Wolfgang Willems

Alexander Zimmermann

ON MORITA THEORY FOR SELF-DUAL MODULES

Let G be a finite group and let k be a field of characteristic p. It is known that a kG-module V carries a non-degenerate G-invariant bilinear form b if and only if V is self-dual. We show that whenever a Morita bimodule M which induces an equivalence between two blocks B(kG) and B(kH) of group algebras kG and kH is self-dual then the correspondence preserves self-duality. Even more, if the bilinear form on M is symmetric then for p odd the correspondence preserves the geometric type of simple modules. In characteristic 2 this holds also true for projective modules.

Introduction

Let k be a field and let G be a finite group. It is well-known that a kG-module V , simply called a G-module, carries a non-degenerate G-invariant k-bilinear form if and only if V is isomorphic to its dual V * = Hom k (V, k). Up to an automorphism of V , a simple self-dual G-module V ∼ = V * carries exactly one G-invariant bilinear form different from the zero. In case k is large enough, for instance k algebraically closed, this means that such a form, if it exists, is unique up to a scalar. If in addition the characteristic p of k is odd this form is symmetric or antisymmetric. The case p = 2 turns out to be more subtle. Note that in this case an antisymmetric form is symmetric. By Fong's Lemma ( [START_REF] Huppert | Finite groups II[END_REF], Chap. VII, Theorem 8.13), a simple self-dual G-module V different from the trivial module always carries a non-degenerate G-invariant symplectic form. However, it may happen that V even has a non-degenerate G-invariant quadratic form. This holds true, for instance, if G is solvable but not in general. What really happens is not an easy task to decide; it is a question in cohomology (see [START_REF] Sin | G-invariant quadratic forms[END_REF]), at least for arbitrary groups. To be brief throughout the paper we call a G-module of symmetric, antisymmetric, symplectic, resp. quadratic type if V carries a non-degenerate G-invariant form of symmetric, antisymmetric, symplectic, resp. quadratic type. The reader not familiar with duality theory of modules may refer Chapter VII of [START_REF] Huppert | Finite groups II[END_REF].

In representation theory of finite groups G we are often interested in the category of modules belonging to a p-block B G of kG where the underlying field k is of characteristic p. Instead of investigating B G we may study a Morita equivalent p-block B H for some other group H. Recall that one calls two blocks Morita equivalent if their module categories are equivalent as exact categories. Morita theory shows that the module categories of the two blocks are equivalent if and only if there is an H × G-bimodule M = H M G where M is a progenerator for B H and for B G . Moreover the functor V → M ⊗ kG V defines an equivalence from the module category of B G onto that of B H . The correspondence preserves all functorial properties of module categories, in particular being simple, indecomposable, projective, Loewy structure and the like. For more details the reader is referred to Chapter 4 in [START_REF] Reiner | Maximal Orders[END_REF]. The first author is supported by a German-French grant "PROCOPE" of the DAAD, the second by a French-German grant "partenariat Hubert Curien PROCOPE dossier 14765WB". Both authors gratefully acknowledge financial support.

In this note we study the behaviour of self-duality including the type of forms under Morita equivalence. Given a group Γ recall that a Γ-module T carries a non degenerate Γ-invariant bilinear form if and only if T is self-dual. A first step to our main result, useful in its own right, is the following.

Theorem A. Let B G and B H be blocks for G and H respectively. Suppose that the H × Gbimodule M = H M G defines a Morita equivalence between B G and B H and suppose that

M is self-dual. Then a B G -left module V is self-dual if and only if M ⊗ kG V is a self-dual B H -left module.
We present now our main result for algebraically closed fields k of odd characteristic. Given a group Γ we call a self-dual kΓ-module T of symmetric type if T carries a non degenerate Γ-invariant symmetric bilinear form. If T is simple and self-dual the Γ-invariant form is unique up to a scalar and the form is then either symmetric or antisymmetric. We call this the type of the simple module T .

Theorem B. Let k be an algebraically closed field of odd characteristic. Let M be an H ×Gbimodule inducing a Morita equivalence between a kG-block B G and a kH-block B H . If M is of symmetric type then the equivalence preserves the type of simple self-dual modules.

The main result for fields of characteristic 2 is weaker. We call a kΓ-module T of quadratic type if T admits a Γ-invariant non degenerate quadratic form.

Theorem C. Let k be an algebraically closed field of characteristic 2. Let M be an H × Gbimodule inducing a Morita equivalence between a kG-block B G and a kH-block B H . If M is of symmetric type then the equivalence preserves the property "being of quadratic type" on projective modules.

Theorem A, Theorem B and Theorem C give invariants under Morita equivalence in case the Morita bimodule which induces the equivalence is self-dual for Theorem A, or even more of symmetric type for Theorem B and Theorem C. Moreover, we show that this is actually a property of the module categories, rather than of the particular bimodule. Indeed Remark 3.5 shows that if M is a self-dual Morita bimodule inducing an equivalence between two blocks of group algebras, then any Morita bimodule between these blocks is self-dual. If the Morita bimodule is moreover of symmetric type, then every Morita bimodule between these blocks is self-dual.

Conditions on the Morita equivalence are actually necessary. The assertions of the theorems are false if we do not require suitable conditions on the Morita module. If the two blocks B G and B H of the group G and the group H are Morita equivalent, it happens that every simple module over B G is self-dual whereas no simple module over B H is self-dual. In particular, no self-dual Morita bimodule exists in this case. An example is given by a non principal 7-block of the triple cover of the McLaughlin group and the principal 7-block of the Mathieu group M 23 . For details see Example 2.3 below. Even worse, two Morita equivalent blocks B G and B H may admit a self-dual Morita bimodule, but the types of the forms on the simple modules of B G and of B H are different. Hence it may happen that no Morita bimodule can be of symmetric type. An example is given by a non principal 3-block of SL 2 (5) and the principal 3-block of A 5 . Details are found in Example 2.2 below.

The paper is organised as follows. In Section 2 the above-cited examples are developed in detail. Section 3 is the technical heart of the paper since there Proposition 3.3 is shown: Given a self-dual, right-projective H × G-bimodule M and a self-dual G module V , Proposition 3.3 gives an explicit isomorphism between the H-module M ⊗ G V and its dual. The proposition applies as well for stable equivalences of Morita type. Except unicity, most of what follows is valid in this more general setting. Moreover, Theorem A is proven there. Section 4 then proves Theorem B and Section 5 proves Theorem C.

Examples

In this section we show that Morita equivalence does not need to preserve self-duality of simple modules. In case it does, examples show that the geometric type of simple modules may vary via the correspondence.

In order to state the examples we need that blocks are determined by their Brauer trees up to Morita equivalence. Kupisch determined the structure of indecomposable modules of a Brauer tree algebra, and hence he implicitly determined their module category, i.e., the Morita equivalence class of a Brauer tree algebra. In modern terms Kupisch's result reads as follows.

Theorem 2.1. (Kupisch [START_REF] Kupisch | Projektive Moduln endlicher Gruppen mit zyklischer p-Sylow-Gruppe[END_REF][START_REF] Kupisch | Unzerlegbare Moduln endlicher Gruppen mit zyklischer p-Sylow-Gruppe[END_REF]) Let A 1 and A 2 be two Brauer tree algebras. Then A 1 and A 2 are Morita equivalent if and only if their Brauer trees, including the exceptional vertex and its multiplicity, coincide.

In the following examples we shall use Brauer trees computed by Hiss and Lux in [START_REF] Hiss | Brauer trees of sporadic groups[END_REF]. The geometric type of simple self-dual modules, i.e. symmetric or alternating, is taken from [START_REF] Conway | Wilson Atlas of finite groups[END_REF] or [START_REF] Jansen | An Atlas of Brauer characters[END_REF].

Example 2.2. Let k be an algebraically closed field of characteristic 3 and let G = SL 2 (5) be the double cover of the alternating group A 5 of degree 5. Then a non principal 3-block, say B 1 (G), of G is a Brauer tree algebra corresponding to the Brauer tree

T : • • •
without an exceptional vertex. The ordinary characters of the algebra have dimensions 2, 4 and 2. Both of the simple modules in characteristic 3 of B 1 (G) admit an, up to scalar unique, G-invariant bilinear form. This form is alternating for both modules. The principal 3-block of the alternating group H = A 5 is also a Brauer tree algebra with Brauer tree T . The ordinary characters of this algebra have dimensions 1, 5 and 4. In characteristic 3 both simple modules admit an, up to scalar unique, A 5 -invariant bilinear form which is symmetric in both cases.

