALGEBRAS 5
This is immediate from the definition of the column-determinant and the triangulardecomposition(1.1) gl N = n − ⊕ h ⊕ n + . Here n − , h , and n + are the subalgebras of gl N spanned by the elements E ij such that i > j , i = j , and i < j respectively. Namely the entries in the lower triangular part,in the diagonal part, and in the upper triangular part of E belong to n − , h , and n + respectively. Considering the action of C gl N ( u ) to the highest weight vector, we caneasily check Theorem 1.2.We can rewrite this Capelli determinant in terms of the “symmetrized determinant”as follows: Theorem 1.3.
We have det( E + u + diag( N − , N − , . . . , E + u ; N − , N − , . . . , . Here the symbol “Det” means the “symmetrized determinant.” Namely, for an N × N matrix Z = ( Z ij ), we putDet Z = 1 N ! X σ,σ ′ ∈ S N sgn( σ ) sgn( σ ′ ) Z σ (1) σ ′ (1) Z σ (2) σ ′ (2) · · · Z σ ( N ) σ ′ ( N ) . Moreover, for N parameters a , . . . , a N ∈ C , we putDet( Z ; a , . . . , a N )= 1 N ! X σ,σ ′ ∈ S N sgn( σ ) sgn( σ ′ ) Z σ (1) σ ′ (1) ( a ) Z σ (2) σ ′ (2) ( a ) · · · Z σ ( N ) σ ′ ( N ) ( a N )with Z ij ( a ) = Z ij + δ ij a . It is obvious that Det Z is equal to the usual determinant,if the entries are commutative. This non-commutative determinant “Det” is useful toconstruct central elements in U ( gl N ). Indeed, we have the following. Proposition 1.4.
For any a , . . . , a N ∈ C , the determinant Det( E ; a , . . . , a N ) is invariant under the adjoint action of GL N ( C ) , and hence this is central in U ( gl N ) . This is immediate from the following two lemmas:
Lemma 1.5.
The symmetrized determinant is invariant under the conjugation by g ∈ GL N ( C ) : Det( gZg − ; a , . . . , a N ) = Det( Z ; a , . . . , a N ) . Here Z is an arbitrary N × N matrix whose entries are elements of an associative C -algebra A . M. ITOH
Lemma 1.6.
The matrix E satisfies the relation Ad( g ) E = t g · E · t g − for any g ∈ GL N ( C ) . Here Ad( g ) E means the matrix (Ad( g ) E ij ) ≤ i,j ≤ N . Lemma 1.6 can be checked by a direct calculation. Lemma 1.5 is also easy from theexpression of “Det” in the framework of the exterior calculus. See [I1] for the details(cf. Section 2 of this paper).For convenience, we consider the symbol ♮ k = ( k − , k − , . . . , C gl N ( u ) = det( E + u + diag ♮ N ) , C ′ gl N ( u ) = Det( E + u ; ♮ N ) . These two expressions play contrast roles. Indeed, it is not so easy to calculate theeigenvalue of C ′ gl N ( u ) directly, but the centrality of C ′ gl N ( u ) is immediate from Propo-sition 1.4, because Det( E + u ; ♮ N ) = Det( E ; u N + ♮ N ). Here u N + ♮ N means thelinear combination of the two vectors 1 N = (1 , . . . ,
1) and ♮ N in C N . Namely we put u N + ♮ N = ( u + N − , u + N − , . . . , u ).Using Theorem 1.3, we can settle the following two problems at the same time: (i) thecentrality of C gl N ( u ), and (ii) the calculation of the eigenvalue of C ′ gl N ( u ). Indeed, asseen above, the eigenvalue of C gl N ( u ) and the centrality of C ′ gl N ( u ) are almost obvious.The proof of Theorem 1.3 will be given in Section 2. Next we recall some generalizations of the Capelli determinant. First, we put C gl N k ( u ) = X ≤ α < ··· <α k ≤ N det( E α + u + diag ♮ k ) . Here we denote by Z α the submatrix ( Z α i α j ) ≤ i,j ≤ k of the matrix Z = ( Z ij ). Obviouslywe have C gl N N ( u ) = C gl N ( u ). This element C gl N k ( u ) is also central in U ( gl N ) for any u ∈ C , and known by the name of the “Capelli elements of degree k .”Moreover we consider the element D gl N k ( u ) = X ≤ α ≤···≤ α k ≤ N α ! per( E α + u α − α diag ♮ k )due to Nazarov [N]. Here the symbol “per” means the “column-permanent.” Namely,for any N × N matrix Z = ( Z ij ), we putper Z = X σ ∈ S N Z σ (1)1 · · · Z σ ( N ) N . Moreover, we put α ! = m ! · · · m N !, where m , . . . , m N are the multiplicities of α =( α , . . . , α k ): α = ( α , . . . , α k ) = ( m z }| { , . . . , , m z }| { , . . . , , . . . , m N z }| { N, . . . , N ) . Note that Z α = ( Z α i α j ) ≤ i,j ≤ k is not a submatrix of Z in general, because α has somemultiplicities. This D gl N k ( u ) is also central in U ( gl N ) for any u ∈ C .We can easily calculate the eigenvalues of these elements C gl N k ( u ) and D gl N k ( u ). Theproof is almost the same as that of Theorem 1.2: WO PERMANENTS FOR THE SYMPLECTIC LIE ALGEBRAS 7
Proposition 1.7.
For the irreducible representation π = π gl N λ of gl N determined bythe partition λ = ( λ , . . . , λ N ) , we have π ( C gl N k ( u )) = X ≤ α < ··· <α k ≤ N ( u + λ α + k − u + λ α + k − · · · ( u + λ α k ) ,π ( D gl N k ( u )) = X ≤ α ≤···≤ α k ≤ N ( u + λ α − k + 1)( u + λ α − k + 2) · · · ( u + λ α k ) . We can rewrite these elements in terms of the “symmetrized determinant” and the“symmetrized permanent”:
Theorem 1.8.
We have X ≤ α < ··· <α k ≤ N det( E α + u + diag ♮ k ) = Det k ( E + u ; ♮ k ) , X ≤ α ≤···≤ α k ≤ N α ! per( E α + u α − α diag ♮ k ) = Per k ( E + u ; − ♮ k ) . Here Det k and Per k are defined as follows. First we putPer Z = 1 N ! X σ,σ ′ ∈ S N Z σ (1) σ ′ (1) · · · Z σ ( k ) σ ′ ( k ) . Noting this, we putDet k ( Z ) = X ≤ α < ··· <α k ≤ N Det Z α = X ≤ α < ··· <α k ≤ N k ! X σ,σ ′ ∈ S k sgn( σ ) sgn( σ ′ ) Z α σ (1) α σ ′ (1) · · · Z α σ ( k ) α σ ′ ( k ) , Per k ( Z ) = X ≤ α ≤···≤ α k ≤ N α ! Per Z α = X ≤ α ≤···≤ α k ≤ N α ! 1 k ! X σ,σ ′ ∈ S k Z α σ (1) α σ ′ (1) · · · Z α σ ( k ) α σ ′ ( k ) . Moreover, for k parameters a , . . . , a k ∈ C , we putDet k ( Z ; a , . . . , a k )= X ≤ α < ··· <α k ≤ N k ! X σ,σ ′ ∈ S k sgn( σ ) sgn( σ ′ ) Z α σ (1) α σ ′ (1) ( a ) · · · Z α σ ( k ) α σ ′ ( k ) ( a k ) , Per k ( Z ; a , . . . , a k )= X ≤ α ≤···≤ α k ≤ N α ! 1 k ! X σ,σ ′ ∈ S k Z α σ (1) α σ ′ (1) ( a ) · · · Z α σ (1) α σ ′ (1) ( a k ) . These Det k and Per k are invariant under the conjugations: M. ITOH
Proposition 1.9.
