New homogeneous Einstein metrics on quaternionic Stiefel manifolds
aa r X i v : . [ m a t h . DG ] O c t NEW HOMOGENEOUS EINSTEIN METRICS ON QUATERNIONIC STIEFELMANIFOLDS
ANDREAS ARVANITOYEORGOS, YUSUKE SAKANE AND MARINA STATHA
Abstract.
We consider invariant Einstein metrics on the quaternionic Stiefel manifolds V p H n of all orthonormal p -frames in H n . This manifold is diffeomorphic to the homogeneous spaceSp( n ) / Sp( n − p ) and its isotropy representation contains equivalent summands. We obtain newEinstein metrics on V p H n ∼ = Sp( n ) / Sp( n − p ), where n = k + k + k and p = n − k . We view V p H n as a total space over the genaralized Wallach space Sp( n ) / (Sp( k ) × Sp( k ) × Sp( k )) andover the generalized flag manifold Sp( n ) / (U( p ) × Sp( n − p )).2010 Mathematics Subject Classification.
Primary 53C25; Secondary 53C30, 13P10, 65H10, 68W30.
Keywords : Homogeneous space, Einstein metric, quaternionic Stiefel manifold, generalized Wallachspace, genaralized flag manifold, isotropy representation, Gr¨obner basis. Introduction
A Riemannian manifold (
M, g ) is called Einstein if the metric g satisfies the condition Ric( g ) = λ · g for some λ ∈ R . We refer to [Be] and [Wa1], [Wa2] for old and new results on homogeneousEinstein manifolds. The structure of the set of invariant Einstein metrics on a given homogeneousspace is still not very well understood in general. The situation is only clear for few classes ofhomogeneous spaces, such as isotropy irreducible homogeneous spaces, low dimensional examples,certain flag manifolds. For an arbitrary compact homogeneous space G/H it is not clear if theset of invariant Einstein metrics (up to isometry and up to scaling) is finite or not. A finitenessconjecture states that this set is in fact finite if the isotropy representation of
G/H consists ofpairwise inequivalent irreducible subrepresentations ([B¨oWaZi]). In the case where the isotropyrepresentation contains some equivalent subrepresentations (isotropy summands) then the diagonalmetrics are not unique. In [ArDzNi1] the authors introduced a method for proving existenceof homogeneous Einstein metrics by assuming additional symmetries. In [St] a systematic andorganized description of such metrics is presented.In the present article we are interested to invariant Einstein metrics on homogeneous spaces
G/H whose isotropy representation is decomposed into a sum of irreducible but possibly equivalentsummands. Typical examples of such homogeneous spaces are the Stiefel manifolds.For new and old results about Einstein metrics on real Stiefel manifolds V p R n = SO( n ) / SO( n − p )we refer to [ArSaSt1]. The first invariant Einstein metrics on the quaternionic Stiefel manifolds V p H n = Sp( n ) / Sp( n − p ) were obtained by G. Jensen in [Je], by using Riemannian submersions.Invariant Einstein metrics on the two marginal cases V H n = S n − and V n H n = Sp( n ) have beenstudied in [Zi] and [ArSaSt2] respectively.In [ArDzNi2] the first author, V.V. Dzhepko and Yu. G. Nikonorov proved that for s > ℓ ≥ k ≥ sk + ℓ ) / Sp( ℓ ) admits at least four Sp( sk + ℓ ) × (Sp( k )) s -invariantEinstein metrics, two of which are Jensen’s metrics. We call the two Einstein metrics, differentfrom Jensen’s metrics, as ADN metrics .In the present paper we obtain new invariant Einstein metrics on V p H n ∼ = Sp( n ) / Sp( n − p )where n = k + k + k and p = n − k . We view V p H n , firstly as a total space over the genaralized Wallach space Sp( n ) / (Sp( k ) × Sp( k ) × Sp( k )) and secondly as total space over the generalizedflag manifold Sp( n ) / (U( p ) × Sp( n − p )), and use invariant metrics described by corresponding innerproducts (7) and (9) (cf. Section 3). Then we use the method in [ArDzNi1] or [St], appropriatelyadjusted.The main results related to the first fibration are the following: Theorem A.
For n = 3 , V H n admits:(1) Eight invariant Einstein metrics which are determined by Ad(Sp(1) × Sp(1) × Sp(1))-invariant inner products of the form (7). Four of them are new, two are Jensen’s metricsand the other two are ADN metrics.(2) Eight invariant Einstein metrics which are determined by Ad(Sp(1) × Sp(1) × Sp(2))-invariant inner products of the form (7). Four of them are new, two are Jensen’s metricsand the other two are ADN metrics.The main result related to the second fibration is the following:
Theorem B.
For ≤ p ≤ n , there exist two invariant Einstein metrics on Sp( n ) / Sp( n − p ) ofthe form (9) which are different from Jensen’s metrics. The proofs of the above theorems are given in Section 4 (Theorems 4.1 and 4.2). We also makea conjecture about Einstein metrics on the Stiefel manifolds V n − H n ( n ≥
3) and V n − H n ( n ≥ Acknowledgements.
The work was supported by Grant E.
037 from the Research Committeeof the University of Patras (Programme K. Karatheodori) and JSPS KAKENHI Grant NumberJP16K05130.2.
A special class of G -invariant metrics on G/H and the Ricci tensor
Let G be a compact Lie group and H a closed subgroup so that G acts transitively on thehomogeneous space G/H . Then the homogeneous space
G/H is reductive, because we can take m = g ⊥ where Ad( H ) m ⊂ m with respect to an Ad-invariant scalar product on g . So the Lie algebra g can be written as g = h ⊕ m . The tangent space of G/H at the o = eH ∈ G/H is canonicallyidentified with m . For G semisimple, the negative of the Killing form B of g is an Ad( G )-invariantscalar product, therefore we can choose the above decomposition with respect to this form. ARiemannian metric g on G/H is called G -invariant if the diffeomorphism τ α : G/H → G/H,τ α ( gH ) = αgH is an isometry. Any such metric is to one-to-one correspondence with an Ad( H )-invariant scalar product h· , ·i on m and fixed points ( M G ) Φ H of the action Φ H = { Ad( h ) | m : h ∈ H }⊂ Φ = { Ad( n ) | m : n ∈ N G ( H ) } ⊂ Aut( m ) on M G . In the special case where H = { e } then N G ( H ) = G , thus the fixed points ( M G ) Φ H are the Ad( G )-invariant inner products on g . Thesecorrespond to the bi-invariant metrics on the Lie group G .The isotropy representation χ : H → Aut( m ) of the reductive homogeneous space G/H coincideswith the restriction of the adjoint representation of H on m . We assume that χ decompose into adirect sum of irreducible subrepresentation χ ∼ = χ ⊕ · · · χ s , so the tangent space splits into a directsum of Ad( H )-invariant subspaces m = m ⊕ · · · ⊕ m s . (1)In this case the G -invariant metrics are determined by the diagonal Ad( H )-invariant scalar productsof the form h· , ·i = x ( − B ) | m + · · · + x s ( − B ) | m s , x i ∈ R + . (2)If some of the subrepresentations χ i are equivalent then decomposition (1) is not unique. Hencethe Ad( H )-invariant scalar product is not necessary diagonal. For this case we can choose a closed ew homogeneous Einstein metrics on quaternionic Stiefel manifolds 3 subgroup K of G such that H ⊂ K ⊂ N G ( H ) and search for Ad( K )-invariant scalar productson m which correspond to a subset M G,K of G -invariant metrics on G/H , sometimes also calledAd( K )-invariant metrics. The benefit of such metrics is that they are diagonal metrics on thehomogeneous space. The next proposition gives a possible way to choose such a subgroup K of G . Proposition 2.1.
