Restricted Invariants on the Space of Elasticity Tensors

Restricted Invariants on the Space of Elasticity Tensors

�� Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20132Milano, Italy

(Received 22 July 2003; accepted 4 March 2004)

Dedicated to Professor Michael Hayes on the occasion of his 65th birthday.

Abstract: A linear function defined on the space of elasticity tensors is a restricted invariant under a groupof rotations G if it has an invariant restriction to a proper subspace which is larger than the set left fixedby the action of G itself. A necessary and sufficient condition for a function to be a restricted invariant isgiven using concepts related with isotypic decomposition, Haar integration and G -dependence. The result isapplied to characterize isotropic and transversely isotropic restricted invariants.

Key Words : elasticity tensor, invariant anisotropic

��

Let � be a (real-valued) linear function defined on ��, the space of elasticity tensors, andlet � be a group of rotations acting on ��. The function � is invariant if it takes constantvalues over each orbit in ��. If not, � is noninvariant. In this case it may happen that �behaves as an invariant when its domain is restricted to a suitable proper subspace �� of��.If �� is larger than the space of tensors which are left fixed by the action of � we say that �is a restricted invariant.

The main questions we want to answer are: (1) How do we characterize the class ofrestricted invariants, for a given group �? (2) Is it possible, and under which conditions,that each linear function over �� is a restricted invariant? (3) Are there functions which areneither invariant nor restricted invariant?

We became interested in this problem after reading a paper by Ting [1], where somefunctions defined over �� which become invariant only when restricted to suitable propersubspaces are found and discussed (see e.g. [1, pp. 515–516]. Ting calls conditional invariantany such function, but we thought that the adjective restricted would give a more descriptiveterminology, since a ‘‘restriction’’ on the domain of definition is needed in order to obtain aninvariant.

The topic of (in our terminology) restricted invariants comes up within the widercontext of the search for so-called structural invariants, sets of relations among the elasticcoefficients which are preserved under the action of a given group [1, 2, 3]. When suchrelations are linear, a structural invariant is just a description of an invariant subspace of

Ma thematics and Mechan ics o f Solid s, 11 : 4 8–8 2, 200 6 DOI: 1 0.117 7/ 1 081 28 650 50 46 483��2006 SAGE Publications

RESTRICTED INVARIANTS 49

��. The connection between the two problems becomes evident when it is noticed thatthe subspace associated to any restricted invariant is by definition itself invariant, thuscorresponding to a set of structural invariants. The search for all such types of invariantsis justified in [1], in view of their physical relevance for analyzing layered composites.

What the problems about restricted and structural invariants have in common is thatboth require a geometric insight into the structure and behaviour of �� under the actionof a rotation group �. Indeed, when we noticed that the results found by Ting were notderived from a general approach but, rather, through clever ad hoc computations, we becameconvinced that a better understanding of the nature of restricted and structural invariants couldonly come from the adoption of a more deeply geometrical approach. Since, as explainedbelow, we later found that the technical tools needed for the discussion were more complexthan previously expected, we decided to limit the content of this paper to the first topic(restricted invariants), being confident that the results presented here would turn out to beuseful also for structural invariants.

In this paper we try to frame the notion of restricted invariants within the well foundedtheory of group representations. To our surprise, we found that in order to understandthis problem a few quite sophisticated and subtle tools are needed. Indeed, the isotypicdecomposition of �� under the action of the group � turned out to be a key ingredient,together with a few ideas based on an application of the Haar integral. Finally, we foundit expedient to introduce a concept, here called �-dependence, which, essentially, is aformulation of the idea of linear dependence for a module over a division algebra, ratherthan for a vector space.

We expected that the abstract results needed for the applications we had in mind couldpossibly be found in books or articles about group representations and, as a consequence, weonly should make suitable adaptations, concentrating our attention to more explicit examplesand applications. In fact, we discovered that the proofs of quite a few results needed for thediscussion were not readily available in the related literature, notwithstanding the enquirieswe made among knowledgeable researchers in the field. Thus, as the reader will see, a partof this paper contains concepts and results which apply to a much more general situation thanlinear elasticity. We make this remark in order to justify its length, which, in our opinion,cannot be easily reduced.

This research is aimed at a readership mainly interested in linear elasticity, albeit froma somewhat abstract point of view. Thus, we decided to keep our discussion consistent withthat framework without unnecessary generalizations. The problem itself, indeed, could beeasily phrased and discussed in a much more abstract setting: the space of elasticity tensorscould be substituted by an arbitrary inner product vector space on which the group � actsorthogonally, without altering the main theoretical results by much. However, we chose tostay close to applications in linear elasticity because that is the field in which we are mainlyinterested.