As a consequence, the type of bilinear forms is usually not preserved by Morita equivalence, even if self-duality is preserved on simple modules.

Example 2.3. Let k be again an algebraically closed field but of characteristic 7 and let G = 3M cL be the triple cover of a the McLaughlin simple group M cL.

Then, a non principal 7-block, say B 2 (3M cL), of G is a Brauer tree algebra corresponding to the Brauer tree

S : • • • • 2
with an exceptional vertex • 2 of multiplicity 2. The ordinary characters of the algebra have dimensions 792, 4752, 6336 and 2376 read from the left in the order of the tree. None of the three simple modules in characteristic 7 is self-dual, hence none of them admit a G-invariant bilinear form. The principal 7-block B 3 (M 23 ) of the Mathieu group H = M 23 is a Brauer tree algebra also with Brauer tree S. The ordinary characters of the algebra have dimensions 1, 1035, 2024 and 990 read from the left. The exceptional vertex corresponds to the two ordinary characters of degree 990. All simple modules in characteristic 7 admit an, up to scalar unique, H-invariant bilinear form which is symmetric in all three cases. As a consequence every Morita bimodule inducing a Morita equivalence between B 3 (M 23 ) and B 2 (3M cL) has the property that each self-dual simple B 3 (M 23 )-module is sent to a simple B 2 (3M cL)-module which is not self-dual.

Self-dual Morita bimodules

Let H and G be finite groups and let k be a field. Furthermore, let M be an H × Gbimodule where H acts on the left and G on the right. Then M * = Hom k (M, k) becomes an H × G-bimodule by setting

(hf g)m = f (h -1 mg -1 )
for h ∈ H, g ∈ G, f ∈ M * and m ∈ M . Indeed, the module structure follows by

((h 2 h 1 )f (g 1 g 2 ))m = f ((h 2 h 1 ) -1 m(g 1 g 2 ) -1 ) = f (h -1 1 h -1 2 mg -1 2 g -1 1 ) = (h 1 f g 1 )(h -1 2 mg -1 2 ) = (h 2 (h 1 f g 1 )g 2 )m. Definition 3.1. Let M be an H × G-bimodule. A k-bilinear form (•, •) on M is called H × G-invariant if (hmg, hng) = (m, n) for all h ∈ H, all g ∈ G and all m, n ∈ M .
The next lemma is well-known. For the readers convenience we give the proof.

Lemma 3.2. Let M be an H × G-bimodule. Then M is isomorphic to M * as an H × G- bimodule if and only if M carries a non-degenerate H × G-invariant bilinear form. More precisely, if ϕ : M → M * is an isomorphism, then (m, m ′ ) := ϕ(m)(m ′ ) is a non-degenerate H × G-invariant bilinear form. If (•, •) is a non-degenerate H × G-invariant bilinear form, then the map M ∋ m → (•, m) ∈ M * is an isomorphism of H × G-bimodules. Proof. If ϕ : M → M * denotes an H × G-isomorphism we put (m, n) := ϕ(m)n for m, n ∈ M . Clearly, the form (•, •) is k-linear, but it is also H × G-invariant since (hmg, hng) = ϕ(hmg)(hng) = (hϕ(m)g)(hng) (since ϕ is H × G-linear) = ϕ(m)(h -1 (hng)g -1 ) (the module structure of M * ) = ϕ(m)n = (m, n). In addition (•, •) is non-degenerate since ϕ is an isomorphism. Conversely, if (•, •) is a non-degenerate H × G-invariant bilinear form on M then we put ϕ(m)n := (m, n) for all m, n ∈ M . Note that ϕ is a k-isomorphism since the bilinear form is non-degenerate. Furthermore, ϕ is H × G-linear since (hmg, n) = ϕ(hmg)n and (hmg, n) = (m, h -1 ng -1 ) = ϕ(m)(h -1 ng -1 ) = (hϕ(m)g)n for all h ∈ H, g ∈ G and m, n ∈ M . Proposition 3.3. Let M be a self-dual H × G-bimodule which is projective as a G-module. If V is a self-dual G-left module then M ⊗ kG V is a self-dual H-left module. More precisely, if β : M → M * is an H × G-linear isomorphism and α : V → V * is a G-linear isomorphism, then M ⊗ kG V -→ (M ⊗ kG V ) * m ⊗ v →   m ′ ⊗ v ′ → g∈G β(mg -1 )(m ′ ) • α(v)(g -1 v ′ )   is an H-linear isomorphism.
Proof. Let β : M → M * denote an H ×G-isomorphism and α : V → V * a G-isomorphism. Therefore, since α and β are kG-linear the map γ

:= β ⊗ kG α : M ⊗ kG V → M * ⊗ kG V * is a well-defined k-vector space isomorphism satisfying (β ⊗ kG α)(m ⊗ kG v) = β(m) ⊗ kG α(v) for all m ∈ M and v ∈ V (cf. e.g. [2, Theorem 12.10]). Now, let m ∈ M, v ∈ V and h ∈ H. Since γ(h(m ⊗ kG v)) = γ(hm ⊗ kG v) = β(hm) ⊗ kG α(v) = hβ(m) ⊗ kG α(v) (since β is H-linear) = hγ(m ⊗ kG v).
the map γ is H-linear.

Finally, we claim that M * ⊗ kG V * ∼ = (M ⊗ kG V ) * as H-left modules. We prove this via a sequence of isomorphisms.

Observe that V * is not only a G-left but in a natural way also a G-right module via the definition

(f g)v = f (gv) for f ∈ V * , g ∈ G and v ∈ V .
Considering V * as G-right module Hom kG (M, V * ) makes sense. With this observation we have

Claim (i) Hom kG (M, V * ) ∼ = Hom k (M ⊗ kG V, k) = (M ⊗ kG V ) * as H-left modules.
The isomorphism is given by sending α ∈ Hom kG (M, V * ) to the map Φ(α) :

M ⊗ kG V → k with Φ(α)(m ⊗ kG v) = α(m)v where m ∈ M and v ∈ V . Note, that Φ is well-defined since α is G-linear, for Φ(α)(mg ⊗ kG v) = α(mg)(v) = (α(m)g) (v) = α(m)(gv) = Φ(α)(m ⊗ kG gv). Furthermore, Φ is H-linear since Φ(hα)(m ⊗ kG v) = (hα)(m)v = α(h -1 m)v = Φ(α)(h -1 m ⊗ kG v) = (hΦ(α))(m ⊗ kG v).
Finally, α → Φ(α) is obviously a monomorphism. In order to show that Φ is an isomorphism we define a k-linear map

Ψ : Hom k (M ⊗ kG V, k)→Hom kG (M, V * ) by putting (Ψ(ϕ)(m))(v) := ϕ(m ⊗ v) for ϕ ∈ Hom k (M ⊗ kG V, k), m ∈ M and v ∈ V . Since ϕ is defined over M ⊗ kG V we have (Ψ(ϕ)(mg))v = ϕ(mg ⊗ kG v) = ϕ(m ⊗ kG gv) = (Ψ(ϕ)(m))(gv) = (Ψ(ϕ)(m)g)v for m ∈ M, g ∈ G and v ∈ V . Thus Ψ(ϕ) ∈ Hom kG (M, V * ). Moreover, ((Φ • Ψ)(ϕ))(m ⊗ -) = Φ(m → ϕ(m ⊗ -)) = ϕ(m ⊗ -)
for m ∈ M . Thus Φ is surjective, hence an isomorphism.

Claim (ii) Hom kG (M, kG) ⊗ kG V * ∼ = Hom kG (M, V * ) as H-left modules:
The isomorphism is given by

α ⊗ kG f → α ⊗ kG f for α ∈ Hom kG (M, kG) and f ∈ V * where ( α ⊗ kG f )(m) (v) = (f α(m))(v) = f (α(m)v)
for m ∈ M and v ∈ V . Here V * is considered as a G-right module by the natural action as defined above. Furthermore, the space Hom kG (M, kG) is an H × G-bimodule via the action

(hf g)(m) := g -1 f (h -1 m) for m ∈ M , g ∈ G, h ∈ H and f ∈ Hom kG (M, kG).
First we show that the map is well-defined.