Let K be a subgroup of G with H ⊂ K ⊂ G and such that K = L × H , for somesubgroup L of G . Then K is contained in N G ( H ) . Now we describe the Ricci tensor for the diagonal metrics of the form (2). Every G -invariant sym-metric covariant 2-tensor on G/H are of the same form as the Riemannian metrics (although theyare not necessarly positive definite). In particular, the Ricci tensor r of a G -invariant Riemannianmetric on G/H is of the same form (2), that is r = r x ( − B ) | m + · · · r s x s ( − B ) | m s . Let { e α } be a ( − B )-orthonormal basis adapted to the decomposition of m , i.e. e α ∈ m i for some i ,and α < β if i < j . We put A γαβ = − B ([ e α , e β ] , e γ ) so that [ e α , e β ] = X γ A γαβ e γ and set A ijk := (cid:20) kij (cid:21) = X ( A γαβ ) , where the sum is taken over all indices α, β, γ with e α ∈ m i , e β ∈ m j , e γ ∈ m k (cf. [WaZi]). Then the positive numbers A ijk are independent of the ( − B )-orthonormal baseschosen for m i , m j , m k , and A ijk = A jik = A kij . Let d k = dim m k . Then we have the following: Lemma 2.2. ([PaSa])
The components r , . . . , r s of the Ricci tensor r of the metric h· , ·i of theform (2) on G/H are given by r k = 12 x k + 14 d k X j,i x k x j x i A jik − d k X j,i x j x k x i A kij ( k = 1 , . . . , s ) , (3) where the sum is taken over i, j = 1 , . . . , s . Since by assumption the submodules m i , m j in the decomposition (1) are matually non equivalentfor any i = j , it is r ( m i , m j ) = 0 whenever i = j . Thus by Lemma 2.2 it follows that G -invariantEinstein metrics on M = G/H are exactly the positive real solutions g = ( x , . . . , x s ) ∈ R s + of thepolynomial system { r = λ, r = λ, . . . , r s = λ } , where λ ∈ R + is the Einstein constant.3. The Stiefel manifold V p H n ∼ = Sp( n ) / Sp( n − p )We embed the Lie algebra sp ( n − p ) = (cid:26)(cid:18) X − ¯ YY ¯ X (cid:19) (cid:12)(cid:12)(cid:12) X ∈ u ( n − p ) ,Y ( n − k ) × ( n − p ) complex symmetric matrix (cid:27) of the Lie group Sp( n − p ) in the Lie algebra sp ( n ) of Sp( n ) as sp ( n − p ) ∋ (cid:18) X − ¯ YY ¯ X (cid:19) ֒ → X − ¯ Y Y X ∈ sp ( n ) . The Killing form of sp ( n ) is B ( X, Y ) = 2( n + 1) tr XY . Then with respect to − B we can find theAd(Sp( n − p ))-invariant subspace m ∼ = T o (Sp( n ) / Sp( n − p )) such that sp ( n ) = sp ( n − p ) ⊕ m .Next we review the isotropy representation of G/H = Sp( n ) / Sp( n − p ) ∼ = V p H n . Let ν n : Sp( n ) → Aut( C n ) be the standard representation of Sp( n ) and Ad Sp( n ) ⊗ C = S ν n the complexified adjoint Andreas Arvanitoyeorgos, Yusuke Sakane and Marina Statha representation of Sp( n ), where S is the second symmetric power. For the isotropy representation χ : Sp( n ) → Aut( m ) of the homogeneous space Sp( n ) / Sp( n − p )we have:Ad Sp ( n ) ⊗ C (cid:12)(cid:12) Sp( n − p ) = S ν n (cid:12)(cid:12) Sp( n − p ) = S ( ν n − p ⊕ p ⊕ p )= S ν n − p ⊕ S ( p ⊕ p ) ⊕ (cid:0) ν n − p ⊗ ( p ⊕ p ) (cid:1) = S ν n − p ⊕ S ( p ⊕ p ) ⊕ ( ν n − p ⊗ p ) ⊕ ( ν n − p ⊗ p )= S ν n − p ⊕ S ( p ⊕ p ) ⊕ ν n − p ⊕ · · · ⊕ ν n − p | {z } p − times ⊕ ν n − p ⊕ · · · ⊕ ν n − p | {z } p − times= S ν n − p ⊕ ⊕ · · · ⊕ | {z } ( p − ) − times ⊕ ν n − p ⊕ · · · ⊕ ν n − p | {z } p − times . In the first line above, p denotes the direct sum 1 ⊕ · · · ⊕ χ ⊗ C = 1 ⊕ · · · ⊕ ⊕ ν n − p ⊕ · · · ⊕ ν n − p . (4)This decomposition induces an Ad(Sp( n − p ))-invariant decomposition of m ⊗ C as m ⊗ C = m ⊕ m ⊕ · · · ⊕ m s , (5)where the first (cid:18) k − (cid:19) Ad(Sp( n − p ))-submodules are 1-dimensional and the rest 2 p are( n − p )-dimensional. Note that the decomposition (4) contains equivalent subrepresentations so acomplete description of all Sp( n )-invariant metrics on Sp( n ) / Sp( n − p ) is not easy.3.1. V p H n as total space over a generalized Wallach space. Let n = k + k + k and p = k + k . We consider the closed subgroup K = Sp( k ) × Sp( k ) × Sp( k ) of Sp( n ). Then fromProposition 2.1 K is contained in N Sp( n ) (Sp( n − p )). We consider the fibrationSp( k ) × Sp( k ) → G/H = Sp( n ) / Sp( n − p ) → Sp( n ) / (Sp( k ) × Sp( k ) × Sp( k ))Let a and p be the orthogonal complements of sp ( k ) in sp ( k ) ⊕ sp ( k ) ⊕ sp ( k ), and of sp ( k ) ⊕ sp ( k ) ⊕ sp ( k ) in sp ( n ), with respect to the negative of the Killing form of sp ( n ). The spaces a and p are called vertical and horizontal subspaces of g . Hence the tangent space of quaternionicStiefel manifold G/H can be written as m = a ⊕ p . We observe that p is the tangent space of thegeneralized Wallach space G/K = Sp( n ) / (Sp( k ) × Sp( k ) × Sp( k )). Actually for i = 1 , ,
3, weembed the Lie subalgebras sp ( k i ) = (cid:26)(cid:18) X i − ¯ Y i Y i ¯ X i (cid:19) (cid:12)(cid:12)(cid:12) X i ∈ u ( k i ) ,Y i is a k i × k i complex symmetric matrix (cid:27) in the Lie algebra sp ( k + k + k ) as follows: X − ¯ Y Y X , X − ¯ Y
00 0 0 0 0 00 0 0 0 0 00 Y X