For g ∈ GL N ( C ) , we have Det k ( gZg − ; a , . . . , a k ) = Det k ( Z ; a , . . . , a k ) , Per k ( gZg − ; a , . . . , a k ) = Per k ( Z ; a , . . . , a k ) . Hence, combining this with Lemma 1.6, we have the following.
Proposition 1.10.
For arbitrary a , . . . , a k ∈ C , the two elements Det k ( E ; a , . . . , a k ) , Per k ( E ; a , . . . , a k ) are invariant under the adjoint action of GL N ( C ) , and hence central in U ( gl N ) . Let us denote by C ′ gl N k ( u ) and D ′ gl N k ( u ) the right hand sides of Theorem 1.8: C ′ gl N k ( u ) = Det k ( E + u ; ♮ k ) = Det k ( E ; u N + ♮ k ) ,D ′ gl N k ( u ) = Per k ( E + u ; − ♮ k ) = Per k ( E ; u N − ♮ k ) . These are obviously central in U ( gl N ) for any u ∈ C . However it is not so easy tocalculate their eigenvalues directly.Theorem 1.8 settles the following two problems at the same time: (i) the central-ities of C gl N k ( u ) and D gl N k ( u ), and (ii) the calculation of the eigenvalues of C ′ gl N k ( u )and D ′ gl N k ( u ).
2. The proof in the case of the general linear Lie algebras.
In this section,we recall the proofs of Theorems 1.3 and 1.8 given in [I1] and [I3]. In these proofs,we express determinants and permanents in the framework of the exterior algebra andthe symmetric tensor algebra. These calculations are the prototypes of the proof of themain theorem.
First, we recall the proof of Theorem 1.3 given in [I1].We can express the column-determinant in the framework of the exterior algebraas follows. Let e , . . . , e N be N anti-commuting formal variables, which generate theexterior algebra Λ N = Λ( C N ). Put η j ( u ) = P Ni =1 e i E ij ( u ) as an element in the extendedalgebra Λ N ⊗ U ( gl N ) in which the two subalgebras Λ N and U ( gl N ) commute with eachother. Then, by a direct calculation, we have the following equality in Λ N ⊗ U ( gl N ):(2.1) η ( a ) η ( a ) · · · η N ( a N ) = e e · · · e N det( E + diag( a , a , . . . , a N )) . The symmetrized determinant is expressed similarly by doubling the anti-commutingvariables. Let e , . . . , e N , e ∗ , . . . , e ∗ N be 2 N anti-commuting formal variables, whichgenerate the exterior algebra Λ N = Λ( C N ⊕ C N ). We put Ξ ( u ) = P Ni,j =1 e i e ∗ j E ij ( u )in Λ N ⊗ U ( gl N ) . Then, by a direct calculation, we have(2.2) Ξ ( a ) Ξ ( a ) · · · Ξ ( a N ) = N ! e e ∗ · · · e N e ∗ N Det( E ; a , a , . . . , a N ) . WO PERMANENTS FOR THE SYMPLECTIC LIE ALGEBRAS 9
Now we can prove Theorem 1.3 using the commutation relation(2.3) η i ( a + 1) η j ( a ) + η j ( a + 1) η i ( a ) = 0 . This relation itself is easy from the relation [ E ij , E kl ] = δ kj E il − δ il E kj . Proof of Theorem
Since Ξ ( u ) = P Ni =1 η i ( u ) e ∗ i , we have Ξ ( u + N − Ξ ( u + N − · · · Ξ ( u )= X ≤ i ,... ,i N ≤ N η i ( u + N − e ∗ i η i ( u + N − e ∗ i · · · η i N ( u ) e ∗ i N = ( − ) N ( N − X ≤ i ,... ,i N ≤ N η i ( u + N − η i ( u + N − · · · η i N ( u ) e ∗ i e ∗ i · · · e ∗ i N . Here the indices i , . . . , i N can be regarded as a permutation of 1 , . . . , N , because e ∗ i ’s are anti-commuting. Moreover the factors η i ( a ) can be reordered by using thecommutation relation (2.3). Thus we have Ξ ( u + N − Ξ ( u + N − · · · Ξ ( u )= ( − ) N ( N − X σ ∈ S N η σ (1) ( u + N − η σ (2) ( u + N − · · · η σ ( N ) ( u ) e ∗ σ (1) e ∗ σ (2) · · · e ∗ σ ( N ) = ( − ) N ( N − N ! η ( u + N − η ( u + N − · · · η N ( u ) e ∗ e ∗ · · · e ∗ N . Comparing this equality with (2.1) and (2.2), we reach to the assertion. (cid:3)
Remark.
From (2.2), we can see that Det( E ; a , . . . , a N ) does not depend on the orderof the parameters a , . . . , a N , because Ξ ( a ) , . . . , Ξ ( a N ) commute with each other.Indeed Ξ ( u ) can be expressed as Ξ ( u ) = Ξ (0) + uτ with τ = P Ni =1 e i e ∗ i , and this τ iscentral in Λ N ⊗ U ( gl N ). Next we go to the proof of Theorem 1.8. We only prove the second relation here,because the proof of the first one is almost the same.We start with the expressions of our permanents in the framework of the symmetrictensor algebra. Let Z = ( Z ij ) be an N × N matrix whose entries are elements of a(non-commutative) associative C -algebra A . Let e , . . . , e N be N commutative formalvariables, which generate the symmetric tensor algebra S N = S ( C N ). We put η j = P Ni =1 e i Z ij as an element in the extended algebra S N ⊗ A in which the two subalgebras S N and A commute with each other. Then, by a direct calculation, we have the relation(2.4) η β = X ≤ α ≤···≤ α k ≤ N α ! e α per( Z αβ ) for 1 ≤ β ≤ · · · ≤ β k ≤ N . Here η β and e α denote η β · · · η β k and e α · · · e α k re-spectively, and Z αβ means the matrix Z αβ = ( Z α i β j ) ≤ i,j ≤ k . Moreover, by putting η j ( u ) = P Ni =1 e i Z ij ( u ), this relation is generalized to(2.5) η β ( a ) · · · η β k ( a k ) = X ≤ α ≤···≤ α k ≤ N α ! e α per( Z αβ + αβ diag( a , . . . , a k )) . The symmetrized permanent is similarly expressed by doubling the commutativevariables. Let e , . . . , e N , e ∗ , . . . , e ∗ N be 2 N commutative formal variables, which gen-erate the symmetric tensor algebra S N = S ( C N ⊕ C N ). We put Ξ = P Ni,j =1 e i e ∗ j Z ij in S N ⊗ A . Then, we have the relation(2.6) Ξ ( k ) = X ≤ α ≤···≤ α k ≤ N ≤ β ≤···≤ β k ≤ N α ! β ! e α e ∗ β Per( Z αβ ) . Here x ( k ) means the divided power: x ( k ) = k ! x k .It is convenient to consider the bilinear form h· | ·i on S N defined by the formula (cid:10) e α e ∗ β (cid:12)(cid:12) e α ′ e ∗ β ′ (cid:11) = δ αα ′ δ ββ ′ α ! β ! . This is known as the “Fischer inner product.” Using this, we can rewrite (2.4) and (2.5)to getper( Z αβ ) = (cid:10) η β (cid:12)(cid:12) e α (cid:11) , per( Z αβ + αβ diag( a , . . . , a k )) = (cid:10) η β ( a ) · · · η β k ( a k ) (cid:12)(cid:12) e α (cid:11) . Similarly (2.6) can be rewritten to getPer( Z αβ ) = (cid:10) Ξ ( k ) (cid:12)(cid:12) e α e ∗ β (cid:11) . Moreover, putting Ξ ( u ) = P Ni,j =1 e i e ∗ j Z ij ( u ) and τ = P Ni =1 e i e ∗ i , we can express Per k as Per k ( Z ) = (cid:10) Ξ ( k ) (cid:12)(cid:12) τ ( k ) (cid:11) , Per k ( Z ; a , . . . , a k ) = (cid:10) k ! Ξ ( a ) · · · Ξ ( a k ) (cid:12)(cid:12) τ ( k ) (cid:11) . These are immediate by noting the relation τ ( k ) = X ≤ α ≤···≤ α k ≤ N α ! e α e ∗ α . We have a similar expression for the column-permanent:per( Z α + u α + α diag( a , . . . , a k )) = (cid:10) η † α ( u + a ) · · · η † α k ( u + a k ) (cid:12)(cid:12) τ ( k ) (cid:11) . WO PERMANENTS FOR THE SYMPLECTIC LIE ALGEBRAS 11
Here we put η † j ( u ) = η j ( u ) e ∗ j .Let us write these expressions simply as(2.7) Per k ( Z ) = (cid:10) Ξ ( k ) (cid:11) , Per k ( Z ; a , . . . , a k ) = (cid:10) k ! Ξ ( a ) · · · Ξ ( a k ) (cid:11) , (2.8) per( Z α + u α + α diag( a , . . . , a k )) = (cid:10) η † α ( u + a ) · · · η † α k ( u + a k ) (cid:11) . Here we put (cid:10) ϕ (cid:11) = ∞ X k =0 (cid:10) ϕ (cid:12)(cid:12) τ ( k ) (cid:11) for ϕ ∈ S N . Note here that the sum is actually finite. Remark.