Indeed, for f ∈ V * , g ∈ G, α ∈ Hom kG (M, kG), v ∈ V and m ∈ M one have ( (αg) ⊗ kG f )(m) (v) = (f ((αg)(m))) (v) (definition of ) = f (((αg)(m)) v) (right module structure of V * ) = f g -1 (α(m)) v (right module structure of Hom kG (M, kG)) = f g -1 (α(m)v) (associativity of the kG-action on V * ) = (gf )(α(m)v) (left module structure of V * ) = ( α ⊗ kG (gf ))(m) (v) (definition of )
Furthermore, the map is H-linear, for

( hα ⊗ kG f )(m) = f (hα)(m) = f α(h -1 m) = ( α ⊗ kG f )(h -1 m) = (h( α ⊗ kG f ))(m).
Thus it remains to prove that is a k-isomorphism. First we consider the case

M = kG. Let α ∈ Hom kG (kG, V * ), hence α( g∈G a g g) = g∈G a g α(g) = g∈G a g α(1)g = f g∈G a g g where α(1) = f ∈ V * and a g ∈ k for g ∈ G. If id denotes the identity in Hom kG (kG, kG) then ( id ⊗ kG f )( g∈G k g g) = f g∈G k g g = α( g∈G k g g),
hence id ⊗ kG f = α. Thus the above map is an epimorphism, and therefore an isomorphism since the dimensions of the spaces in (ii) are obviously equal for M = kG. Since any free G-module is a direct sum of modules isomorphic kG the map is also an isomorphism if M is a free module. Since a projective module is a direct summand of a free module and since the isomorphisms are all compatible with taking direct summands the proof of (ii) is complete.

Claim (iii) Hom k (M, k) ∼ = Hom kG (M, kG) as H × G-bimodules:
Abstractly this follows by Frobenius reciprocity [9, Chapter VI, formula (8.7)] and the fact that kG is a symmetric algebra. However, we need the isomorphism explicit. Let (•, •) be the symmetrising bilinear form making kG into a symmetric algebra. For all x, y ∈ kG the value of (x, y) is the coefficient of 1 in xy ∈ kG (cf. e.g. [2, Theorem 62.1]). Then the map

Hom kG (M, kG) ρ M -→ Hom k (M, k) f → (m → (f (m), 1)) is a k-linear. If ρ M (f ) = 0 we get 0 = (f (mx), 1) = (f (m)x, 1) = (f (m), x)
for all m ∈ M and all x ∈ kG. Since (•, •) is non degenerate f (m) = 0 for all m ∈ M and hence f = 0. Therefore ρ M is injective.

Moreover, ρ M is a morphism of H × G-bimodules. Indeed, for g ∈ G and h ∈ H, for f ∈ Hom kG (M, kG) and m ∈ M , we have

(ρ M (hf g)) (m) = ((hf g)(m), 1) (definition of ρ M ) = (g -1 f (h -1 m), 1) (H × G-bimodule structure of Hom kG (M, kG)) = (1, g -1 f (h -1 m)) (since (•, •) is symmetric) = (g -1 , f (h -1 m)) (since (•, •) is associative) = (f (h -1 m), g -1 ) (since (•, •) is symmetric) = (f (h -1 m)g -1 , 1) (since (•, •) is associative) = (f (h -1 mg -1 ), 1) (f is kG-linear) = ρ M (f )(h -1 mg -1 ) (definition of ρ M ) = (h (ρ M (f )) g) (m) (H × G-bimodule structure of Hom k (M, k))
and so ρ M (hf

g) = h (ρ M (f )) g which shows that ρ M is H × G-linear.
The inverse isomorphism is given by sending

ϕ ∈ Hom k (M, k) to Λ M (ϕ) ∈ Hom kG (M, kG) defined by Λ M (ϕ)(m) = x∈G ϕ(mx -1 )x
for m ∈ M : First note, that Λ M (ϕ) is G-linear since it is the trace under the G-action. Furthermore,

((ρ M • Λ M )(ϕ)) (m) = ( x∈G ϕ(mx -1 )x, 1) = x∈G ϕ(mx -1 )(x, 1) = x∈G ϕ(mx -1 )δ x,1 = ϕ(m)
for m ∈ M and ϕ ∈ Hom k (M, k) where δ denotes the Kronecker symbol. Thus, ρ M is surjective and [START_REF] Curtis | Representation theory of finite groups and associative algebras[END_REF]Theorem 12.10]). Summarising (i), (ii) and (iii) we get explicit isomorphisms

ρ -1 M = Λ M is an isomorphism of H × G-bimodules. Since Λ M is H × G-linear, Λ M ⊗ kG id V * is well defined (cf. e.g.
M ⊗ kG V β⊗α -→ M * ⊗ kG V * Λ M ⊗id V * -→ Hom kG (M, kG) ⊗ kG V * -→ Hom kG (M, V * ) Φ -→ Hom k (M ⊗ kG V, k) = (M ⊗ kG V ) *
as H-left modules.

We need to verify the formula given in the Proposition.

m ⊗ v β⊗α → β(m) ⊗ kG α(v) Λ M ⊗id → g∈G β(mg -1 )g ⊗ kG α(v) = g∈G β(mg -1 ) ⊗ kG gα(v) → m ′ →   v ′ →   g∈G gα(v) β(mg -1 )(m ′ )v ′     = m ′ →   v ′ →   g∈G β(mg -1 )(m ′ ) • gα(v)(v ′ )     Φ →   (m ′ ⊗ v ′ ) →   g∈G β(mg -1 )(m ′ ) • α(v)(g -1 v ′ )    
This complets the proof.

The following is the translation of the isomorphism given in Proposition 3.3 in terms of bilinear forms. Corollary 3.4. Let M be an H × G-bimodule which is projective as a G-module. Suppose that M carries a non-degenerate H × G-invariant non-degenerate bilinear form b. Let V be a G-left module with a non-degenerate G-invariant bilinear form B. Then, for m, m

′ ∈ M and v, v ′ ∈ V , B(m ⊗ kG v, m ′ ⊗ kG v ′ ) = g∈G b(mg -1 , m ′ )B(v, g -1 v ′ ) defines a non-degenerate H-invariant bilinear form on M ⊗ kG V . Proof. By Lemma 3.2, the bilinear form b is associated to an H × G-isomorphism β : M → M * via b(m, m ′ ) = β(m)m ′ for m, m ′ ∈ M . Similarly, B is associated to a G-isomorphism α : V → V * via B(v, v ′ ) = α(v)v ′ for v, v ′ ∈ V . Proposition 3.3 shows that an isomorphism Ψ : M ⊗ kG V -→ (M ⊗ kG V ) * is given by (Ψ(m ⊗ v)) (m ′ ⊗ v ′ ) = g∈G β(mg -1 )(m ′ ) • α(v)(g -1 v ′ ) = g∈G b(mg -1 , m ′ )B(v, g -1 v ′ ). Thus, by Lemma 3.2, the bilinear form B defined on M ⊗ kG V by B(m ⊗ kG v, m ′ ⊗ kG v ′ ) = g∈G b(mg -1 , m ′ )B(v, g -1 v ′ )
is non-degenerate and H-invariant.

Proof of Theorem A. By Proposition 3.3, the functor M ⊗ kG -maps self-dual G-modules to self-dual H-modules.

On the other hand,

Hom kH (M, kH) becomes a G × H-bimodule via (gαh)(m) = α(mg)h for all g ∈ G, h
g • m • h = h -1 mg -1 then the inverse of the given isomorphism H M G -→ H M * G defines an isomorphism from G (M * ) H to M ∼ = ( G (M * ) H
) * and we are done.

Example 2.3 shows that not all Morita bimodules do admit a non-degenerate invariant bilinear form. Remark 3.5. Given two Morita equivalences between two blocks B G and B H of group algebras kG and kH, the composition of the one with the inverse of the other gives a Morita self-equivalence of the block B G . Now, the isomorphism classes of Morita self-equivalences of B G form the Picard group P ic(B G ) which is isomorphic to the outer automorphism group Out(B G ) := Aut k-alg (B G )/Inn(B G ) where as usual Inn(B G ) denotes the inner automorphisms of B G . Each α ∈ Aut k-alg (B G ) induces an equivalence by sending a module V to α V , which is defined to be V as a k-vector space, but where the action of B G on α V is given by a

• v = α(a)v for a ∈ B G and v ∈ V . Since ( α V ) * ∼ = α (V *
) the automorphism α preserves the property of self-duality on modules. Since B H ≃ End B G (M ), whatever isomorphism is taken to identify B H with End B G (M ) the question whether a given module M ⊗ kG V carries a non degenerate H-invariant bilinear form is independent of this choice.