00 0 0 0 0 0 , X − ¯ Y Y X . Then the tangent space p of G/K is given by k ⊥ in g = sp ( k + k + k ) with respect to theAd( G )-invariant inner product − B . If we denote by M ( a, b ) the set of all a × b matrices, then we ew homogeneous Einstein metrics on quaternionic Stiefel manifolds 5 see that p is given by p = A A − ¯ B − ¯ B − t ¯ A A − t ¯ B − ¯ B − t ¯ A − t ¯ A − t ¯ B − t ¯ B B B A ¯ A t B B − t A A t B t B − t A − t A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A ij , B ij ∈ M ( k i , k j )(1 ≤ i < j ≤ . If k , k , k are distinct then the isotropy representation of G/K can be written as a direct sumof three non equivalent subrepresentations. More precisely, let p i = ν k i ◦ σ k i be the standardrepresentation of K i.e.Sp( k ) × Sp( k ) × Sp( k ) σ ki −→ Sp( k i ) ν ki −→ Aut( C k i ) . By using the relation Ad G ⊗ C | K = (Ad K ⊗ C ) ⊕ ( χ ⊗ C ) we have thatAd G ⊗ C (cid:12)(cid:12) K = S ν k + k + k (cid:12)(cid:12) K = S ( p k ⊕ p k ⊕ p k )= S p k ⊕ S p k ⊕ S p k ⊕ ( p k ⊗ p k ) ⊕ ( p k ⊗ p k ) ⊕ ( p k ⊗ p k )= (Ad K ⊗ C ) ⊕ ( p k ⊗ p k ) ⊕ ( p k ⊗ p k ) ⊕ ( p k ⊗ p k ) . Hence χ ⊗ C = ( p k ⊗ p k ) ⊕ ( p k ⊗ p k ) ⊕ ( p k ⊗ p k ) where the dimensions of the subrepresentations arerespectively 2 k k , k k and 2 k k . Therefore, the complexified tangent space of G/K is expressedas a direct sum of three non equivalent irreducible submodules as n ⊗ C = n ⊕ n ⊕ n . Hence, thereal tangent space can be written as p = p ⊕ p ⊕ p , where p ⊗ C = n , p ⊗ C = n , p ⊗ C = n and the dimensions of p ij , i = j are dim( p ij ) = 4 k i k j . We set p = A − ¯ B − t ¯ A − t ¯ B B A t B − t A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A , B ∈ M ( k , k ) , p = A − ¯ B − t ¯ A − t ¯ B B A t B − t A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A , B ∈ M ( k , k ) , p = A − ¯ B − t ¯ A − t ¯ B
00 0 0 0 0 00 0 B A t B − t A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A , B ∈ M ( k , k ) . Hence for the tangent space m of G/H we have the decomposition m = a ⊕ p = sp ( k ) ⊕ sp ( k ) ⊕ p ⊕ p ⊕ p . (6) Andreas Arvanitoyeorgos, Yusuke Sakane and Marina Statha
So for k i distinct the G -invariant metrics ˇ g on G/K are determined by the following Ad( K )-invariantscalar products on p : ( · , · ) = x ( − B ) | p + x ( − B ) | p + x ( − B ) | p . Also, any Ad( H )-invariant inner product on a defines a K -invariant metric ˆ g on K/H . The directsum of these inner products on m = a ⊕ p defines a G -invariant metric g = ˆ g + ˇ g on the Stiefelmanifold G/H , called submersion metric . This metric is determined by the following Ad( K )-invariant inner product on m : h· , ·i = x ( − B ) | sp ( k ) + x ( − B ) | sp ( k ) + x ( − B ) | p + x ( − B ) | p + x ( − B ) | p , (7)where x , x , x ij , i, j = 1 , , i = j , belong to R + . In general the submersion metric corre-sponds to an element of ( M G ) Φ K , as defined in Section 2.We set in the decomposition (6) sp ( k ) = p and sp ( k ) = p . Then we see that the followingrelations hold: Lemma 3.1.
The submodules in the decomposition (6) satisfy the following bracket relations :[ p , p ] = p , [ p , p ] = p , [ p , p ] = p , [ p , p ] = p , [ p , p ] = p , [ p , p ] = p , [ p , p ] ⊂ p , [ p , p ] ⊂ p , [ p , p ] ⊂ p , [ p , p ] ⊂ p ⊕ p , [ p , p ] ⊂ p ⊕ p , [ p , p ] ⊂ p ⊕ p ,and the other bracket relations are zero. V p H n as total space over a generalized flag manifold. Let n = k + k + k and p = k + k .Now we consider the closed subgroup K = U( p ) × Sp( n − p ) of Sp( n ). From Proposition 2.1 wehave that K ⊂ N Sp( n ) (Sp( n − p )). We consider the fibrationU( p ) → G/H = Sp( n ) / Sp( n − p ) → Sp( n ) / (U( p ) × Sp( n − p )) . The fiber U ( p ) is diffeomorphic to the Lie group U(1) × SU( p ) so the vertical subspace h of sp ( n )is written as direct sum h = h ⊕ h , where h is the center of U( p ). We set d = dim( h ) = 1and d = dim( h ) = p −
1. We also observe that the horizontal subspace p of sp ( n ) is the tangentspace of generalized flag manifold Sp( n ) / (U( p ) × Sp( n − p )). So the isotropy representation of thisspace given as follows:Ad Sp( n ) ⊗ C (cid:12)(cid:12) U( p ) × Sp( n − p ) = S ν n (cid:12)(cid:12) U( p ) × Sp( n − p ) = S ( µ p ⊕ ¯ µ p ⊕ ν n − p )= S ν n − p ⊕ ( µ p ⊗ ¯ µ p ) ⊕ S µ p ⊕ S ¯ µ p ⊕ ( µ p ⊗ ν n − p ) ⊕ (¯ µ p ⊗ ν n − p )= Ad U( p ) × Sp( n − p ) ⊗ C ⊕ S µ p ⊕ S ¯ µ p ⊕ ( µ p ⊗ ν n − p ) ⊕ (¯ µ p ⊗ ν n − p ) . In the above calculations µ p : U( p ) → Aut( C p ) is the standard representation of Lie group U( p ) andAd U( p ) ⊗ C = µ p ⊗ ¯ µ p the complexified adjoint representation of U( p ). Therefore the complexifiedisotropy representation of the generalized flag manifold G/K is χ ⊗ C = S µ p ⊕ S ¯ µ p ⊕ ( µ p ⊗ ν n − p ) ⊕ (¯ µ p ⊗ ν n − p ). The dimension of the first two subrepresentations is 2 p ( n − p ) and of the rest twois (cid:0) p +12 (cid:1) . Also the representations µ p ⊗ ν n − p and ¯ µ p ⊗ ν n − p are conjugate to each other and thesame holds for the representations S µ p and S ¯ µ p . Thus p decomposes in two real Ad( K )-invariantirreducible submodules p and p of dimension d = dim( p ) = 4 p ( n − p ) and d = dim( p ) = p ( p + 1). So the tangent space m of the Stiefel manifold Sp( n ) / Sp( n − p ) can be expressed as m = h ⊕ p = h ⊕ h ⊕ p ⊕ p = n ⊕ n ⊕ n ⊕ n . (8) ew homogeneous Einstein metrics on quaternionic Stiefel manifolds 7 In this case the submersion metric on the Stiefell manifold
G/H is determined by the followingAd( K )-invarinat inner product on m : h· , ·i = u ( − B ) | n + u ( − B ) | n + u ( − B ) | n + u ( − B ) | n , (9)where u i , i = 0 , , , R + . It is easy to see that the following relations hold:[ n , n ] ⊂ h ⊕ n , [ n , n ] ⊂ h , [ n , n ] ⊂ n . (10)3.3. The Ricci tensor for metrics corresponding to inner products (7) and (9).