We can see that Per k ( Z ; a , . . . , a k ) does not depend on the order the param-eters a , . . . , a k , because Ξ ( a ) , . . . , Ξ ( a N ) commute with each other. Indeed, we canexpress Ξ ( u ) as Ξ (0) + uτ , and τ is central in S N ⊗ A . Similarly, Det k ( Z ; a , . . . , a k )does not depend on the order the parameters.Using these expressions, we can prove the second relation of Theorem 1.8 as follows: Proof of the second relation of Theorem
We put η i ( u ) = P Nj =1 e j E ij ( u ). Thecommutation relation η i ( a ) η j ( a + 1) − η j ( a ) η i ( a + 1) = 0is easy from the relation [ E ij , E kl ] = δ kj E il − δ il E kj . In particular, η † i ( u ) = η i ( u ) e ∗ i satisfies the relation(2.9) η † i ( a ) η † j ( a + 1) − η † j ( a ) η † i ( a + 1) = 0 . Since Ξ ( u ) = P Ni,j =1 e i e ∗ j E ij ( u ) is written as Ξ ( u ) = P Ni =1 ˜ η i ( u ), we have Ξ ( u − k + 1) Ξ ( u − k + 2) · · · Ξ ( u )= X ≤ i ,... ,i k ≤ N η † i ( u − k + 1) η † i ( u − k + 2) · · · η † i k ( u ) . The factors η † i ( a ) can be reordered by using the commutation relation (2.9). Thus wehave Ξ ( u − k + 1) Ξ ( u − k + 2) · · · Ξ ( u )= X ≤ α ≤···≤ α k ≤ N k ! α ! η † α ( u − k + 1) η † α ( u − k + 2) · · · η † α k ( u ) . Comparing this equality with (2.7) and (2.8), we reach to the assertion. (cid:3)
Next, let us prove Proposition 1.9. This is an application of the following lemma, anelementary fact for the “Fischer inner product” h· | ·i : Lemma 2.1.
Consider the standard action of g ∈ GL N on the vector space C N ⊕ C N ,which is naturally extended to an automorphism of S N . In this situation, we have (cid:10) ϕ (cid:12)(cid:12) ϕ ′ (cid:11) = (cid:10) g ( ϕ ) (cid:12)(cid:12) t g − ( ϕ ′ ) (cid:11) for ϕ, ϕ ′ ∈ S N . Proof of the second relation of Proposition
For g ∈ GL N , we put h g = diag( g, t g − ) = (cid:18) g t g − (cid:19) ∈ GL N , and consider its natural action on S N ⊗ A . By direct calculations, we have the relations h g ( Ξ Z ( u )) = Ξ gZg − ( u ) and t h − g ( τ ) = τ for Ξ Z ( u ) = P Ni,j =1 e i e ∗ j Z ij ( u ) and τ = P Ni =1 e i e ∗ i . Since h g and t h − g are automorphisms of S N ⊗ A , we have (cid:10) Ξ gZg − ( u ) · · · Ξ gZg − ( u k ) (cid:12)(cid:12) τ ( k ) (cid:11) = (cid:10) h g ( Ξ Z ( u ) · · · h g Ξ Z ( u k )) (cid:12)(cid:12) t h − g ( τ ) ( k ) (cid:11) = (cid:10) Ξ Z ( u ) · · · Ξ Z ( u k ) (cid:12)(cid:12) τ ( k ) (cid:11) . Here, we used Lemma 2.1 for the second equality. By (2.7) this implies our assertion. (cid:3)
The relations for determinants in Theorem 1.8 and Proposition 1.9 can be provedsimilarly by considering the exterior algebra instead of the symmetric tensor algebra(see [I3]).
3. The case of the orthogonal Lie algebras.
Before going to the main result in thecase of the symplectic Lie algebra sp N , we recall the case of the orthogonal Lie algebra o N . In this case, two analogues of the Capelli determinant are known. One was givenby R. Howe and T. Umeda [HU], and the other was recently given by A. Wachi [W]. First we see the general realization of o N . Let S ∈ Mat N ( C ) be a nondegeneratesymmetric matrix of size N . We can realize the orthogonal Lie group as the isometrygroup with respect to the bilinear form determined by S : O ( S ) = { g ∈ GL N | t gSg = S } . The corresponding Lie algebra is expressed as o ( S ) = { Z ∈ gl N | t ZS + SZ = 0 } . As generators of this o ( S ), we can take F o ( S ) ij = E ij − S − E ji S , where E ij is thestandard basis of gl N . We consider the N × N matrix F o ( S ) = ( F o ( S ) ij ) ≤ i,j ≤ N whose( i, j )th entry is this generator F o ( S ) ij . By a direct calculation, this F o ( S ) satisfies thefollowing relation: WO PERMANENTS FOR THE SYMPLECTIC LIE ALGEBRAS 13
Lemma 3.1.
For any g ∈ O ( S ) , we have Ad( g ) F o ( S ) = t g · F o ( S ) · t g − . Here
Ad( g ) F o ( S ) means the matrix (Ad( g ) F o ( S ) ij ) ≤ i,j ≤ N . Combining this with Proposition 1.9, we have the following.
Proposition 3.2.