There are Morita equivalences which are induced by self-dual bimodules. Clearly, e is the block idempotent of the principal block B 0 of kG and ekG ∼ = kG/N = kH as H ×G-bimodules where N obviously acts trivially on both sides. The bimodule H M G = ekG is a Morita bimodule which induces an equivalence between the principal block of kG and the group algebra kH. In fact End kG (ekG) ∼ = ekGe ∼ = kG/N = kH, and obviously ekG is a progenerator of ekG = B 0 . Moreover, ekG ∼ = (ekG) * as H × Gbimodules since, by a well-known result on duality, (ekG) * ∼ = ēkG = ekG where -denotes the anti-automorphism of kG defined by g → ḡ = g -1 . Thus the equivalence preserves self-duality, by Proposition 3.3. Clearly, this fact is obvious since N acts trivially on all modules in B 0 .

Morita equivalence and forms in odd characteristic

Throughout this section let k be a field of odd characteristic (which includes the characteristic 0 case as well). A G-module V is called of symmetric resp. antisymmetric type if V carries a non-degenerate G-invariant symmetric resp. antisymmetric bilinear form.

= ± B(m ′ ⊗ kG v ′ , m ⊗ kG v),
where the + sign appears for symmetric and the -sign for antisymmetric type modules V .

Proof of Theorem B. This is an immediate consequence of Proposition 4.1.

Example 4.2. We consider again Example 3.6. Note that the Morita module ekG is of symmetric type. As a form one can take the standard non-degenerate G-invariant symmetric form on kG defined by b(g, g ′ ) = δ g,g ′ for g, g ′ ∈ G restricted to ekG. The restriction is indeed non-degenerate as one can check easily. Thus Proposition 4.1 applies. Now let V be a simple self-dual kG-module. Thus V carries a G-invariant non-degenerate bilinear form. If k is algebraically closed then -up to scalars -there is exactly one nondegenerate G-invariant form which is symmetric or antisymmetric since the characteristic is odd ([4],Chap. VII, 8.12). Thus, for k algebraically closed, a simple self-dual G-module is either of symmetric or antisymmetric type but not of both. In particular, the Morita module M = ekG in Example 4.2 is never of antisymmetric type since otherwise, by Proposition 4.1, there are simple H-modules which are of symmetric and antisymmetric type. Indeed it is well-known, that a group algebra kH ∼ = ekG in odd characteristic never has a non-degenerate H-invariant antisymmetric form. Remark 4.3. Let k be an algebraically closed field of odd characteristic. By definition, a Morita bimodule M = H M G is a projective kG-right and a projective kH-left module. If M carries a non-degenerate H × G-invariant symmetric form then a projective indecomposable kG-resp. kH-module of antisymmetric type occurs with even multiplicity in M . Note that, by ([12], Proposition 2.2), an indecomposable projective module is of antisymmetric type if and only if its head is of antisymmetric type.

Morita equivalence and forms in even characteristic

In characteristic 2 the geometry of self-dual modules turns out to be more subtle. In this case an alternating form is symmetric. By Fong's Lemma ( [START_REF] Huppert | Finite groups II[END_REF], Chap. VII, Theorem 8.13), a simple non-trivial self-dual kG-module V always carries a non-degenerate G-invariant symplectic form, but it may happen that V even has a non-degenerate G-invariant quadratic form. This happens if H 1 (G, V ) = 0 ([11], Proposition 2.4) and a connecting homomorphism to the first cohomology group carries the essential information what really happens. We call an arbitrary module V ∼ = V * of quadratic type if it carries a non-degenerate G-invariant quadratic form, say Q. The corresponding G-invariant non-degenerate symplectic form B defined by

B(v, v ′ ) = Q(v + v ′ ) -Q(v) -Q(v ′ ) for v, v ′ ∈ V is called the Polarisation of Q. Note that the quadratic form Q is non-degenerate if rad Q := {v | v ∈ rad B, Q(v) = 0} = 0.
Throughout the following k is assumed to be a perfect field of characteristic 2. The next Lemma can be found as Proposition 2.2 in [START_REF] Gow | Quadratic geometries, projective modules, and idempotents[END_REF].

Lemma 5.1. Let P be a projective kG-module not containing the projective cover of the trivial module. Then each G-invariant symmetric form on P is the Polarisation of a Ginvariant quadratic form on P .

We are now ready to prove Theorem C.

Proof of Theorem C. Let P be a projective kG-module of quadratic type. Then P is an orthogonal sum of orthogonal indecomposable modules. By ( [START_REF] Willems | Duality and forms in representation theory[END_REF], Lemma 3.6), an orthogonal indecomposable module is either indecomposable or of the form S ⊕ S * where S is indecomposable. Since for m, m ′ ∈ M and p, p ′ ∈ P is a non-degenerate H-invariant bilinear form on M ⊗ kG P . Since b and B are symmetric the form B is symmetric as well. Thus, if the projective module M ⊗ kG P does not contain the projective cover of the trivial kH-module, say P H (1), then M ⊗ kG P is of quadratic type, by Lemma 5.1. In case it contains P H (1) it is either P H (1) or a direct sum of two copies of P H (1). But, by ([12], Remark 3.5 (b)), P H (1) and hence any finite direct sum of it is always of quadratic type if 2 | |H|. In case 2 ∤ |H| the trivial module k is projective and obviously of quadratic type. This completes the proof.

At the moment we are not able to prove an analogue of Theorem C for simple self-dual modules with the methods we have developed so far.

Date:

  May 13, 2007; revised March 25, 2008.

  ∈ H and α ∈ Hom kH (M, kH). Then Hom kH (M, kH) is the inverse Morita bimodule to M and hence Hom kH (M, kH) defines the inverse functor from the category of B H -modules onto the category of B G -modules. In order to prove the only if part it is, by Proposition 3.3, enough to show that the G × H-bimodule Hom kH (M, kH) is self-dual as well. First note that Hom kH (M, kH) ∼ = Hom k (M, k) = G (M * ) H as G × H-bimodules (by Claim (iii) in the proof of Proposition 3.3, changing left-and right structures there). If we define a G × H-structure on M by setting

Example 3 . 6 .

 36 Let G be a finite p-solvable group and letN = O p ′ (G). We put H = G/N and e = 1 |O p ′ (G)| g∈O p ′ (G)g.

Proposition 4 . 1 .

 41 Let M be an H × G-bimodule inducing a Morita equivalence between a kG-block B G and a kH-block B H . Suppose that M is of symmetric type. Let V be a self-dual G-left module. If V is of symmetric resp. antisymmetric type then M ⊗ kG V is of symmetric resp. antisymmetric type.Proof. Suppose b is a symmetric bilinear form and the bilinear form B is symmetric resp. antisymmetric if V is of symmetric resp. antisymmetric type. With the notation of Corollary 3.4, we have B(m⊗ kG v, m ′ ⊗ kG v ′ ) = g∈G b(mg -1 , m ′ )B(v, g -1 v ′ ) = g∈G b(m, m ′ g)B(gv, v ′ ) (since b and B are G-invariant) = ± g∈G b(m ′ g, m)B(v ′ , gv) (by symmetry of b and B)

  (M ⊗ kG (S ⊕ S * )) * ∼ = M * ⊗ kG (S ⊕ S * ) * ∼ = M ⊗ kG (S ⊕ S * )by Claim (i), (ii) and (iii) in the proof of Proposition 3.3 we need to prove the Theorem only for orthogonal indecomposable projective modules.So let P be an orthogonal indecomposable projective kG-module in B G of quadratic type. Thus P carries a non-degenerate G-invariant symplectic form B which is also symmetric. According to Corollary 3.4 the form B defined byB(m ⊗ kG p, m ′ ⊗ kG p ′ ) = g∈G b(mg -1 , m ′ )B(p, g -1 p ′ )


Elisabetta Caffau

L Sbordone

H.-G Ludwig

P Bonifacio

M Steffen

N T Behara

The solar photospheric abundance of hafnium and thorium. Results from CO 5 BOLD 3D hydrodynamic model atmospheres

Keywords:

The solar photospheric abundance of hafnium and thorium. Results from CO5BOLD 3D hydrodynamic model atmospheres.

Introduction

The determination of the ages of the older stellar populations in the Galaxy is fundamental for constraining its buildup mechanisms and determining the associated timescales. A variety of dating techniques have been developed, the majority relying on comparing the colour-magnitude-diagram of a stellar population with theoretical isochrones or "fiducial lines" derived from other observed populations (typically globular clusters).