FromLemma 3.1 we see that the only non zero triples (up to permutation of indices) for the metriccorresponding to (7) are A , A , A , A , A , A , A (12)(23)(13) (11)We recall the following result by A. Arvanitoyeorgos, V.V. Dzhepko and Yu.G. Nikonorov: Lemma 3.2. ([ArDzNi1], [ArDzNi2])
For a, b, c = 1 , , and ( a − b )( b − c )( c − a ) = 0 the followingrelations hold : A aaa = k a ( k a + 1)(2 k a + 1) n + 1 , A ( ab )( ab ) a = k a k b (2 k a + 1)( n + 1) , A ( ab )( bc )( ac ) = 2 k a k b k c n + 1 . Now for the metric corresponding to (9) we see that from relations (10) and [[ArMoSa], Propo-sition 6, p. 269] the only non zero triples are: A , A , A , A , A , A . From A. Arvanitoyeorgos, K. Mori and Y. Sakane we have the following:
Lemma 3.3. ([ArMoSa])
The triples A ijk are given as follows: A = d ( d + 4 d ) , A = 4 d ( d + 4 d ) A = 2 d (2 d + 2 − d )( d + 4 d ) A = d d ( d + 4 d ) A = 2 d ( d − d + 2 d ) A = d d ( d + 4 d ) . By using the above lemmas, we obtain the components of the Ricci tensor for the metrics (7)and (9).
Proposition 3.4.
The components of the Ricci tensor r for the invariant metric h· , ·i on Stiefelmanifold G/H defined by (7) are given as follows : r = k + 14( n + 1) x + k n + 1) x x + k n + 1) x x ,r = k + 14( n + 1) x + k n + 1) x x + k n + 1) x x ,r = 12 x + k n + 1) (cid:18) x x x − x x x − x x x (cid:19) − k + 18( n + 1) x x − k + 18( n + 1) x x ,r = 12 x + k n + 1) (cid:18) x x x − x x x − x x x (cid:19) − k + 18( n + 1) x x ,r = 12 x + k n + 1) (cid:18) x x x − x x x − x x x (cid:19) − k + 18( n + 1) x x . (12) Andreas Arvanitoyeorgos, Yusuke Sakane and Marina Statha
To find Einstein metrics of the form (7) reduces to find positive solutions of the system r = r , r = r , r = r , r = r . (13) Proposition 3.5.
The components of the Ricci tensor r for the invariant metric h· , ·i on Stiefelmanifold G/H defined by (9) are given as follows : r = u u d ( d + 4 d ) + u u d ( d + 4 d ) r = 14 d u d (2 d + 2 − d )( d + 4 d ) + u u d ( d + 4 d ) + u d u d ( d − d + 4 d ) r = 12 u − u u d ( d + 4 d ) − u (cid:18) u d + 4 d ) + u d ( d + 4 d ) (cid:19) r = 1 u (cid:18) − d ( d + 4 d ) (cid:19) + u u d ( d + 4 d ) − u (cid:18) u d + 4 d ) + u d − d + 4 d ) (cid:19) (14)The metric of the form (9) is Einstein if and only if the system: r = r , r = r , r = r , (15)has positive solutions. 4. Einstein metrics on V p H n In this section we solve the systems (13) and (15) for the various values of k i , i = 1 , , Einstein metrics for the inner products (7).Theorem 4.1.
For n = 3 , the Stiefel manifold V H n admits: (1) Eight invariant Einstein metrics which are determined by
Ad(Sp(1) × Sp(1) × Sp(1)) -invariant inner products of the form (7). Four of them are new, two are Jensen’s metricsand the other two are ADN metrics. (2)
Eight invariant Einstein metrics which are determined by
Ad(Sp(1) × Sp(1) × Sp(2)) -invariant inner products of the form (7). Four of them are new, two are Jensen’s metricsand the other two are ADN metrics.Proof.
For (1) we see that from Proposition 3.4 the Ricci components of the metric correspondingto Ad(Sp(1) × Sp(1) × Sp(1))-invariant inner products of the form (7) are given as follows: r = x x + x x + 18 x , r = x x + x x + 18 x r = − x x − x x + 116 (cid:18) x x x − x x x − x x x (cid:19) + 12 x ,r = − x x + 116 (cid:18) − x x x − x x x + x x x (cid:19) + 12 x ,r = 116 (cid:18) − x x x + x x x − x x x (cid:19) − x x + 12 x . ew homogeneous Einstein metrics on quaternionic Stiefel manifolds 9 We consider the system of equations (13) for n = 3. Then the metric corresponding to Ad(Sp(1) × Sp(1) × Sp(1))-invariant inner products of the form (7) is Einstein if the system (13) has positivesolutions. We normalize our equations by putting x = 1. Then we obtain the system of equations: f = x x x + x x x − x x x x − x x x − x x x + 2 x x x = 0 f = 3 x x x − x x + 2 x x x + 4 x x + 2 x x x − x x x +2 x x + 5 x x = 0 f = 3 x x − x x + 4 x x − x x − x x + 16 x x − x x = 0 f = − x x + 3 x x x − x x + 16 x x + 4 x − x = 0 (16)We consider a polynomial ring R = Q [ x , x , x , x ] and an ideal I generated by { f , f , f , f ,z x x x x − } to find non zero solutions of equations (16). We take a lexicographic order > with z > x > x > x > x for a monomial ordering on R . Then, by the aid of computer, wesee that a Gr¨obner basis for the ideal I contains the polynomial( x − h ( x ) , where h ( x ) is a polynomial of x given by h ( x ) = 26264641347161101886463 x − x +4744919846389271826285855 x − x +123660202199445490641611164 x − x +1125839862148616037654823494 x − x +4689788438841035164098977324 x − x +9384221512521610297473108154 x − x +7301130495287173304405589627 x − x − x + 3187252032549620236743974472 x − x − x +7301130495287173304405589627 x − x +9384221512521610297473108154 x − x +4689788438841035164098977324 x − x +1125839862148616037654823494 x − x +123660202199445490641611164 x − x +4744919846389271826285855 x − x +26264641347161101886463 . By solving the equation h ( x ) = 0 numerically, we obtain four positive solutions which are givenapproximately as 0 . , . , . , . I contains the polynomials x − α ( x ) , x − α ( x , x ) , x − α ( x , x )where α , α , α are polynomials of x and x with rational coefficients. By substituting thesolution of x into α , α and α we obtain four positive solutions of the system of equations { f = 0 , f = 0 , f = 0 , f = 0 } approximately as ( x , x , x , x , x ) ≈ (0 . , . , . , . , , (1 . , . , . , . , , (0 . , . , . , . , , (0 . , . , . , . , . Now we consider the case where x = 1. By substituting x = 1 into (16) and solving againnumerically, we obtain solutions approximately as ( x , x , x , x , x ) ≈ (0 . , . , . , , , (1 . , . , . , , , (0 . , . , . , , , (0 . , . , . , , . (17)The first two of the solutions (17) are Jensen’s metrics and the other two are ADN metrics.To prove part (2) of the theorem, we can work analogously. In this case the system (13) for k = k = 1, k = 2 and x = 1 is the following: g = 2 x x x + x x x − x x x x − x x x − x x x + 2 x x x = 0 g = 3 x x x − x x + 4 x x x + 4 x x + 4 x x x − x x x +4 x x + 5 x x = 0 g = 3 x x − x x + 6 x x − x x − x x + 20 x x − x x − x x = 0 g = − x x + 3 x x x − x x + 20 x x + 4 x − x = 0 . (18)A Gr¨obner basis for the ideal I of the polynomial ring R = Q [ x , x , x , x ] generated by { g , g , g , g , z x x x x − } and equipped with the lexicographic order > with z > x >x > x > x for a monomial ordering on R , contains the polynomial ( x − h ( x ). Thedegree of h ( x ) is 30. By performing analogous computations as in the proof of part (1) we obtainthe following Einstein metrics which correspond to the Ad(Sp(1) × Sp(1) × Sp(2))-invariant innerproducts of the form (7): The new metrics ( x , x , x , x , x ) ≈ (0 . , . , . , . , , (1 . , . , . , . , . , . , . , . , , (0 . , . , . , . , . the Jensen’s metrics: ( x , x , x , x , x ) ≈ (0 . , . , . , , , (1 . , . , . , , x , x , x , x , x ) ≈ (0 . , . , . , , , (0 . , . , . , , . (cid:3) By working in a similar manner as in the above proof, we conjecture the existence of new Einsteinmetrics on V n − H n ( n ≥
3) and V n − H n ( n ≥
5) as shown in Table 1. V k + k H k + k + k Jensen’s metrics New metrics k = n − , k = k = 13 ≤ n ≤ ≤ n ≤
29 2 830 ≤ n k = n − , k = 1 , k = 25 ≤ n ≤ n = 10 2 811 ≤ n ≤
27 2 628 ≤ n ≤
40 2 841 ≤ n Table 1.