The two elements
Det k ( F o ( S ) ; a , . . . , a k ) , Per k ( F o ( S ) ; a , . . . , a k ) are invariant under the adjoint action of O ( S ) , and in particular these are centralin U ( o ( S )) : Thus, as in the case of gl N , the symmetrized determinant and the symmetrizedpermanent are useful to construct central elements of U ( o ( S )). On the other hand,unfortunately, it seems not easy to construct central elements of U ( o ( S )) similarly usingthe column-determinant or the column-permanent at least for general S .However, for some special S ( S = and S = S = ( δ i,N +1 − j ) ≤ i,j ≤ N ), we haveanalogues of the Capelli determinant expressed in terms of the column-determinant. First, let us consider the case that S is equal to the unit matrix . Namely weconsider the Lie algebra consisting of all alternating matrices: o ( ) = { Z ∈ gl N | Z + t Z = 0 } . In this case, R. Howe and T. Umeda gave an analogue of the Capelli determinant interms of the column-determinant:
Theorem 3.3 ( [HU] ). The following element is central in U ( o ( )) for any u ∈ C : C o ( ) ( u ) = det( F o ( ) + u + diag ♮ N ) . As in the case of gl N , we can rewrite this in terms of the symmetrized determinant: Theorem 3.4 ( [IU] ). We have det( F o ( ) + u + diag ♮ N ) = Det( F o ( N ) + u ; ♮ N ) . Theorem 3.3 is immediate from this Theorem 3.4. Indeed, by Proposition 3.2, C ′ o ( ) ( u ) = Det( F o ( ) + u ; ♮ N ) = Det( F o ( ) ; u N + ♮ N )is central in U ( o ( )) for any u ∈ C . Remarks. (1) As in the case of gl N , we have the following generalization of C o ( ) ( u ): C o ( ) k ( u ) = X ≤ α < ··· <α k ≤ N det( F o ( ) α + u + diag ♮ k ) . This can be rewritten in terms of “Det k ” as X ≤ α < ··· <α k ≤ N det( F o ( ) α + u + diag ♮ k ) = Det k ( F o ( N ) + u ; ♮ k ) . (2) These elements are quite similar to the Capelli elements C gl N k ( u ). However, it is noteasy to calculate their eigenvalues. Indeed, for this realization o ( ), we can not take itstriangular decomposition so simply as (1.1). Next, we see the case S = S = ( δ i,N +1 − j ), namely we consider the split realizationof the orthogonal Lie algebra: o ( S ) = { Z = ( Z ij ) ∈ gl N | Z ij + Z N +1 − j,N +1 − i = 0 } . A central element of U ( o ( S )) was recently given in terms of the column-determinant: Theorem 3.5 ( [W] ). The element C o ( S ) ( u ) = det( F o ( S ) + u + diag ˜ ♮ N ) is central in U ( o ( S )) for any u ∈ C . Here ˜ ♮ N is the following sequence of length N : ˜ ♮ N = (cid:26) ( N − , N − , . . . , , , . . . , − N + 1) , N : even , ( N − , N − , . . . , , , − , . . . , − N + 1) , N : odd . The proof of this theorem is not so easy (Wachi showed the commutativity with thegenerators of o ( S ) by employing the exterior calculus). On the other hand, we caneasily calculate its eigenvalue: Theorem 3.6 ( [W] ). Let π o ( S ) λ be the irreducible representation of o ( S ) determinedby the partition λ = ( λ , . . . , λ [ N/ ) , where [ N/ means the greatest integer not ex-ceeding N/ . Then we have π o ( S ) λ ( C o ( S ) ( u )) = ( ( u − l )( u − l ) · · · ( u − l N/ ) , N : even ,u ( u − l )( u − l ) · · · ( u − l N/ ) , N : odd . Here we put l i = λ i + N/ − i . The proof is almost the same as that of Theorem 1.2. Namely this is easy from thedefinition of the column-determinant and the triangular decomposition o ( S ) = n − ⊕ h ⊕ n + . WO PERMANENTS FOR THE SYMPLECTIC LIE ALGEBRAS 15
Here n − , h , and n + are the subalgebras of o ( S ) spanned by the elements F o ( S ) ij suchthat i > j , i = j , and i < j respectively. Namely, the entries in the lower triangularpart, in the diagonal part, and in the upper triangular part of the matrix F o ( S ) belongto n − , h , and n + respectively.We can also rewrite C o ( S ) ( u ) in terms of the symmetrized determinant. To see thiswe put C ′ o ( S ) ( u ) = Det( F o ( S ) + u ; ˜ ♮ N ) = Det( F o ( S ) ; u N + ˜ ♮ N ) . We can easily check that this C ′ o ( S ) ( u ) is central in U ( o ( S )) for any u ∈ C . On theother hand, it is not so easy to calculate its eigenvalue. However this was given througha hard and complicated calculation: Theorem 3.7 ( [I1] ). We have π o ( S ) λ ( C ′ o ( S ) ( u )) = ( ( u − l )( u − l ) · · · ( u − l N/ ) , N : even ,u ( u − l )( u − l ) · · · ( u − l N/ ) , N : odd . Comparing this with Theorem 3.6, we have C o ( S ) ( u ) = C ′ o ( S ) ( u ) (recall that anycentral element in the universal enveloping algebras of semisimple Lie algebras is deter-mined by its eigenvalue): Theorem 3.8 ( [W] ). We have det( F o ( S ) + u + diag ˜ ♮ N ) = Det( F o ( S ) + u ; ˜ ♮ N ) . This equality was first shown by A. Wachi in this way. Namely this proof dependson the two non-trivial results Theorems 3.5 and 3.7.However, we can also prove Theorem 3.8 directly not using Thereoms 3.5 and 3.7(see [I4]; this is similar to the proof of the main theorem in this paper, but easier).Conversely, Theorems 3.5 and 3.7 follow from this Theorem 3.8 immediately.
These results can be generalized in terms of minors:
Theorem 3.9 ( [W] ). The following element is central in U ( o ( S )) for any u ∈ C : C o ( S ) k ( u ) = X ≤ α < ··· <α k ≤ N det( e F o ( S ) α + u + diag( k − , k − , . . . , − k )) . Here we put e F o ( S ) = (cid:26) F o ( S ) + diag(0 , . . . , , , . . . , , N : even ,F o ( S ) + diag(0 , . . . , , , , . . . , , N : odd , where the numbers of ’s and ’s are equal to [ N/ . This central element can be rewritten in terms of the symmetrized determinant:
Theorem 3.10 ( [W] ). We have X ≤ α < ··· <α k ≤ N det( e F o ( S ) α + u + diag( k − , k − , . . . , − k )) = Det k ( F o ( S ) + u ; ˜ ♮ k ) . These can be deduced from Theorem 3.8. See [W] for the details.
Remarks. (1) We can also express C o ( S ) k ( u ) as C o ( S ) k ( u ) = X ≤ α < ··· <α k ≤ N det( b F o ( S ) α + u + diag( k , k − , . . . , − k + 1)) . Here b F o ( S ) is defined by b F o ( S ) = (cid:26) F o ( S ) − diag(1 , . . . , , , . . . , , N : even ,F o ( S ) − diag(1 , . . . , , , , . . . , , N : odd , where the numbers of 1’s and 0’s are equal to [ N/ C o ( S ) ( u ) is also equal to the central element given in [M] in terms ofthe Sklyanin determinant. This is seen by comparing their eigenvalues. See [M], [MN],[MNO], [IU], [I1], [W] for the details.(3) The following relation holds for general S [IU]:(3.1) Det k ( F o ( S ) ; ˜ ♮ k ) = X ≤ α < ··· <α k ≤ N Pf( F o ( S ) S ) α Pf( S − F o ( S ) ) α . Here we define the Pfaffian Pf Z for an alternating matrix Z = ( Z ij ) of size 2 k byPf Z = 12 k k ! X σ ∈ S k sgn( σ ) Z σ (1) σ (2) Z σ (3) σ (4) · · · Z σ (2 k − σ (2 k ) .
4. The case of the symplectic Lie algebras.
In this section, we introduce the mainobject of this paper, namely an analogue of the Capelli determinant for the symplecticLie algebra sp N . We can regard this as the direct counterpart of the element C o ( S ) k ( u )due to A. Wachi, but this element is given in terms of the column-permanent not interms of the column-determinant. First we see the general realization of sp N . Let J ∈ Mat N ( C ) be a non-degeneratealternating matrix of size N (hence N must be even; let us put n = N/ J : Sp ( J ) = { g ∈ GL N | t gJ g = J } . The corresponding Lie algebra is expressed as sp ( J ) = { Z ∈ gl N | t ZJ + J Z = 0 } . As generators of this sp ( J ), we can take F sp ( J ) ij = E ij − J − E ji J . We consider the N × N matrix F sp ( S ) = ( F sp ( J ) ij ) ≤ i,j ≤ N whose ( i, j )th entry is this generator F sp ( J ) ij .By a direct calculation, this F sp ( J ) satisfies the following relation: WO PERMANENTS FOR THE SYMPLECTIC LIE ALGEBRAS 17
Lemma 4.1.