Long lived unstable isotopes of heavy elements such as 238 U and 232 Th provide an independent method of determining stellar ages. Such elements are generally produced through neutron capture (specifically rapid neutron capture or r-process) and then decay. Their abundance at the present time thus provides a direct estimate of the time passed since they were produced, if their initial abundance can somehow be inferred. This is typically accomplished by measuring some other stable element with a common nucleosynthetic origin together with the radioactive element. Europium (Eu) has long been a

Send offprint requests to: Elisabetta.Caffau@obspm.fr typical choice, since its abundance relative to the abundance of thorium can be established from r-process enrichment models (see Francois et al. 1993, Cowan et al. 1999, Ivans et al. 2006). Sneden et al. (2000) used the [Th/Eu] abundance ratio to estimate the age of the old, metal-poor globular cluster M15 to 14 ±3 Gyr. The study was made easier by the fact that M15 shows strong r-process element enhancement ([Eu/Fe]=1.15, [Th/Fe]=0.81, Sneden et al. 2000). Nevertheless, Eu presents some drawbacks as a cosmochronometric reference element: while easily measurable, its atomic mass is substantially different from the one of Th and U. This generally increases uncertainties. In fact, decaying and stable reference elements should both be formed in the same event, which -for heavy n-capture species -can be safely assumed only if the atomic masses of the two elements are similar. This means in practise that, when a stable reference element is used, it should be as massive as possible. Alternatively, where U is measured, the U/Th ratio can be used since the two have very different half-life times. The measurement of U is nevertheless extremely diffi-Fig. 1. Four different observed solar spectra, displayed for four different spectral windows, each containing a Hf  line, indicated by a thin dashed vertical line. For clarity, the intensity spectra, labelled as DI (Delbouille Intensity) and NI (Neckel Intensity), are shifted down by -0.2 units, while the diskintegrated spectra, KF (Kurucz Flux) and NF (Neckel Flux), are shifted up by 0.2 units for the 356.1 nm and 391.8 nm lines. Except for the DI spectrum, the original normalisation has been retained. Note that the different spectra do not always agree perfectly.

401.90

401.91 401.92 Wavelength (nm) 0.5 0.6 0.7 0.8 0.9

1.0 Normalised Flux 3D Fe+Ni 1D Fe+Ni 3D Th 3D-1D Fe+Ni
Fig. 2. The Fe +Ni  blend in the 3D simulation (solid line) is compared to the 1D LHD profile (dash-dotted line), scaled in order to have the same central residual intensity as the 3D profile. The difference of the two spectra is shown as well as a Th 3D line profile. Fig. 3. The 3D profile of (Fe +Ni ) + (Th +Co +V ) blend in solid black is over-imposed on the 1D LHD fit (green/grey dashed line). In the plot is also over-imposed (black dashed line) the 1D LHD with the same abundance as the 3D profile as well as the 3D Th +Co +V  profile (dotted line). cult and has been possible only in two metal-poor r-enhanced stars (Cayrel et al. 2001;Hill et al. 2002;Frebel et al. 2007).

Recently, hafnium (Hf, Z=72) has been suggested as a suitable stable reference element. The Hf production rate via the r-process is believed to be strictly tied to the one of thorium, so that their abundance ratio is almost constant regardless of the neutron density in the generating Super Novae (SN) (Kratz et al. 2007). Hafnium is more massive than Eu (stable Eu isotopes are 151 and 153, while A Hf ≈ 178) and can be measured from a set of lines in the blue part of the visual spectrum (the most suitable ones being at 391.8 and 409.3 nm) for which accurate transition data have been recently measured (Lawler et al. 2007). On the down-side, Hf is believed to be efficiently produced by s-process (about 55% of the meteoritic Hf content in the solar system is believed to have been produced by the s-process, see Arlandini et al. 1999). As a consequence, it can be effectively employed as a reference only where s-process enrichment can be considered negligible, or if the s-process contribution can be reliably modelled.

Although in principle promising, Hf/Th dating is not free of drawbacks. In the first place an uncertainty of 0.1 dex in [Th/Hf] implies about 4.7 Gyr (see also Ludwig et al. 2008a) uncertainty in the dating. Second, the theoretical prediction of strong ties between Th and Hf production needs to be verified by observations. Third, the applicability of the method to normal (i. e. not r-process enhanced) stars needs some testing: especially Th lines become relatively weak and significant blends can be prejudicial especially in more metal-rich objects.

With the prospect of using hafnium and thorium for dating purposes, we decided to determine the solar hafnium and thorium abundances using hydrodynamical model atmospheres. We anticipate that 3D effects are found to be non-negligible for Th, and mainly due to the convection-induced asymmetry of blending lines.

In the solar spectrum, the measurable lines of Hf are from the singly ionised atom, and are often blended with other Thorium is measurable in the Sun from one single line of Th  at 401.9 nm, which is heavily blended. Solar thorium abundances in literature are very rare. Holweger (1980) derived a Th abundance of 0.16, Anders & Grevesse (1989) report abundances of 0.12±0.06 from the 401.9 nm line due to an unpublished measurement of Grevesse. Lodders (2003) recommends the meteoritic value A(Th)=0.09 ± 0.04 from Grevesse et al. (1996). The meteoritic value is 0.08 ± 0.02 (Anders & Grevesse 1989), inferred from the mass-spectroscopically measured ratio of 0.0329 Th atoms to 10 6 Si atoms (Rocholl & Jochum 1993) in carbonaceous chondrites.

Atomic data

We considered the same four Hf  lines with the same log g f as Lawler et al. (2007) have been considered (see Table 1). Line parameters for the 401.9 nm Th  and blending lines in this range are from del Peloso et al. (2005a).

Models

Our analysis is based on a 3D model atmosphere computed with the CO 5 BOLD code (Freytag et al. 2002;Wedemeyer et al. 2004). In addition to the CO 5 BOLD hydrodynamical simulation we use several 1D models. More details about the applied models can be found in Caffau et al. (2007) and Caffau & Ludwig (2007). As in previous works we call 3D the 1D model derived by a horizontal and temporal averaging of the 3D CO 5 BOLD model. The reference 1D model for the computation of the 3D abundance corrections is a hydrostatic 1D model atmosphere computed with the LHD code (see Caffau & Ludwig 2007), hereafter 1D LHD model. 1D LHD employs the same micro-physics (opacities, equation-of-state, radiative transfer scheme) as CO 5 BOLD, and treats the convective energy transport with mixing-length theory. Here, we set the mixing-length parameter to 1.5. We also use the solar ATLAS 9 model computed by Fiorella Castelli1 and the Holweger-Müller solar model (Holweger 1967;Holweger & Müller 1974). The spectrum synthesis code we employ is Linfor3D2 for all models.

The 3D CO 5 BOLD solar model covers a time interval of 4300 s, represented by 19 snapshots. This time interval should be compared to a characteristic time scale related to convection. Here we choose the sound crossing time over one pressure scale height at the surface t c = H P /c (H P : pressure scale height at τ ross = 1, c: sound speed) which amounts to 17.8 s in the solar case. Note, that this time scale is not the time over which the convective pattern changes its appearance significantly. This time scale is significantly longer than t c ; e.g., Wedemeyer et al. (2004) obtained from simulations a autocorrelation life time of about 120 s for the convection pattern. However, while t c does not measure the life time of the convective pattern as such, it provides a reasonably accurate scaling of its life time with atmospheric parameters (e.g., [START_REF] Svensson | 13th Cambridge Workshop on Cool Stars[END_REF]). Hence, it can be used for the inter-comparison of different 3D models.

The particular solar model used in this paper has a higher wavelength resolution (12 opacity bins) with respect to the model we used to study the solar phosphorus abundance (Caffau et al. 2007). The basic characteristics of the 3D models we consider in this work can be found in Table 2.

As in our study of sulphur (see Caffau & Ludwig 2007) and phosphorus (see Caffau et al. 2007), we define as "3D correction" the difference in the abundance derived from the 3D CO 5 BOLD model and the 1D LHD model, both synthesised with Linfor3D. We consider also the difference in the abundance de-rived from the 3D CO 5 BOLD model and from the 3D model. Since the 3D CO 5 BOLD and 3D models have, by construction, the same mean temperature structure, this allows us to single out the effects due to the horizontal temperature fluctuations.

Observational data

The observational data we use consists of two high-resolution, high signal-to-noise ratio spectra of centre disc solar intensity: that of Neckel & Labs (1984) (hereafter referred to as the "Neckel intensity spectrum") and that of [START_REF] Delbouille | Photometric Atlas of the Solar Spectrum from λλ3000 to λλ10000[END_REF]) (hereafter referred to as the "Delbouille intensity spectrum"3 ). We further use the disc-integrated solar flux spectrum of Neckel & Labs (1984) and the solar flux spectrum of Kurucz (2005) 4 , referred to as the "Kurucz flux spectrum".