Conjectured number of Einstein metrics corresponding to Ad(Sp( k ) × Sp( k ) × Sp( k ))-invariant inner products of the form (7) ew homogeneous Einstein metrics on quaternionic Stiefel manifolds 11 Einstein metrics for inner products (9).
For the invariant metrics on Sp( n ) / Sp( n − p )determined by the Ad(U( p ) × Sp( n − p ))-invariant scalar products (9), if either u = u , u = u or u = u , we see that invariant Einstein metrics on Sp( n ) / Sp( n − p ) of the form (9) are Jensen’smetrics. Theorem 4.2.
For ≤ p ≤ n , there exist two invariant Einstein metrics on Sp( n ) / Sp( n − p ) ofthe form (9) which are different from Jensen’s metrics.Proof. From Proposition 3.5 we see that the components of the Ricci tensor for the metric of theform (9) are given by r = ( n − p )4( n + 1) u u + p + 14( n + 1) u u , r = p n + 1) 1 u + ( n − p )( n + 1) u u + ( p + 2)8( n + 1) u u ,r = 12 u − p + 18( n + 1) u u − p ( n + 1) u (cid:0) u + ( p − u (cid:1) ,r = 1 u p + 12( n + 1) + ( n − p )4( n + 1) u u − p ( n + 1) u (cid:18) u + u ( p − p + 2)2 (cid:19) . We put u = 1 so the system { r = r , r = r , r = r } is given by f = 2 u u ( n − p ) − u ( n − p ) + 2( p + 1) u u u − ( p + 2) u u − pu = 0 f = u ((2 n − p ) p − − n + 1) pu u + p u + p ( p + 2) u u + p ( p + 1) u + u u = 0 f = 4( n + 1) pu + p ( − n + p −
1) + 2( p − p + 2) u u − ( p − p + 1) u − p ( p + 1) u + 4 u u − u = 0 . (19)We consider a polynomial ring R = Q [ z, u , u , u ] and an ideal I generated by { f , f , f , z u u u − } to find non zero solutions of the above equations. We take a lexicographic order > with z > u > u > u for a monomial ordering on R . Then, by the aid of computer, we see thata Gr¨obner basis for the ideal I contains the polynomial ( u − U ( u ) , where U is a polynomialof u given by: U ( u ) = (4 n − p + 1) ( p − ( p + 2) u − n − p + 1) ( p − p + 2) (cid:0) − p + 2 np − p − p + 8 n (cid:1) u +(4 n − p + 1) (140 p − np + 726 p − n p − np + 1057 p + 352 n p − n p − np + 178 p + 896 n p + 1248 n p − np − p + 640 n p + 832 n p − np − p + 512 n + 1088 n + 608 n − u − n − p + 1)( − p − np − p + 2020 n p + 3174 np − p − n p +2412 n p + 10762 np − p + 576 n p − n p − n p + 9190 np + 1377 p +1024 n p − n p − n p − np − p + 512 n p − n p − n p − np − p − n − n − n + 128 n − u +(46 p + 2524 np + 2706 p − n p − np + 7173 p + 6624 n p − n p − np − p + 31936 n p + 160544 n p + 138044 n p − np − p − n p − n p + 60992 n p + 300208 n p + 126100 np + 14418 p + 24576 n p − n p − n p − n p − n p − np − p + 57344 n p +202752 n p + 196608 n p + 19072 n p − n p − np − p + 16384 n + 65536 n +90112 n + 52992 n + 17344 n + 3264 n + 248) u − − p + 274 np − p + 3408 n p + 8656 np − p − n p − n p +13352 np + 2146 p + 11264 n p + 10304 n p − n p + 8004 np + 5788 p − n p − n p − n p − n p − np − p + 4096 n p +7168 n p + 17664 n p + 52928 n p + 52752 n p + 13308 np + 1114 p + 4096 n p − n p − n p − n p − n p − np − p + 8192 n + 32768 n +47104 n + 31616 n + 12096 n + 2576 n + 256) u +(444 p − np + 1226 p + 1452 n p − np − p + 1856 n p + 11032 n p +1416 np − p − n p − n p + 1788 n p + 10132 np + 439 p + 2560 n p +2560 n p − n p − n p − np + 646 p + 3072 n p + 14336 n p +14080 n p − n p − np − p + 5120 n p + 14336 n p + 11072 n p +1200 np − p + 1792 n + 3584 n + 1696 n + 320) u +2(3 p − np − p − p − np + 5 p + 124 n p + 34 np − p − n p − n p − np − p + 64 n p + 144 n p + 104 n p + 106 np + 9 p + 32 n p +64 n p + 88 np + 56 p + 16) u + (cid:0) np − p + p + 2 (cid:1) (cid:0) np − p + p + 1 (cid:1) . If u = 1 then U ( u ) = 0. We will prove that the equation U ( u ) = 0 has at least two positiveroots. Observe that U (0) = (cid:0) np − p + p + 2 (cid:1) (cid:0) np − p + p + 1 (cid:1) is positive for all p ≤ n and U (1) = 64 (cid:0) n p − n p + 32 n p + 4 n − np + 4 np + 4 p + 12 p − p (cid:1) × (cid:0) n p + 48 n − n p − n p + 112 n − np − np + 72 n + 4 p + 12 p + 5 p − p + 9 (cid:1) = (cid:0) (32 p + 32 p + 4)( n − p ) + (16 p + 24 p + 12 p )( n − p ) + 16 p ( n − p ) + 4 p + 4 p + p (cid:1) × (cid:0) (16 p + 48)( n − p ) + (32 p + 128 p + 112)( n − p ) + (16 p + 96 p + 168 p + 72)( n − p )+4 p + 28 p + 61 p + 42 p + 9 (cid:1) is positive for all p ≤ n . We also see that U (1 /
5) = − u ( n, p ) , where u ( n, p ) = 160000 n p − n p + 640000 n − n p + 230400 n p − n p + 2548800 n − n p + 582912 n p + 2333952 n p − n p +3695424 n + 1771264 n p − n p + 827840 n p + 8098416 n p − n p +2114464 n − n p − n p − n p − n p + 9653884 n p − n p + 135164 n + 1436864 np + 1036416 np + 1028576 np − np − np + 4423856 np − np − n − p − p − p +859256 p + 313895 p − p + 477967 p + 177080 p − . We claim that, for p ≤ n , U (1 /
5) is negative. Indeed, we expand u ( n, p ) at n = 4 p u ( n, p ) = (cid:0) p − p + 640000 (cid:1) (cid:18) n − p (cid:19) + (cid:0) p − p + 2744000 p + 2548800 (cid:1) (cid:18) n − p (cid:19) + 643 (cid:0) p − p + 166904 p + 427242 p + 173223 (cid:1) (cid:18) n − p (cid:19) ew homogeneous Einstein metrics on quaternionic Stiefel manifolds 13 + 1627 (cid:0) p − p + 2322548 p + 19232541 p + 15903405 p + 3568158 (cid:1) (cid:18) n − p (cid:19) + 427 (2691728 p − p − p + 46116396 p + 53547669 p + 20344662 p + 912357) (cid:18) n − p (cid:19) + 481 (1202864 p + 3605920 p − p + 38327736 p + 51328395 p +23978268 p − p − (cid:18) n − p (cid:19) + 1729 (383344 p + 23887424 p − p + 136017720 p + 182269791 p +96117138 p − p − p − . Then we see that the coefficients are polynomials of p and are positive for p ≥