For any g ∈ Sp ( J ) , we have Ad( g ) F o ( J ) = t g · F sp ( J ) · t g − . Here
Ad( g ) F sp ( J ) means the matrix (Ad( g ) F sp ( J ) ij ) ≤ i,j ≤ N . Combining this with Proposition 1.9, we have the following proposition:
Proposition 4.2.
The two elements
Det k ( F sp ( J ) ; a , . . . , a k ) , Per k ( F sp ( J ) ; a , . . . , a k ) are invariant under the adjoint action of Sp ( J ) , and in particular this is central in U ( sp ( J )) . Thus, the symmetrized determinant and the symmetrized permanent are useful toconstruct central elements of U ( sp ( J )) as in the case of gl N . However, unfortunately, thecolumn-determinant and the column-permanent do not seem so useful for this purposeat least for general J . Let us consider the split realization of the symplectic Lie algebra. Namely weconsider the case J = J = . . . − . . . − . It is convenient to introduce the symbols i ′ = N + 1 − i, ε ( i ) = (cid:26) − , ≤ i ≤ n, +1 , n + 1 ≤ i ≤ N, so that J = ( ε ( j ) δ ij ′ ) ≤ i,j ≤ N and F sp ( J ) ij = E ij − ε ( i ) ε ( j ) E j ′ i ′ . Moreover the commu-tation relation of F sp ( J ) ij is given by[ F sp ( J ) ij , F sp ( J ) kl ] = F sp ( J ) il δ kj − F sp ( J ) kj δ il + ε ( k ) ε ( l ) F sp ( J ) l ′ j δ ik ′ + ε ( i ) ε ( j ) F sp ( J ) ki ′ δ j ′ l . (4.1)This realization sp ( J ) is important in the representation theory. Indeed, we can takea triangular decomposition of sp ( J ) simply as follows:(4.2) sp ( J ) = n − ⊕ h ⊕ n + . Here n − , h , and n + are the subalgebra of sp ( J ) spanned by the elements F sp ( J ) ij suchthat i > j , i = j , and i < j respectively. We call this sp ( J ) be the “split realization”of the symplectic Lie algebra. The main object of this paper is the following element of U ( sp ( J )): D sp ( J ) k ( u ) = X ≤ α ≤···≤ α k ≤ N α ! per( e F sp ( J ) α + u α − α diag( k − , k − , . . . , − k )) . Here e F sp ( J ) means the matrix e F sp ( J ) = F sp ( J ) − diag(0 , . . . , , , . . . , , where the numbers of 0’s and 1’s are equal to n . Theorem 4.3.
The element D sp ( J ) k ( u ) is central in U ( sp ( J )) for any u ∈ C . The eigenvalue of D sp ( J ) k ( u ) on the irreducible representations of sp ( J ) can be cal-culated easily by noting the triangular decomposition (4.2): Theorem 4.4.
For the representation π sp ( J ) λ of sp ( J ) determined by the partition λ = ( λ , . . . , λ n ) , we have π sp ( J ) λ ( D sp ( J ) k ( u )) = k X l =0 X ≤ α ≤···≤ α l ≤ nn +1 ≤ α l +1 ≤···≤ α k ≤ N ( u + λ α − k + 1)( u + λ α − k + 2) · · · ( u + λ α l − k + l ) · ( u − λ α ′ l +1 − k + l )( u − λ α ′ l +2 − k + l + 1) · · · ( u − λ α ′ k + k − . To prove Theorem 4.3, we additionally consider the following element: D ′ sp ( J ) k ( u ) = Per k ( F sp ( J ) + u ; ˜ ♮ k ) = Per k ( F sp ( J ) ; u N + ˜ ♮ k ) . It is obvious from Proposition 4.2 that this D ′ sp ( J ) k ( u ) is central in U ( sp ( J )) for any u ∈ C . However it is not so easy to calculate the eigenvalue of D ′ sp ( J ) k ( u ) directly.Actually these two elements D sp ( J ) k ( u ) and D ′ sp ( J ) k ( u ) are equal: Theorem 4.5.
We have D sp ( J ) k ( u ) = D ′ sp ( J ) k ( u ) , namely X ≤ α ≤···≤ α k ≤ N α ! per( e F sp ( J ) α + u α − α diag( k − , k − , . . . , − k ))= Per k ( F sp ( J ) + u ; ˜ ♮ k ) . Theorem 4.3 is immediate from this. Moreover, using Theorems 4.4 and 4.5, we caneasily see the eigenvalue of D ′ sp ( J ) k ( u ). Thus, as in the case of o ( S ), this Theorem 4.5settles two problems at the same time. WO PERMANENTS FOR THE SYMPLECTIC LIE ALGEBRAS 19
The remainder of this paper is devoted to the proof of Theorem 4.5.
Remarks. (1) We can also express D sp ( J ) k ( u ) as D sp ( J ) k ( u ) = X ≤ α ≤···≤ α k ≤ N α ! per( b F sp ( J ) α + u α − α diag( k , k − , . . . , − k + 1)) . Here we put b F sp ( J ) = F sp ( J ) + diag(1 , . . . , , , . . . , D sp ( J ) k ( u ), we can rewrite Theorem 4.4 moresimply (see [I3]). Moreover the right hand side of Theorem 4.4 can be regarded as thecomplete symmetric polynomials associated to a kind of factorial power. See [I5] for thedetails.(3) From Theorem 4.4, we see that D sp ( J ) k (0) is equal to D k defined in [MN]. Moreover,as the counterpart of (3.1), we have the relationPer k ( F sp ( J ) ; ˜ ♮ k ) = X ≤ α ≤···≤ α k ≤ N α ! Hf( F sp ( J ) J ) α Hf( J − F sp ( J ) ) α for general J . Here we define the Hafnian Hf Z for a symmetric matrix Z = ( Z ij ) ofsize 2 k by Hf Z = 12 k k ! X σ ∈ S k Z σ (1) σ (2) Z σ (3) σ (4) · · · Z σ (2 k − σ (2 k ) . This is deduced from Theorem 5.1 in [MN].