Data analysis

Hafnium

The computation of a 3D synthetic spectrum with Linfor3D is rather time consuming and the spectral-synthesis code can handle easily only a few tens of lines. The log g f values of the lines blending the Hf lines and lying nearby are not well known, and the synthetic spectra do not reproduce the observed solar spectrum well in that region. To fit the line profiles Lawler et al. (2007) changed the log g f of some lines blending the Hf  lines, but did not make available the values used. In any case, solar log g f values derived from 1D and 3D fitting are not guaranteed to be the same. Our computational power is insufficient at the moment to enable us to fit 3D synthetic spectra to an observed spectrum by changing the log g f values of various lines. Consequently, we could not produce solar log g f for the blending lines and fit the line profiles with a 3D model grid. Instead of fitting astrophysical solar log g f of the blending features, we introduced blending components in the process of measuring EW. We thus took advantage of the deblending mode of the IRAF task splot. The corresponding Hf abundance is then derived from the curves of growth computed with Linfor3D for the various lines. The results of Lawler et al. (2007) are used as a check for our measurements. The abundances we obtain from the Holweger-Müller (HM) model are in agreement with their measurement. The results for all spectra and individual lines are presented in Table 3.

We determine the following hafnium abundances: 0.870 ± 0.038 if we consider only the disc-centre intensity spectra, 0.818 ± 0.005 according to the solar flux spectra, and 0.856 ± 0.040 if we consider both the intensity and the flux spectra. As final uncertainty we took the line to line RMS scatter over all spectra. Equivalent widths can be measured in both flux spectra only for the 409.3 nm hafnium line, which is nearly free of blends. Only the Neckel flux spectrum can be considered for the Hf line at 391.8 nm, due to the fact that the larger broadening of flux spectra with respect to intensities makes the measurement more difficult, if not impossible. Additionally, NLTE corrections may be different in intensity and flux spectra. For these reasons, and to have a consistent comparison with the results of Lawler et al. (2007), we determine the solar hafnium abundance from the average value stemming from the disccentre spectra. Our results are in good agreement with the results of Lawler et al. (2007); we obtain A(Hf)=0.878 ± 0.040 from the Holweger-Müller model, considering only disc-centre spectra, which is to compare to their value of 0.88 ± 0.02.

The 3D abundance corrections for Hf are very small. The highest value, 0.037 dex, is the one for the strongest line. As previously mentioned, a 3D correction is the difference in abundance obtained from the 3D model and the 1D LHD model. If one chooses an ATLAS or the Holweger-Müller model as a 1D reference model, the 3D correction would be even smaller or negative, respectively.

The 3D and 1D LHD solar model temperature structures are similar. The relative contribution to the 3D correction due to different average temperature profiles is small. The 3D correction is larger for stronger lines, quite insensitive to microturbulence, of the same order of magnitude for flux and intensity, and slightly higher for intensity.

Thorium

Singly ionised thorium is the dominant ionisation stage in the solar atmosphere, and thorium is measurable in the Sun only through the 401.9 nm Th  resonance line. Holweger (1980) detected also the 408.6 nm Th  line, however in our view this line is too weak and blended to be used for quantitative analysis. The 401.9 nm line happens to fall on the wing of a strong blend of neutral iron and nickel. Additionally, it is blended with a Co  line (about 25% of the EW of the Th  line) and a weaker V  line (1/5 of the Co line, 1/9 of the Th line in EW). The Co  and V  lines are very close in wavelength to the Th  line, and it is therefore practically impossible to disentangle the blend. Consequently, we do not measure the contribution in EW due to thorium, but instead use the EWs measured by Lawler et al. (1990) to obtain A(Th) (see Table 4, first two lines). However, while Lawler et al. (1990) considered only the contribution of Co  to the blend, we consider also the contribution of V . To do so we computed from the CO 5 BOLD solar model the EW of the Co +V  blend, using the atomic data from del Peloso et al. (2005a), and subtracted this value from the EW of Lawler et al. (1990). In this manner, we obtained a smaller solar A(Th) (see last two lines of Table 4). The 3D corrections listed in Table 4 are deduced from the EWs alone, and we shall call them in the following "intrinsic" 3D corrections. These are the corrections that would be found if the Th  line were isolated. Since it is instead found in the middle of a rather complex spectral region, other 3D-related effects arise, which are discussed later.

The last two columns of Table 4 provide estimates of the statistical uncertainty of the theoretical EW predicted by the 3D model and the associated uncertainty in the hafnium abundance. The 3D model provides only a statistical realisation of the stellar atmosphere, and is thus subject to limited statistics (see Ludwig et al. 2008b). However, as necessary for a reliable derivation of abundances these uncertainties are insignificant and only added for completeness. Lawler et al. (1990). Notes: The contribution to the EW of Co and V, according to the 3D simulation, has been been from the EWs in the last two rows (labelled with *). ( 1) identification flag to distinguish disc-centre to flux spectrum results;

spec EW A(Th) σ 3D-3D 3D-1D LHD σ EW σ A pm 3D 3D 1D LHD 1D HM % dex (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
(3) thorium abundance, A(Th), according to the CO 5 BOLD 3D model; ( 4)-( 6) A(Th) from 1D models, with ξ, micro-turbulence, of 1.0 km s -1 ; (7) the uncertainty due to the EW; ( 8)-(9) 3D corrections;

(10) statistical uncertainty of the theoretical EW; (11) corresponding uncertainty in abundance. 4) results of the fitting: A(Th), shift, and broadening using a 3D grid; ( 5)-( 7) same for 1D LHD grid; (8) gives the difference in abundance A(Th) 3D -A(Th) 1D LHD .

Since the Th  line lies on the red wing of the stronger Fe +Ni  blend, one can expect that the line asymmetry of the latter contributes in a non-negligible manner to the measured EW. Up to now it has generally been considered that convective line asymmetries have no impact on the derived abundances, which is true, if the line is isolated. Cayrel et al. (2007) have demonstrated that such asymmetries are non-negligible for the derivation of the 6 Li/ 7 Li isotopic ratio. From the purely morphological point of view, the case of the Th  resonance line is similar to that of the 6 Li  resonance line: a weak line which falls on the red wing of a stronger line, which is asymmetric due to the effects of convection. In order to investigate this effect one has to consider the full line profile and resort to spectrum synthesis.

To gain some insight into the problem we began by comparing synthetic profiles, which included only the Fe +Ni  blend and the Th +Co +V  blend, computed from the CO 5 BOLD and 1D LHD models. In Fig. 2, the 1D LHD flux profile of the Fe +Ni  blend has been scaled (since the line strength is different in 1D and 3D) in order to have the same central depth of the 3D profile. The asymmetry of the red wing is obvious. The difference between the 3D and the 1D LHD profiles is shown in the dotted line of the figure. The 3D thorium profile is also shown as a dashed line and it makes clear that the line asymmetry does indeed contribute to the measured EW. The EW of this difference is 0.134 pm, which translates into a difference in abundance of +0.13 dex for flux. For intensity, ∆(EW) = 0.088 pm implies ∆A(Th)=+0.10 dex in Th abundance.

On a level of slightly higher sophistication with respect to the above naive argument, we fitted the 3D synthetic profile with a grid of 1D LHD synthetic spectra of differing Th abundance. We scaled the 1D LHD Fe+Ni blend in order to induce it to have the same central residual intensity as the 3D profile. One of the results of the fitting is shown in Fig. 3: the 3D profile (black solid line) with A(Th)=0.09 when fitted with a 1D LHD grid (dashed green/grey line) gives a higher abundance A(Th)=0.171. The use of a 3D grid leads to an even higher thorium abundance: A(Th)=0.227. In the plot the 1D LHD flux profile, with the same abundance as the 3D synthetic spectrum, is shown, as well as the 3D thorium profile. For the disc-centre intensity, with the same procedure as used for flux, the fitting of a 3D profile assuming a thorium abundance of 0.09 with a 1D LHD grid implies A(Th)=0.110, while with a 3D grid A(Th)=0.155. The difference of the result of the fit when using 3D or 1D LHD grids is due to the superposition of the effect of the line asymmetry and the "intrinsic" 3D corrections (A(Th) 3D -A(Th) 1D ).

The Fe +Ni  line asymmetry and the 3D corrections act in opposite directions: the line asymmetry produces a higher A(Th) from 1D analysis, while the 3D correction is positive, thus A(Th) from a 1D analysis is smaller. However, the 3D correction in intensity is between 3 and 4 times larger than for flux (see Table 4).