1. Hence we seethat the equation U ( u ) = 0 has at least two positive solutions u = α , α with 0 < α < / / < α < > with z > u > u > u ) for theideal J generated by the polynomials { f , f , f , z u u u ( u − − } . This basis contains thepolynomial U ( u ) and the polynomials a ( n, p ) u + W ( u , n, p ) , a ( n, p ) u + W ( u , n, p )where a i ( n, p ) ( i = 1 ,
2) are polynomials of n and p , and W i ( u , n, p ) ( i = 1 ,
2) are polynomials of u , n and p . For 1 ≤ p < n we can see that the polynomials a i ( n, p ) ( i = 1 ,
2) are positive. Thus,for the positive values u = α , α we obtain the real values u = γ , γ and u = β , β as solutionsof the system (19). We prove next that these solutions are positive. We consider the ideal J andwe take the lexicographic order > with z > u > u > u for a monomial ordering on R . Then wesee that the Gr¨obner basis for the ideal J contains the polynomial U ( u ): U ( u ) = 2( p + 1)( p + 2) (cid:0) p + 8 p + p + 2 (cid:1) u − n + 1)( p + 2)(3 p + 1) (cid:0) p + p + 2 (cid:1) u + (cid:0) n (cid:0) p + 58 p + 61 p + 20 p + 4 (cid:1) − p − p − p + 11 p + 28 p + 4 (cid:1) u − n + 1)( n − p ) (cid:0) p + 9 p + 9 p + 6 (cid:1) u + 2( p + 1)( n − p ) (cid:0) n (21 p + 26) p − p − p +16 p + 12) u − n + 1)(7 p + 8 p + 4)( n − p ) (34 np + 116 np + 122 np + 40 np + 8 n − p − p − p + 11 p + 28 p + 4) u + ( n − p ) (cid:0) np + 24 np − p − p + 20 p + 14 (cid:1) u − n + 1) p ( n − p ) u + 2( n − p ) (cid:0) np − p + p + 1 (cid:1) . The above polynomial can be written as follows: U ( u ) = 2( p + 1)( p + 2) (cid:0) p + 8 p + p + 2 (cid:1) u − (cid:0)(cid:0) p + 40 p + 60 p + 64 p + 16 (cid:1) ( n − p ) + 12 p + 52 p + 100 p + 124 p + 80 p + 16 (cid:1) u + (cid:0)(cid:0) p + 116 p + 122 p + 40 p + 8 (cid:1) ( n − p ) + 4 p + 15 p + 34 p + 51 p + 36 p + 4 (cid:1) u − (cid:0)(cid:0) p + 72 p + 72 p + 48 (cid:1) ( n − p ) + (cid:0) p + 104 p + 144 p + 120 p + 48 (cid:1) ( n − p ) (cid:1) u + (cid:0)(cid:0) p + 94 p + 52 p (cid:1) ( n − p ) + (cid:0) p + 28 p + 50 p + 56 p + 24 (cid:1) ( n − p ) (cid:1) u − (cid:0)(cid:0) p + 32 p + 16 (cid:1) ( n − p ) + (cid:0) p + 60 p + 48 p + 16 (cid:1) ( n − p ) (cid:1) u + (cid:0)(cid:0) p + 24 p (cid:1) ( n − p ) + (cid:0) p + 15 p + 20 p + 14 (cid:1) ( n − p ) (cid:1) u − (cid:0) (cid:0) p + p (cid:1) ( n − p ) + 8 p ( n − p ) (cid:1) u + 4 p ( n − p ) + 2 (cid:0) p + p + 1 (cid:1) ( n − p ) . Then we see that the coefficients of the polynomial U ( u ) are positive for even degree andnegative for odd degree terms. Thus, if the equation U ( u ) = 0 has real solutions, then these are all positive. So the solutions u = γ , γ are positive. Now if we take the lexicographic order > with z > u > u > u for a monomial ordering on R we see that the Gr¨obner basis for ideal J contains the polynomial U ( u ): U ( u ) = X j =0 b j ( n, p ) u j , where b j ( n, p ) ( j = 0 , . . . ,
8) are polynomials of n, p given by b ( n, p ) = (4 n − p + 1) (cid:0) np − p + p + 2 (cid:1) , − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − b ( n, p ) = − n − p + 1) (cid:0) np − p + p + 2 (cid:1) (cid:0) p + p − p (cid:1) (cid:18) n − p (cid:19) + 23 (cid:0) p + 35 p + p − (cid:1) (cid:18) n − p (cid:19) + 19 (cid:0) p + 80 p + 64 p + 81 p + 15 p (cid:1)(cid:1) , − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − b ( n, p ) = 2(4 n − p + 1) (cid:16) p (cid:0) p + 3 p + 8 p + 4 (cid:1) (cid:18) n − p (cid:19) + 1283 p (cid:0) p + 99 p (cid:1) +251 p + 229 p + 105 p +24) (cid:18) n − p (cid:19) + 169 (cid:0) p + 5055 p + 12251 p + 14491 p +11064 p + 5682 p +1620 p + 144) (cid:18) n − p (cid:19) + 827 (7682 p + 29130 p + 69385 p +93512 p + 91377 p + 66996 p + 34938 p + 11556 p +1836) (cid:18) n − p (cid:19) + 481 (cid:0) p +70326 p + 170383 p + 251516 p + 281415 p + 238755 p + 159678 p + 82647 p +29700 p +6156) (cid:18) n − p (cid:19) + 1243 (cid:0) p + 101688 p + 246461 p + 374056 p +441102 p + 375186 p + 248805 p + 99990 p − p − p − (cid:17) , − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − b ( n, p ) = − n − p ) (cid:16) (cid:0) ( p − + 18( p − + 145( p − + 665( p − +1841( p − + 3036( p − + 2740( p −
2) + 1040 (cid:1) (cid:18) n − p (cid:19) + (cid:0) p − +59392( p − + 539648( p − + 2855808( p − + 9527680( p − +20345728( p − + 26992640( p − + 20261376( p −
2) + 6572032 (cid:1) (cid:18) n − p (cid:19) + 323 (cid:0) p − + 14659( p − + 148284( p − + 886309( p − + 3429435( p − +8861566( p − + 15226429( p − + 16721530( p − + 10623556( p − (cid:1) (cid:18) n − p (cid:19) + 827 (cid:0) p − + 2863646( p − + 11577253( p − +30611791( p − + 54558303( p − + 65610147( p − + 51897508( p − +25321372( p − + 6590944( p −
1) + 611632 (cid:1) (cid:18) n − p (cid:19) + 827 (cid:0) p − +46794397( p − + 228835788( p − + 781428193( p − + 1896404686( p − +3261741064( p − + 3884236767( p − + 3038186126( p − + 1397340138( p − (cid:1) (cid:18) n − p (cid:19) + 281 (cid:0) p − + 1292327534( p − +4925962096( p − + 13582984080( p − + 27208043881( p − + 39118134870( p − ew homogeneous Einstein metrics on quaternionic Stiefel manifolds 15 +39117496470( p − + 25567903940( p − + 9637883888( p −