5. Proof of the main theorem.
Let us show Theorem 4.5 using the symmetric tensoralgebra. This proof is similar to that of Theorem 1.8, but more complicated. Namelywe need the variable transformation method developed in [IU], [I2], [I3].Hereafter, we omit the superscript sp ( J ). Namely we denote F sp ( J ) , F sp ( J ) ij , D sp ( J ) k ( u ), and D ′ sp ( J ) k ( u ) simply by F , F ij , D k ( u ), and D ′ k ( u ) respectively. First, let us express both sides of Theorem 4.5 using the symmetric tensor algebra S N = S ( C N ⊕ C N ). Let e , . . . , e N , e ∗ , . . . , e ∗ N be the standard generators of S N .In the extended algebra S N ⊗ U ( sp ( J )), we put Ξ = N X i,j =1 e i e ∗ j F ij , Ξ ( u ) = N X i,j =1 e i e ∗ j F ij ( u ) , τ = N X i =1 e i e ∗ i , so that Ξ ( u ) = Ξ + uτ . Then, by (2.7), we can express D ′ k ( u ) as D ′ k ( u ) = Per k ( F + u ; − k + 1 , − k + 2 , . . . , k − , k ! (cid:10) Ξ ( u − k + 1) Ξ ( u − k + 2) · · · Ξ ( u + k − · Ξ ( u ) (cid:11) = 1 k ! (cid:10) Ξ k − ( u − k + 1) · Ξ ( u ) (cid:11) . (5.1) Here Ξ k ( u ) means the “rising factorial power” Ξ k ( u ) = Ξ ( u ) Ξ ( u + 1) · · · Ξ ( u + k − . Let us express D k ( u ) similarly. We put η j ( u ) = P Ni =1 e i F ij ( u ) and η † j ( u ) = η j ( u ) e ∗ j ,so that P Nj =1 η † j ( u ) = Ξ ( u ). Moreover, we put ˜ η j ( u ) = P Ni =1 e i e F ij ( u ) and ˜ η † j ( u ) =˜ η j ( u ) e ∗ j . Then η † j ( u ) and ˜ η † j ( u ) are related by(5.2) ˜ η † i ( u ) = ( η † i ( u ) , ≤ i ≤ n,η † i ( u − , n + 1 ≤ i ≤ N. In this notation, we haveper( e F α + u α − α diag( k − , k − , . . . , − k ))= (cid:10) ˜ η † α ( u − k + 1)˜ η † α ( u − k + 2) · · · ˜ η † α k ( u + k ) (cid:11) by (2.8). Thus we can express D k ( u ) as(5.3) D k ( u ) = X ≤ α ≤···≤ α k ≤ N α ! (cid:10) ˜ η † α ( u − k + 1)˜ η † α ( u − k + 2) · · · ˜ η † α k ( u + k ) (cid:11) . Remark.
Recall that Ξ ( u ) and Ξ ( w ) are commutative for any u , w ∈ C , because τ iscentral. By (5.1) and (5.3), our goal D k ( u ) = D ′ k ( u ) can be expressed as X ≤ α ≤···≤ α k ≤ N k ! α ! (cid:10) ˜ η † α ( u − k +1)˜ η † α ( u − k +2) · · · ˜ η † α k ( u + k ) (cid:11) = (cid:10) Ξ k − ( u − k +1) · Ξ ( u ) (cid:11) . Replacing u by u + k −
1, we can rewrite this simply as X ≤ α ≤···≤ α k ≤ N k ! α ! (cid:10) ˜ η † α ( u )˜ η † α ( u + 1) · · · ˜ η † α k ( u + k − (cid:11) = (cid:10) Ξ k − ( u ) · Ξ ( u + k − (cid:11) . Let us prove Theorem 4.5 in this form. Namely, we hereafter aim the following relation:
Lemma 5.1.
We have (cid:10) W k ( u ) (cid:11) = (cid:10) W ′ k ( u ) (cid:11) . Here we put W k ( u ) = X ≤ α ≤···≤ α k ≤ N k ! α ! ˜ η † α ( u )˜ η † α ( u + 1) · · · ˜ η † α k ( u + k − ,W ′ k ( u ) = Ξ k − ( u ) · Ξ ( u + k −
1) = Ξ k ( u ) − k Ξ k − ( u ) τ. WO PERMANENTS FOR THE SYMPLECTIC LIE ALGEBRAS 21
In Sections 5.3–5.5, we will study the relation between W k ( u ) and the factorialpowers of Ξ ( u ). First, W k ( u ) is expressed in terms of η † i ( u ) as W k ( u ) = k X l =0 X ≤ α ≤···≤ α l ≤ nn +1 ≤ α l +1 ≤···≤ α k ≤ N k ! α ! η † α ( u ) η † α ( u + 1) · · · η † α l ( u + l − · η † α l +1 ( u + l − η † α l +2 ( u + l ) · · · η † α k ( u + k − . (5.4)Note that η i ( u ) and η † i ( u ) satisfy the following commutation relation. This is deducedfrom (4.1) by a direct calculation. Lemma 5.2.
We have η j ( u ) η l ( u + 1) − η l ( u ) η j ( u + 1) = ΘJ jl = ε ( j ) Θδ j ′ l ,η † j ( u ) η † l ( u + 1) − η † l ( u ) η † j ( u + 1) = ΘJ jl e ∗ j e ∗ l = ε ( j ) Θδ j ′ l e ∗ j e ∗ l . Here J ij means the ( i, j ) th entry of the matrix J − , and Θ is defined by Θ = N X a,b =1 ε ( b ) e a e b F ab ′ . Remark.
For any u ∈ C , we have Θ = N X b =1 ε ( b ) η b ( u ) e b . Corollary 5.3.
When ≤ i, j ≤ n or n + 1 ≤ i, j ≤ N , we have η † i ( u ) η † j ( u + 1) = η † j ( u ) η † i ( u + 1) . Noting this relation, we consider the two elements Ξ − ( u ) = n X j =1 η † j ( u ) = N X i =1 n X j =1 e i e ∗ j F ij ( u ) ,Ξ + ( u ) = N X j = n +1 η † j ( u ) = N X i =1 N X j = n +1 e i e ∗ j F ij ( u ) . Then we have Ξ − ( u ) + Ξ + ( u ) = Ξ ( u ). Moreover we put Ξ k − ( u ) = Ξ − ( u ) Ξ − ( u + 1) · · · Ξ − ( u + k − ,Ξ k + ( u ) = Ξ + ( u ) Ξ + ( u + 1) · · · Ξ + ( u + k − . By Corollary 5.3, these factorial powers can be expanded as Ξ k − ( u ) = X ≤ α ≤···≤ α k ≤ n k ! α ! η † α ( u ) η † α ( u + 1) · · · η † α k ( u + k − ,Ξ k + ( u ) = X n +1 ≤ β ≤···≤ β k ≤ N k ! β ! η † β ( u ) η † β ( u + 1) · · · η † β k ( u + k − . Thus we can rewrite (5.4) simply as W k ( u ) = X l ≥ (cid:18) kl (cid:19) Ξ l − ( u ) Ξ k − l + ( u + l − . Let us consider an analogue of W k ( u ): V k ( u ) = X l ≥ (cid:18) kl (cid:19) Ξ l − ( u ) Ξ k − l + ( u + l ) . This V k ( u ) is related to W k ( u ) as follows: Lemma 5.4.
We have V k ( u ) = W k ( u ) + kV k − ( u ) τ + . Here we define τ − and τ + by τ − = n X i =1 e i e ∗ i , τ + = N X i = n +1 e i e ∗ i , so that τ = τ − + τ + , Ξ − ( u ) = Ξ − (0) + uτ − , and Ξ + ( u ) = Ξ + (0) + uτ + . Remark.
These τ − and τ + are obviously central. Hence Ξ − ( u ) and Ξ − ( w ) commutewith one another for any u , w ∈ C . Similarly Ξ + ( u ) and Ξ + ( w ) are commutative. Proof of Lemma
Since Ξ + ( u ) = Ξ + (0) + uτ + , we have Ξ k + ( u ) − Ξ k + ( u −
1) = kΞ k − ( u ) τ + . Hence, we have V k ( u ) − W k ( u ) = X l ≥ (cid:18) kl (cid:19) Ξ l − ( u ) · { Ξ k − l + ( u + l ) − Ξ k − l + ( u + l − } = X l ≥ (cid:18) kl (cid:19) Ξ l − ( u ) · ( k − l ) Ξ k − l − ( u + l ) τ + = k X l ≥ (cid:18) k − l (cid:19) Ξ l − ( u ) Ξ k − l − ( u + l ) τ + = kV k − ( u ) τ + . (cid:3) Let us study the relation between V k ( u ) and the factorial powers of Ξ ( u ). First,the following commutation relations are easy from Lemma 5.2: WO PERMANENTS FOR THE SYMPLECTIC LIE ALGEBRAS 23
Lemma 5.5.
We have Ξ + ( u − Ξ − ( u ) − Ξ − ( u − Ξ + ( u ) = Θρ ∗ . Here we put ρ ∗ = P ni =1 e ∗ i e ∗ i ′ . Lemma 5.6.