In Fig. 4 we show the results of fitting the whole spectral region both with 3D (left hand side) and 1D LHD (right hand side) profiles. The synthetic profiles have been built by adding 5 components: a Mn I+Fe I blend centred at 401.89 nm, the Fe +Ni  blend, the Th +Co +V  blend, an artificial Fe  line at 401.9206, initially suggested by Francois et al. (1993) and adopted also by del Peloso et al. (2005a), and the two Co  lines at 401.93. Of these components the Th +Co +V  blend was computed for three different Th abundances, all the other components were instead computed for a single abundance and were scaled by the fitting procedure in order to best reproduce the observed spectrum.

The results of the fitting are summarised in Table 5. From the table one sees that the total 3D correction, taking into account both line asymmetry and "intrinsic" 3D correction is from -0.07 to -0.09 dex for the intensity spectra and -0.11 dex for the flux spectra. This difference is expected, given that the "intrinsic" 3D correction, as given in Table 4, is about +0.02 dex for intensity and about +0.005 dex for flux.

It is quite interesting to note that in the case of Th the line asymmetry implies a total 3D effect which is not only much larger, but also in a different direction, from what would have been implied by the "intrinsic" 3D correction. To our knowledge this is the second time such an effect has been highlighted. These "total" 3D corrections are only slightly larger than what was estimated above from the fit of the 3D Fe +Ni  blend plus Th +Co +V  blend, with 1D LHD profiles. This suggests that the use of the simpler procedure is still capable of providing the correct order of magnitude for the "total" 3D correction. Finally we point out that, as can be seen from the left hand side of Fig. 4, it is obvious that the 1D LHD profiles are unable to provide the correct shape for the Th feature. The shape is reproduced in a much more satisfactory way by the 3D profiles.

We conclude this section by investigating the iron line at 401.9 nm, which is the stronger component of the Fe +Ni  blend. The 3D synthetic line shows a strong asymmetry; the difference 3D-1D is very pronounced on the red wing if we make an appropriate shift and broadening to the 1D profile so as to over-impose the blue wings of the 3D and 1D synthetic spectra. While the flux of the 3D synthetic spectrum is shifted to the blue by 0.1 km s -1 , the intensity spectra, corresponding to different inclination angles, show a shift ranging from -0.5 km s -1 to +0.15 km s -1 .

6. The 3D effects for other models.

Hafnium

Beyond the solar model as such we investigated the hydrodynamical effects on some other 3D models of dwarf and subgiant stars (see Table 2). We restricted our investigation to solar or -1.0 metallicity, because for lower metallicity these hafnium lines are no longer measurable in dwarfs. Even the detection at -1.0 metallicity is only possible for the strongest of the hafnium lines, but the two metallicities can be used for interpolation to intermediate metallicity. We find that 3D and 1D abundances are generally very similar, making the 3D corrections insignificant for all models and all lines we consider.

Hf  is the dominant species of hafnium in the photosphere of these models. Both the 5900 K models predict that at least 98% of the Hf is ionised. The hotter models foresee an even lower fraction of Hf . The coolest models at 5000 K predict that at least 75% of the Hf is ionised. The results are similar when considering the 3D or the 1D LHD temperature profile for the atmosphere.

If we compare the horizontal average temperature profile of the 3D model to the 1D LHD model we can see that they are very similar. There is no pronounced cooling of the 3D atmospheres relative to 1D LHD in radiative equilibrium for the same atmospheric parameters which is known to be prominent in more metal-poor models.

Thorium

As for hafnium and as in the solar case, the "intrinsic" 3D corrections are small for the other models. The hydrodynamical effects are instead due to the line asymmetry, like in the Sun. When we fit, as we do for the Sun, a 3D flux profile, with meteoritic or scaled meteoritic A(Th) for the metal-poor models, with a 1D LHD grid we obtain for A(Th) a value higher by 0.08-0.12 dex considering the models: 5500 K/3.50/0.0, 5500 K/3.50/-1.0 5900 K/4.00/0.0, 5900 K/4.00/-1.0, 5900 K/4.50/0.0, and 5900 K/4.50/-1.0 (see Table 2). One has to keep in mind that such differences translate into systematic age differences of almost 5 Gyr in the context of nucleocosmochronology.

Conclusions

Our determination of the photospheric hafnium abundance, A(Hf)=0.87 ± 0.04, derived from a 3D hydrodynamical CO 5 BOLD solar model, is very close to the abundance determination of Lawler et al. (2007) obtained using the Holweger-Müller solar model. The EWs of the hafnium lines are not very sensitive to the hydrodynamical effects. A difference of the same amount of the 3D correction can be found if we compare the A(Hf) obtained using two different 1D models, such as ATLAS and Holweger-Müller model or 1D LHD and the Holweger-Müller model. These difference are related to different opacities and physics used to compute the models. Synthetic profiles of hafnium lines computed from 3D models show a pronounced asymmetry, but this effect is not relevant for abundance determination as long as only the EW is considered.

For thorium, as for hafnium, 3D effects on the EWs are less than or comparable to the uncertainties related to the EW measurement, so that they are negligible when determining A(Th). However, for the solar photospheric thorium determination the asymmetry of the strong Fe +Ni  line blending the 401.9 nm Th  weak line implies a non-negligible effect on the derived Th abundance and is in fact the dominant source of the "total" 3D effect. The asymmetry is a hydrodynamical effect, and we find a difference in the A(Th) determination of about -0.1 dex when considering this asymmetry. Our results for the Sun and for other 3D dwarfs and sub-giants models imply that Th cannot be reliably measured without making use of hydrodynamical simulation if the Th line is weak and blended. A reappraisal of existing Th measurements in the light of hydrodynamical simulations is warranted for dwarfs and sub-giants.

Fig. 4 .

 4 Fig.4. Fits of the synthetic spectra to the observed solar spectra. On the left hand side, the synthetic spectra have been computed from the CO 5 BOLD model, on the right hand side from the corresponding 1D LHD model. On either side, from top to bottom, the spectra are: the Delbouille disc centre intensity, the Neckel disc centre intensity, the Neckel flux and the Kurucz flux. On each panel the sub-components used to synthesise the feature are shown individually: the light dotted line is the Th +Co +V  blend, the darker dotted line is the Ni +Fe  blend, the black dashed lines are, from blue to red, the Mn +Fe  blend the artificial Fe  line and the Co  401.93 nm lines.

Table 1 .

 1 Hf  lines considered in this work.

	Wavelength Transition E low log g f
	nm	eV
	338.983	a 4 F-z 2 P 0 0.45	-0.78
	356.166	a 2 D-z 4 F 0 0.00	-0.87
	391.809	a 4 F-z 4 D 0 0.45	-1.14
	409.315	a 4 F-z 4 P 0 0.45	-1.15
	photospheric absorption lines. As a consequence, solar pho-
	tospheric abundance determinations are scarce in the litera-
	ture. Russell (1929) determined the photospheric hafnium solar
	abundance to be 0.90. Recently, Lawler et al. (2007) measured
	Hf lifetimes for several lines, four of which were used to deter-
	mine a solar photospheric hafnium abundance of 0.88 ± 0.08.
	Andersen et al. (1976) considered nine lines of Hf  (two of
	which are in common with Lawler et al. 2007) and obtained
	from a Kitt Peak National Observatory centre-disc spectrum a
	solar photospheric hafnium abundance of 0.88 ± 0.08. Values
	of the meteoritic hafnium abundance cited in literature include:
	0.73 ± 0.01, Anders & Grevesse (1989); 0.75 ± 0.02 Grevesse
	& Sauval (1998); and 0.77 ± 0.04 Lodders (2003).

Table 2 .

 2 The CO 5 BOLD models considered in this work.

	T eff	log g [M/H]	N t	time	t c
	(K) (cm s -2 )			(s)	(s)
	(1)	(2)	(3) (4)	(5)	(6)
	5770	4.44	0.0 19	4300	17.8
	4980	4.50	0.0 20	30800	14.8
	5060	4.50	-1.0	19	65600	15.1
	5430	3.50	0.0 18 144000 151.0
	5480	3.50	-1.0	19 106800 149.3
	5930	4.00	0.0 18	26400	49.8
	5850	4.00	-1.0	18	36600	48.9
	5870	4.50	0.0 19	16900	16.0
	5920	4.50	-1.0	8	7000	15.7
	6230	4.50	0.0 20	35400	16.3
	6240	4.50	-1.0	20	25000	16.1
	6460	4.50	0.0 20	17400	16.4
	6460	4.50	-1.0	20	36200	16.3

Note: Cols.(

1

), (2), and (3) state the atmospheric parameters of the models; col(4) the number of snapshots N t considered; col(5) the time interval covered by the selected snapshots; col(6) a characteristic timescale t c of the evolution of the granular flow.