2) + 1529459552 (cid:1) (cid:18) n − p (cid:19) + 1729 (cid:0) p − + 16666847996( p − + 50919921029( p − +114707778059( p − + 189167284379( p − + 222947172468( p − +178718186969( p − + 88184923605( p − + 21024910874( p −
2) + 644929528 (cid:1)(cid:17) , − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − b ( n, p ) = (cid:0) p + 24576 p + 86016 p + 131072 p + 245760 p + 294912 p + 65536 p (cid:1) × (cid:18) n − p (cid:19) + 20483 (cid:0) p + 321 p + 1233 p + 2261 p + 3750 p + 4782 p + 2740 p + 768 p +96 p (cid:1) (cid:18) n − p (cid:19) + 2569 (cid:0) p + 28932 p + 118842 p + 247316 p + 400641 p + 523896 p +417748 p + 207240 p + 67440 p + 11808 p + 576 (cid:1) (cid:18) n − p (cid:19) + 25627 (cid:0) p + 180444 p +778551 p + 1777939 p + 2922618 p + 3878928 p + 3651965 p + 2429397 p + 1195056 p +403596 p + 79488 p + 6912 (cid:1) (cid:18) n − p (cid:19) + 12881 (cid:0) p + 1372854 p + 6131907 p +15018679 p + 25376340 p + 34044954 p + 35115293 p + 27920751 p + 17703822 p +8548038 p + 2855736 p + 589464 p + 57024 (cid:1) (cid:18) n − p (cid:19) + 32243 (cid:0) p +13148268 p + 60007947 p + 154833551 p + 269354535 p + 363913566 p + 393754726 p +350780631 p + 265670607 p + 163840680 p + 75798837 p + 24414912 p + 4825656 p +402408 (cid:1) (cid:18) n − p (cid:19) + 16729 (cid:0) p + 38994072 p + 179809266 p + 481672982 p +860592570 p + 1167204042 p + 1287983179 p + 1232426736 p + 1067917563 p +795423942 p + 465504435 p + 200889342 p + 59178519 p + 10162260 p + 790236 (cid:1) (cid:18) n − p (cid:19) + 82187 (cid:0) p + 65954616 p + 304768332 p + 838750310 p + 1537511100 p +2100643851 p + 2341188529 p + 2357827446 p + 2283500553 p + 1980427914 p +1382011065 p + 732725784 p + 280881675 p + 72047799 p + 11519658 p + 892296 (cid:1) (cid:18) n − p (cid:19) + 16561 (cid:0) p + 196359072 p + 904687368 p + 2541976256 p + 4803063945 p +6757501974 p + 7870887961 p + 8606912064 p + 9296805819 p + 8981127162 p +6981506001 p + 4203824292 p + 1914330996 p + 634122108 p + 148364622 p +22149936 p + 1627128 (cid:1) , − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − b ( n, p ) = − (cid:16) (cid:0) p + 26624 p + 92160 p + 161792 p + 106496 p + 16384 p + 16384 p (cid:1) × (cid:18) n − p (cid:19) + 5123 (cid:0) p + 1307 p + 4563 p + 8813 p + 8036 p + 3304 p + 1240 p + 432 p +96 p (cid:1) (cid:18) n − p (cid:19) + 1289 (cid:0) p + 54451 p + 192546 p + 397024 p + 434848 p + 259613 p +112820 p + 44496 p + 15864 p + 4752 p + 288 (cid:1) (cid:18) n − p (cid:19) + 8729 (cid:0) p − +162347188( p − + 1543182304( p − + 9135183025( p − + 37388517475( p − +111258986177( p − + 247002835859( p − + 413979949333( p − +524851826468( p − + 499980342751( p − + 352066978464( p − +177974328764( p − + 61419614640( p − + 13149220928( p −
1) + 1359883264 (cid:1) × (cid:18) n − p (cid:19) + 22187 (cid:0) p − + 866418488( p − + 14699562236( p − +154716831646( p − + 1128622433268( p − + 6037636612741( p − +24442007879239( p − + 76163670514689( p − + 183984545006253( p − +344157568970319( p − + 493885552028239( p − + 533343522444589( p − +419010880295843( p − + 225773544980448( p − + 74489301958682( p − (cid:1) (cid:18) n − p (cid:19) + 12827 (cid:0) p + 310214 p + 1106496 p + 2378078 p +2901275 p + 2104141 p + 1107319 p + 510813 p + 218826 p + 93474 p + 26676 p + 3456 (cid:1) × (cid:18) n − p (cid:19) + 3281 (cid:0) p + 4280770 p + 15245253 p + 33332221 p + 42879460 p +33980882 p + 19792277 p + 10467657 p + 5460186 p + 2954718 p + 1326888 p + 385992 p +59616 (cid:1) (cid:18) n − p (cid:19) + 8243 (cid:0) p + 37017932 p + 129677616 p + 279941813 p +358391423 p + 276796984 p + 149657704 p + 79836921 p + 54122511 p + 42334110 p +27686430 p + 12549384 p + 3621996 p + 480168 (cid:1) (cid:18) n − p (cid:19) + 16561 (cid:0) p − +594225632( p − + 10682883096( p − + 119460467888( p − +929124476957( p − + 5324019350794( p − + 23224120188504( p − +78573533111735( p − + 208092670261551( p − + 432175097670381( p − +700253199489961( p − + 873949665477915( p − + 821258270676976( p − +559567017398773( p − + 259270566907163( p − + 72376779210930( p − (cid:1)(cid:17) , − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − b ( n, p ) = 2 (cid:16)(cid:0) p + 163840 p + 245760 p + 262144 p + 65536 p (cid:1) (cid:18) n − p (cid:19) + (cid:0) p + 1851392 p + 3360768 p + 3985408 p + 2187264 p + 376832 p (cid:1) (cid:18) n − p (cid:19) + 10243 (cid:0) p + 25063 p + 52965 p + 70829 p + 53652 p + 19528 p + 2496 p − p (cid:1) × (cid:18) n − p (cid:19) + 2569 (cid:0) p + 755751 p + 1803240 p + 2693636 p + 2483131 p +1301357 p + 340194 p + 27276 p − p − p (cid:1) (cid:18) n − p (cid:19) + 12827 (cid:0) p +6900068 p + 18123783 p + 29729521 p + 31437354 p + 20579762 p + 7722099 p +1317717 p − p − p − p (cid:1) (cid:18) n − p (cid:19) + 6481 (cid:0) p − +118846646( p − + 986151751( p − + 4971372165( p − + 16952334168( p − +41153296820( p − + 72822618167( p − + 94485465333( p − + 89054474614( p − +59359625980( p − + 26515906016( p − + 7115536800( p −
1) + 866081984 (cid:1) (cid:18) n − p (cid:19) + 16243 (cid:0) p − + 893886380( p − + 8009668926( p − + 43931708085( p − +164515487580( p − + 443853483382( p − + 886597009628( p − +1325763408157( p − + 1481799690902( p − + 1220624912368( p − +719031079864( p − + 286287517072( p − + 68910586496( p −