We have η j ( u ) Θ = Θη j ( u + 2) , and in particular Ξ ( u ) Θ = ΘΞ ( u + 2) , Ξ − ( u ) Θ = ΘΞ − ( u + 2) , Ξ + ( u ) Θ = ΘΞ + ( u + 2) . Note that ρ ∗ is central in S N ⊗ U ( sp ( J )). Thus, the following relation is obtainedfrom Lemmas 5.5 and 5.6 by a simple calculation. Lemma 5.7.
We have Ξ k + ( u ) Ξ − ( u + k ) − Ξ − ( u ) Ξ k + ( u + 1) = kΞ k − ( u ) Θρ ∗ . Moreover, we have the following relation:
Lemma 5.8.
We have V k ( u ) Ξ ( u + k ) − V k +1 ( u ) = kV k − ( u ) Θρ ∗ . Proof of Lemma
First, we have V k ( u ) Ξ − ( u + k ) − Ξ − ( u ) V k ( u + 1)= X l ≥ (cid:18) kl (cid:19) Ξ l − ( u ) Ξ k − l + ( u + l ) Ξ − ( u + k ) − X l ≥ (cid:18) kl (cid:19) Ξ l +1 − ( u ) Ξ k − l + ( u + l + 1)= X l ≥ (cid:18) kl (cid:19) Ξ l − ( u ) · { Ξ k − l + ( u + l ) Ξ − ( u + k ) − Ξ − ( u + l ) Ξ k − l + ( u + l + 1) } = X l ≥ (cid:18) kl (cid:19) Ξ l − ( u ) · ( k − l ) Ξ k − l − ( u + l ) Θρ ∗ = X l ≥ k (cid:18) k − l (cid:19) Ξ l − ( u ) Ξ k − l − ( u + l ) Θρ ∗ = kV k − ( u ) Θρ ∗ . Here we used Lemma 5.7 for the third equality. Moreover, we have V k ( u ) Ξ + ( u + k ) + Ξ − ( u ) V k ( u + 1) = X l ≥ (cid:18) kl (cid:19) Ξ l − ( u ) Ξ k − l +1+ ( u + l ) + X l ≥ (cid:18) kl (cid:19) Ξ l +1 − ( u ) Ξ k − l + ( u + l + 1)= X l ≥ (cid:18) kl (cid:19) Ξ l − ( u ) Ξ k +1 − l + ( u + l ) + X l ≥ (cid:18) kl − (cid:19) Ξ l − ( u ) Ξ k +1 − l + ( u + l )= X l ≥ (cid:18) k + 1 l (cid:19) Ξ l − ( u ) Ξ k +1 − l + ( u + l )= V k +1 ( u ) , because (cid:0) kl (cid:1) + (cid:0) kl − (cid:1) = (cid:0) k +1 l (cid:1) . Thus we have V k ( u ) Ξ ( u + k ) − V k +1 ( u )= V k ( u ) { Ξ − ( u + k ) + Ξ + ( u + k ) } − { V k ( u ) Ξ + ( u + k ) + Ξ − ( u ) V k ( u + 1) } = V k ( u ) Ξ − ( u + k ) − Ξ − ( u ) V k ( u + 1)= kV k − ( u ) Θρ ∗ . (cid:3) Using this, we can show the following expansion:
Lemma 5.9.
We have Ξ k ( u ) = X l ≥ R kl V k − l ( u ) Θ l ρ ∗ l . Here we put R kl = (cid:18) k l (cid:19) (2 l − with (2 l − l − l − · · · we put ( − when l = 0) . Proof of Lemma
This can be proved by induction on k . First, the case k = 0 iseasy. Next, by assuming the case k = m , the case k = m + 1 is deduced as follows: Ξ m +1 ( u ) = Ξ m ( u ) Ξ ( u + m )= X l ≥ R ml V m − l ( u ) Θ l ρ ∗ l Ξ ( u + m )= X l ≥ R ml V m − l ( u ) Ξ ( u + m − l ) Θ l ρ ∗ l = X l ≥ R ml V m − l +1 ( u ) Θ l ρ ∗ l + X l ≥ ( m − l ) R ml V m − l − ( u ) Θ l +1 ρ ∗ l +1 = X l ≥ R ml V m − l +1 ( u ) Θ l ρ ∗ l + X l ≥ ( m − l + 2) R ml − V m − l +1 ( u ) Θ l ρ ∗ l = X l ≥ R m +1 l V m − l +1 ( u ) Θ l ρ ∗ l . WO PERMANENTS FOR THE SYMPLECTIC LIE ALGEBRAS 25
Here we used Lemma 5.8 for the fourth equality. To show the last equality, we also usedthe relations R m +10 = R m = 1 , R m +1 l = R ml + ( m − l + 2) R ml − . These relations themselves are immediate from the definition of R kl . (cid:3) The coefficient R kl also appears in the expansions(5.5) u k = X l ≥ ( − ) l R kl u k − l , u k = X l ≥ R k + l − l u k − l . Here u k and u k mean the two factorial powers u k = u ( u + 1)( u + 2) · · · ( u + k − , u k = u ( u + 2)( u + 4) · · · ( u + 2 k − . In particular, we have X l ≥ ( − ) l R kl R k − lm − l = δ m, . By noting this, the following is immediate from Lemma 5.9:
Lemma 5.10.
We have V k ( u ) = X l ≥ ( − ) l R kl Ξ k − l ( u ) Θ l ρ ∗ l . Next we consider the following relations:
Lemma 5.11.
We have k (cid:10) Ξ k − ( u ) Θ l ρ ∗ l τ m (cid:11) + l (cid:10) Ξ k ( u ) Θ l − ρ ∗ l − τ m ω (cid:11) = 0 . Lemma 5.12.
We have k (cid:10) W ′ k − ( u ) Θ l ρ ∗ l (cid:11) = l (cid:10) Ξ k ( u ) Θ l − ρ ∗ l − ω (cid:11) . Here ω is the central element defined by ω = N X i =1 ε ( i ) e i e ∗ i = − τ − + τ + . Moreover Ξ k ( u ) means the factorial power Ξ k ( u ) = Ξ ( u ) Ξ ( u + 2) Ξ ( u + 4) · · · Ξ ( u + 2 k − . To prove Lemma 5.11, we use a variable transformation and Lemma 2.1. We put g = (cid:18) a b t J c t J − d (cid:19) ∈ GL N , so that t g − = 1 ad − bc (cid:18) d − cJ − − bJ a (cid:19) . These g and t g − naturally act on C N , the space spanned by the formal variables e , . . . , e N , e ∗ , . . . , e ∗ N . Let us consider their extended actions on S N = S ( C N ) andmoreover on S N ⊗ U ( sp ( J )) as automorphisms. Then, we have the relations(5.6) g ( τ ) = ( ad − bc ) τ,g ( ρ ) = a ρ + c ρ ∗ + acω,g ( Ξ ) = ( ad + bc ) Ξ + abΘ + cdΘ ∗ ,g ( Θ ) = a Θ + c Θ ∗ + 2 acΞ, g ( ω ) = ( ad + bc ) ω + 2 abρ + 2 cdρ ∗ ,g ( ρ ∗ ) = b ρ + d ρ ∗ + bdω,g ( Θ ∗ ) = b Θ + d Θ ∗ + 2 bdΞ. Here we define Θ ∗ and ρ by Θ ∗ = − N X i,j =1 ε ( i ) e ∗ i e ∗ j F i ′ j , ρ = − n X i =1 e i e i ′ . Moreover we have t g − ( τ ) = ( ad − bc ) − τ. Remark.