Table 3 .

 3 Solar hafnium abundances from the various observed spectra.

	spec wavelength	EW			A(Hf)			σ Ref. 3D-1D LHD σ EW	σ A
		nm	pm	3D	3D	1D LHD 1D ATLAS 1D HM	dex	dex	%	dex
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9) (10)	(11)	(12)	(13)
	NI	338.9	0.489 0.911 0.895	0.883	0.900 0.914 0.02	0.029	0.29 0.0013
	DI	338.9	0.477 0.899 0.883	0.871	0.886 0.903 0.02 0.91	0.029
	NI	356.2	0.845 0.819 0.801	0.781	0.799 0.821 0.01	0.037	0.31 0.0014
	DI	356.2	0.851 0.822 0.805	0.785	0.803 0.824 0.02 0.85	0.037
	NF	391.8	0.318 0.823 0.827	0.815	0.836 0.848 0.05	0.008	0.26 0.0011
	NI	391.8	0.302 0.911 0.899	0.893	0.912 0.925 0.02	0.018	0.27 0.0012
	DI	391.8	0.274 0.867 0.854	0.849	0.868 0.881 0.05 0.91	0.018
	KF	409.3	0.317 0.814 0.818	0.805	0.826 0.838 0.07	0.010	0.26 0.0011
	NF	409.3	0.318 0.816 0.820	0.806	0.828 0.840 0.07	0.010
	NI	409.3	0.292 0.886 0.874	0.865	0.884 0.899 0.02	0.020	0.26 0.0011
	DI	409.3	0.268 0.847 0.835	0.827	0.846 0.860 0.04 0.86	0.020

Note: Abundances and their uncertainties -if not noted otherwise -are given on the usual logarithmic spectroscopic scale where A(H)=12. Col. (1) is an identification flag, DI means Delbouille and NI Neckel disc-centre spectrum, NF Neckel and KF Kurucz flux spectrum. Col. (4): hafnium abundance, A(Hf), according to the CO 5 BOLD 3D model. Cols. (5)-(8): A(Hf) from 1D models, with ξ, micro-turbulence, of 1.0 km s -1 . Col. (9): uncertainties related to EW measurement. Col. (10) Lawler et al. (2007). Col. (11): 3D corrections. Col. (12): uncertainty of the theoretical equivalent width due to the limited statistics obtained from the 3D model. Col. (13): uncertainty in abundance due to the model uncertainty in equivalent width.

Table 4 .

 4 Solar thorium abundances derived from the EWs of

Table 5 .

 5 Solar thorium abundance derived from line profile fitting.

			3D			LHD
		A(Th) SHIFT	V br A(Th) SHIFT	V br 3D-1D LHD
		dex	km s -1	km s -1	dex	km s -1	km s -1	dex
	Int. (Neckel)	0.088	-0.05	1.49	0.160	-0.53	2.77	-0.720
	Int. (Delbouille)	0.080	+0.01	1.86	0.168	-0.47	2.99	-0.088
	Flux (Neckel)	0.056	-0.39	1.68	0.168	-0.47	4.11	-0.112
	Flux (Kurucz)	0.067	-0.38	2.82	0.175	-0.46	4.71	-0.108
	Note: (2)-(

Table 6 .

 6 Non-solar CO 5 BOLD models considered in this work. The atmospheric parameters are given as T eff /log g/[M/H]. cols. (3)-(5) are the A(Hf), according to CO 5 BOLD 3D, 3D and 1D LHD model; cols. (6)-(7) are 3D corrections, 3D-3D and 3D-1D LHD respectively; col. (8) statistical uncertainty of the theoretical EW; col. (9) corresponding uncertainty in abundance.

	Wavelength	EW		A(Hf)		3D-3D	3D-1D LHD σ EW	σ A
	nm	pm	3D	3D	1D LHD			%	dex
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
	5060/4.5/0.0
	339.0 0.718 0.870 0.842	0.829	0.028	0.041	0.05 0.0002
	356.2 1.550 0.872 0.809	0.796	0.062	0.076	0.09 0.0003
	391.8 0.405 0.870 0.854	0.841	0.016	0.029	0.07 0.0003
	409.3 0.410 0.870 0.855	0.841	0.016	0.029	0.07 0.0003
	5060/4.5/-1.0
	339.0 0.151 0.870 0.866	0.852	0.004	0.018	0.11 0.0005
	356.2 0.395 0.870 0.864	0.852	0.007	0.018	0.12 0.0005
	391.8 0.080 0.870 0.870	0.855	0.001	0.015	0.11 0.0005
	409.3 0.081 0.868 0.867	0.853	0.001	0.015	0.11 0.0005
	5870/4.5/0.0
	339.0 0.551 0.870 0.862	0.858	0.008	0.011	0.20 0.0008
	356.2 1.150 0.870 0.857	0.849	0.012	0.020	0.22 0.0010
	391.8 0.330 0.871 0.867	0.869	0.004	0.002	0.18 0.0008
	409.3 0.336 0.870 0.867	0.867	0.003	0.003	0.18 0.0008
	5920/4.5/-1.0
	3390. 0.095 0.871 0.872	0.868	-0.001	0.003	0.57 0.0025
	3562. 0.230 0.870 0.880	0.881	-0.010	-0.011	0.67 0.0029
	3918. 0.053 0.869 0.872	0.873	-0.003	-0.004	0.55 0.0024
	4093. 0.054 0.866 0.869	0.870	-0.003	-0.004	0.56 0.0026
	6230/4.5/0.0
	339.0 0.444 0.870 0.865	0.859	0.005	0.012	0.22 0.0009
	356.2 0.916 0.870 0.865	0.855	0.004	0.015	0.25 0.0011
	391.8 0.275 0.870 0.866	0.870	0.004	-0.001	0.19 0.0008
	409.3 0.281 0.870 0.866	0.869	0.003	0.001	0.20 0.0009
	6240/4.5/-1.0
	339.0 0.073 0.869 0.876	0.870	-0.007	-0.001	0.19 0.0008
	356.2 0.173 0.871 0.889	0.889	-0.018	-0.018	0.20 0.0009
	391.8 0.042 0.866 0.874	0.876	-0.008	-0.010	0.17 0.0007
	409.3 0.044 0.875 0.884	0.885	-0.009	-0.010	0.17 0.0007
	6460/4.5/0.0
	339.0 0.374 0.871 0.869	0.864	0.002	0.006	0.23 0.0010
	356.2 0.764 0.870 0.871	0.863	0.000	0.007	0.26 0.0011
	391.8 0.239 0.869 0.868	0.877	0.002	-0.008	0.20 0.0009
	409.3 0.245 0.871 0.869	0.877	0.002	-0.006	0.20 0.0009
	6460/4.5/-1.0
	339.0 0.059 0.867 0.883	0.875	-0.016	-0.007	0.25 0.0011
	356.2 0.137 0.868 0.897	0.897	-0.029	-0.029	0.27 0.0012
	391.8 0.035 0.865 0.881	0.882	-0.016	-0.017	0.21 0.0009
	409.3 0.036 0.866 0.883	0.883	-0.016	-0.017	0.21 0.0009
	5500/3.5/0.0
	339.0 1.500 0.868 0.855	0.850	0.014	0.019	0.23 0.0010
	356.2 2.780 0.871 0.845	0.830	0.026	0.041	0.29 0.0012
	391.8 0.987 0.870 0.865	0.867	0.005	0.003	0.21 0.0009
	409.3 1.000 0.868 0.864	0.863	0.004	0.005	0.22 0.0009
	5500/3.5/-1.0
	339.0 3.160 0.870 0.870	0.863	0.001	0.007	0.29 0.0013
	356.2 0.758 0.870 0.874	0.862	-0.003	0.008	0.34 0.0015
	391.8 0.182 0.871 0.874	0.872	-0.003	-0.001	0.28 0.0012
	409.3 0.186 0.871 0.875	0.870	-0.004	0.001	0.29 0.0013
	Note:

http://wwwuser.oats.inaf.it/castelli/sun/ap00t5777g44377k1asp.dat

http://www.aip.de/ mst/Linfor3D/linfor

3D manual.pdf

http://bass2000.obspm.fr/solar spect.php

http://kurucz.harvard.edu/sun.html

Acknowledgements. The authors L.S., H.-G.L., P.B, and N.T.B. acknowledge financial support from EU contract MEXT-CT-2004-014265 (CIFIST).