1) + 7554724096 (cid:1) × (cid:18) n − p (cid:19) + 16729 (cid:0) p − + 1042222652( p − + 9979234256( p − +58831030897( p − + 238508655079( p − + 702977316389( p − +1551902086965( p − + 2603556297720( p − + 3330426101016( p − ew homogeneous Einstein metrics on quaternionic Stiefel manifolds 17 +3226707834546( p − + 2325808006688( p − + 1206279110888( p − +424160371456( p − + 90114540928( p −
1) + 8673425408 (cid:1) (cid:18) n − p (cid:19) + 42187 (cid:0) p − + 2722292056( p − + 27568091156( p − +172620320070( p − + 747161024728( p − + 2366345497299( p − +5659184311954( p − + 10391520423657( p − + 14742382956732( p − +16117344646500( p − + 13420607976808( p − + 8317499536784( p − +3686792531040( p − + 1090498987456( p − + 188447185920( p − (cid:1) (cid:18) n − p (cid:19) + 119683 (393910608( p − + 15229116448( p − +275646193992( p − + 3100284455968( p − + 24249634772905( p − +139844686276206( p − + 614944316806352( p − + 2102754931172164( p − +5648681203773822( p − + 11955552983145732( p − + 19859975096618684( p − +25602952630231152( p − + 25089976494997385( p − + 18047962971277118( p − +8974256392885572( p − + 2750739644643676( p −
2) + 390518428891112) (cid:17) , − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − b ( n, p ) = − p ) (cid:16) (cid:0) p + 163840 p + 98304 p + 65536 p (cid:1) (cid:18) n − p (cid:19) + (cid:0) p + 1867776 p + 1867776 p + 1179648 p + 393216 p (cid:1) (cid:18) n − p (cid:19) + (cid:0) p + 8690688 p + 11844096 p + 9195520 p + 4462592 p + 966656 p (cid:1) (cid:18) n − p (cid:19) + 1289 (cid:0) p + 1551023 p + 2630223 p + 2519177 p + 1551591 p + 567228 p + 89244 p +144 p (cid:1) (cid:18) n − p (cid:19) + 2569 (cid:0) p + 1198141 p + 2397385 p + 2739873 p + 2044315 p +996450 p + 282204 p + 34920 p + 288 p (cid:1) (cid:18) n − p (cid:19) + 3227 (cid:0) p + 28318098 p +64527951 p + 85185889 p + 74405822 p + 44421800 p + 17327475 p + 3959649 p +424512 p + 8532 p (cid:1) (cid:18) n − p (cid:19) + 8243 (cid:0) p + 644522804 p + 1629982860 p +2420187287 p + 2406568896 p + 1679806635 p + 813030876 p + 259492401 p + 50300136 p +4946589 p + 85536 p − p (cid:1) (cid:18) n − p (cid:19) + 8243 (cid:0) p + 253174084 p +696564112 p + 1138779993 p + 1260935722 p + 998072517 p + 566765046 p +225114543 p + 59846670 p + 9584811 p + 536058 p − p − p (cid:1) (cid:18) n − p (cid:19) + 4729 (cid:0) p + 335998500 p + 989279874 p + 1750818979 p + 2123240909 p +1867929560 p + 1203732324 p + 558263160 p + 178897338 p + 34688142 p + 1198962 p − p − p − p (cid:1) (cid:18) n − p (cid:19) + 119683 (489042736( p − +10741721536( p − + 109618245032( p − + 690084927664( p − +2999595396751( p − + 9542417798552( p − + 22963667848149( p − +42583235164476( p − + 61361832092016( p − + 68713003505656( p − +59299589725888( p − + 38721416081312( p − + 18510971432512( p − +6112599731968( p − + 1245725978624( p −
1) + 117991145472) (cid:17) , − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − b ( n, p ) = (2 + p + 8 p + 5 p ) (cid:16) (cid:0) p + 65536 p (cid:1) (cid:18) n − p (cid:19) + (cid:0) p + 786432 p +393216 p (cid:1) (cid:18) n − p (cid:19) + (cid:0) p + 3801088 p + 3506176 p + 983040 p (cid:1) (cid:18) n − p (cid:19) + 2569 (cid:0) p + 350806 p + 455973 p + 244692 p + 47088 p + 216 p (cid:1) (cid:18) n − p (cid:19) + (cid:0) p + 143439872 p p p + 7532544 p + 1134592 p +24576 p (cid:1) (cid:18) n − p (cid:19) + 25627 (cid:0) p + 1714250 p + 3364529 p + 3336384 p + 1797066 p +516528 p + 71712 p + 3888 p (cid:1) (cid:18) n − p (cid:19) + 32243 (cid:0) p + 81095072 p + 183025959 p +218550762 p + 151598655 p + 62898390 p + 15659973 p + 2346894 p + 195129 p + 1458 p (cid:1) × (cid:18) n − p (cid:19) + 32243 (cid:0) p + 33305912 p + 84256741 p + 116455788 p + 97611129 p +52111152 p + 18296307 p + 4361364 p + 706887 p + 68040 p + 2916 p (cid:1) (cid:18) n − p (cid:19) + 32729 (cid:0) p + 23347544 p + 64988012 p + 101259894 p + 98684001 p + 63802980 p +28499985 p + 9061956 p + 2056995 p + 325620 p + 40095 p + 4374 p (cid:1) (cid:18) n − p (cid:19) + 119683 (289238768 p + 2039206624 p + 6165039240 p + 10647882432 p + 11790525807 p +8913187026 p + 4784163588 p + 1862446284 p + 528299010 p + 109752408 p + 15825132 p +236196 p − p + 39366) (cid:17) . Thus, for 2 ≤ p ≤ n , we see that the coefficients b j ( n, p ) ( j = 0 , . . . ,
8) of the polynomial U ( u )are positive for even degree and negative for odd degree terms and hence, if the equation U ( u ) = 0has real solutions, then these are all positive. In particular, the solutions u = β , β are positive.Hence we see that the solutions of the system (19) are of the form: { u = β , u = α , u = γ , u = 1 } and { u = β , u = α , u = γ , u = 1 } and satisfy α , α = 1 (cid:3) References [ArDzNi1] A. Arvanitoyeorgos, V.V. Dzhepko and Yu. G. Nikonorov:
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University of Patras, Department of Mathematics, GR-26500 Rion, Greece
E-mail address : [email protected] Osaka University, Department of Pure and Applied Mathematics, Graduate School of InformationScience and Technology, Toyonaka, Osaka 560-0043, Japan
E-mail address : [email protected] University of Patras, Department of Mathematics, GR-26500 Rion, Greece
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