To show (5.6), it is convenient to consider the “row vectors” e = ( e , . . . , e N )and e ∗ = ( e ∗ , . . . , e ∗ N ), so that τ = e t e ∗ , Ξ = eF t e ∗ , Θ = eF J t e, Θ ∗ = e ∗ t J − F t e ∗ ,ω = eK t e ∗ , ρ = 12 eK J t e, ρ ∗ = 12 e ∗ t J − K t e ∗ . Here K means the matrix K = diag( − , . . . , − , , . . . , Proof of Lemma
Let us suppose that a = b = d = 1 and c = 0. Then (5.6) isrewritten as g ( τ ) = τ, g ( ω ) = ω + 2 ρ, g ( ρ ) = ρ, g ( ρ ∗ ) = ρ + ρ ∗ + ω,g ( Ξ ) = Ξ + Θ, g ( Θ ) = Θ, g ( Θ ∗ ) = Θ + Θ ∗ + 2 Ξ. In particular we have g ( Ξ ( u )) = Ξ ( u ) + Θ, g ( Θ ) = Θ, g ( ω − ρ ∗ ) = − ( ω + 2 ρ ∗ ) , g ( τ ) = τ. WO PERMANENTS FOR THE SYMPLECTIC LIE ALGEBRAS 27
Using Lemma 5.6, we can show a kind of binomial expansion: g ( Ξ k ( u )) = ( Ξ ( u ) + Θ )( Ξ ( u + 2) + Θ ) · · · ( Ξ ( u + 2 k −
2) + Θ )= X s ≥ (cid:18) ks (cid:19) Ξ ( u ) Ξ ( u + 2) · · · Ξ ( u + 2 s − · Θ k − s = X s ≥ (cid:18) ks (cid:19) Ξ s ( u ) · Θ k − s . Moreover, expanding the equality g (( ω − ρ ∗ ) l ) = ( − ) l ( ω + 2 ρ ∗ ) l , we have g (cid:0) X r ≥ (cid:18) lr (cid:19) ω r ( − ρ ∗ ) l − r (cid:1) = ( − ) l X r ≥ (cid:18) lr (cid:19) ω r (2 ρ ∗ ) l − r . We also have g ( Θ l − ) = Θ l − and g ( τ m ) = τ m . Multiplying these equalities, we have g (cid:0) Ξ k ( u ) Θ l − X r ≥ (cid:18) lr (cid:19) ω r ( − ρ ∗ ) l − r τ m (cid:1) = ( − ) l X s ≥ (cid:18) ks (cid:19) Ξ s ( u ) Θ k − s + l − X r ≥ (cid:18) lr (cid:19) ω r (2 ρ ∗ ) l − r τ m . Take the inner product with τ ( k +2 l − , and apply Lemma 2.1. Then, since t g − ( τ ) = τ ,we have X r ≥ (cid:18) lr (cid:19)(cid:10) Ξ k ( u ) Θ l − ω r ( − ρ ∗ ) l − r τ m (cid:11) = X s ≥ X r ≥ ( − ) l (cid:18) ks (cid:19)(cid:18) lr (cid:19)(cid:10) Ξ s ( u ) Θ k + l − − s ω r (2 ρ ∗ ) l − r τ m (cid:11) . (5.7)Note that (cid:10) Ξ a ( u ) Θ b ω c ρ ∗ d τ e (cid:11) is equal to zero, unless b = d (compare the order of e , . . . , e N with the order of e ∗ , . . . , e ∗ N ). Hence, the left hand side of (5.7) is equal tozero, unless r = 1. Similarly, the right hand side is equal to zero, unless s = k , r = 1 or s = k − r = 0. Thus we have l (cid:10) Ξ k ( u ) Θ l − ω ( − ρ ∗ ) l − τ m (cid:11) = ( − ) l l (cid:10) Ξ k ( u ) Θ l − ω (2 ρ ∗ ) l − τ m (cid:11) + ( − ) l k (cid:10) Ξ k − ( u ) Θ l (2 ρ ∗ ) l τ m (cid:11) . Simplifying this equality, we have k (cid:10) Ξ k − ( u ) Θ l ρ ∗ l τ m (cid:11) + l (cid:10) Ξ k ( u ) Θ l − ρ ∗ l − τ m ω (cid:11) = 0 . (cid:3) Proof of Lemma . By the expansion (5.5), we have(5.8) Ξ k ( u ) = X r ≥ ( − ) r R kr Ξ k − r ( u ) τ r . Moreover, we have kW ′ k − ( u ) = k ( Ξ k − ( u ) − k − Ξ k − ( u ) τ )= X r ≥ ( − ) r kR k − r Ξ k − − r ( u ) τ r − k ( k − X r ≥ ( − ) r R k − r Ξ k − − r ( u ) τ r +1 . Since R kr = 0 for r <
0, this is equal to X r ≥ ( − ) r kR k − r Ξ k − − r ( u ) τ r − k ( k − X r ≥ ( − ) r − R k − r − Ξ k − − r ( u ) τ r = X r ≥ ( − ) r ( kR k − r + k ( k − R k − r − ) Ξ k − − r ( u ) τ r = X r ≥ ( − ) r ( k − r ) R kr Ξ k − − r ( u ) τ r . Here we used the relation kR k − r + k ( k − R k − r − = ( k − r ) R kr + rR kr = ( k − r ) R kr . Comparing this with (5.8) and applying Lemma 5.11, we have the assertion. (cid:3)
Combining Lemmas 5.4 and 5.9, we can write W k ( u ) as follows. Using kR k − l =( k − l ) R kl , we have W k ( u ) = V k ( u ) − kV k − ( u ) · τ + (5.9) = X l ≥ ( − ) l R kl Ξ k − l ( u ) Θ l ρ ∗ l − k X l ≥ ( − ) l R k − l Ξ k − l − ( u ) Θ l ρ ∗ l τ + = X l ≥ ( − ) l R kl Ξ k − l ( u ) Θ l ρ ∗ l − X l ≥ ( − ) l ( k − l ) R kl Ξ k − l − ( u ) Θ l ρ ∗ l τ + = X l ≥ ( − ) l R kl Ξ k − l ( u ) Θ l ρ ∗ l − X l ≥ ( − ) l ( k − l ) R kl Ξ k − l − ( u ) Θ l ρ ∗ l ω − X l ≥ ( − ) l ( k − l ) R kl Ξ k − l − ( u ) Θ l ρ ∗ l τ WO PERMANENTS FOR THE SYMPLECTIC LIE ALGEBRAS 29 = X l ≥ ( − ) l R kl W ′ k − l ( u ) Θ l ρ ∗ l − X l ≥ ( − ) l ( k − l ) R kl Ξ k − l − ( u ) Θ l ρ ∗ l ω = W ′ k ( u ) + X l ≥ ( − ) l +1 R kl +1 W ′ k − l − ( u ) Θ l +1 ρ ∗ l +1 − X l ≥ ( − ) l ( k − l ) R kl Ξ k − l − ( u ) Θ l ρ ∗ l ω. By Lemma 5.12, we have( k − l − (cid:10) W ′ k − l − ( u ) Θ l +1 ρ ∗ l +1 (cid:11) = ( l + 1) (cid:10) Ξ k − l − ( u ) Θ l ρ ∗ l ω (cid:11) . Multiplying this by k − l − R kl +1 = k − l l +2 R kl , we have R kl +1 (cid:10) W ′ k − l − ( u ) Θ l +1 ρ ∗ l +1 (cid:11) = ( k − l ) R kl (cid:10) Ξ k − l − ( u ) Θ l ρ ∗ l ω (cid:11) . Combining this with (5.9), we have (cid:10) W k ( u ) (cid:11) = (cid:10) W ′ k ( u ) (cid:11) , namely Theorem 5.1. Thisfinishes the proof of the main theorem. Remark.
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