We also hope that this paper might have a useful role in bridging a gap. Problems arisingin the literature on linear elasticity are sometimes presented and discussed with techniqueswhich lack sufficient generality and thus make it difficult to understand the real mathematicalstructure which lies underneath the question under discussion. Admittedly, most of the timethis is not so important, since questions find answers, anyway. There are cases, however, whenthis is not so: without an approach which is based on a much wider theoretical perspective itturns out to be difficult to fully comprehend what the real question is and find a satisfactory

50 S. FORTE and M. VIANELLO

answer. We hope to convince the reader that the problem of restricted invariants is a typicalexample of such a situation.

It is important to realize that, here, we stand on a thin threshold: one more step inthe direction of abstract algebra and group theory and this paper would perhaps become apresentation of general results focused towards an application to linear elasticity. We triednot to cross this line, both by keeping our notation consistent with what can be found in theliterature on linear elasticity and by trying to keep always at the reader’s mind the problemwhich first motivated this research.

��

The space of classical elasticity tensors is denoted by ��. An element � of �� is a fourth-order tensor of the three-dimensional Euclidean space which, after the introduction of anorthonormal basis �� fixed once and for all, we write as

� � �� (1)

where � denotes the tensor product and sum over repeated indices is understood. Thecomponents �� satisfy the following index symmetries:

�� (2)

Elasticity tensors are written with capital blackboard boldface letters, such as �, �, �, �,while we use boldface capital letters for second-order tensors. Subspaces of �� have animportant role in our paper and are mostly denoted by italic letters, such as �,�. Occasionally,in order to make easier comparisons with related results found in the current literature, weshall also use the well known Voigt contracted notation, in which the components of � aredescribed by a symmetric �� matrix �� (� � �� ).

We briefly recall the definition of a (left and linear) action of a Lie group � on a finite-dimensional vector space � (see e.g. [4, 5]). In our discussion � stands for any compactsubgroup of ��, the group of rotations in the three-dimensional Euclidean space (later onwe show that � might otherwise be chosen more generally as a compact subgroup of ��,with no essential modification of our results). We say that � acts (from the left) on� if foreach� � � there is a linear map� of� into itself such that��

� ��

. Moreover,the map �� defined by �� is required to be continuous.

A subspace of � is invariant if �� for each � � � and each � � �. Theaction of � is irreducible if� has no proper invariant subspace.

Let�� and�� be two vector spaces on which � acts through � and �. A linear map� from�� into�� is �-invariant if it commutes with the actions of �. This means that

�� (3)

for each� in � and � in��. We say that�� is a copy of�� (and vice versa) if there is a�-invariant isomorphism between�� and��. For this, which is obviously an equivalencerelation, we write�� .


Let � be a �-invariant linear map of an irreducible space �� into a copy ��. Sincethe kernel of � is an invariant subspace of��, which is irreducible, we deduce that either� �� or � �� . In the former case � is bijective and, since�� and�� haveequal dimension, we conclude that � is an invariant isomorphism. In the latter case, if thekernel of � is the whole of��, then � is trivially the null map. This is a useful result whichwe make formal, for later reference.

�� A �-invariant linear map of an irreducible space� into a copy of itself iseither a null map or an invariant isomorphism.

It is also known that the set of �-invariant linear maps of an irreducible space intoitself is an associative algebra isomorphic to one among: the set of real numbers, the set ofcomplex numbers, the set of quaternions (see e.g. [4, Chapter XII]). A space� is absolutelyirreducible with respect to the action of � if the only invariant linear maps of� into itselfare scalar multiples of the identity.

A left action of a subgroup � of the group of rotations �� on �� is defined as

�� (4)

where

�� (5)

for each� � � (�� are the components of� and sum over repeated indexes is understood).For simplicity of notation, we shall simply write� � for ��.

This action of � on �� is quite standard and central to all discussions about symmetrygroups and symmetry classes for elasticity tensors (see e.g. [6, 7, 8]).

The action of � is orthogonal with respect to the standard inner product

� � � �� (6)

Thus,

�� (7)

Notice that

�� (8)

The orbit of � � �� is the set of all elements obtained from � through the action of �.We denote this orbit by � �:

� � � �� (9)

A subset � of �� is invariant (with respect to the action of �) if � � � � for each� � �.


An important concept is given by the introduction of the set �� of all elements of ��which are left fixed by the action of �

�� (10)

Notice that orbits, invariant sets and �� are concepts which depend on the group �.Thus, strictly speaking, one should always specify this dependence, writing that a certain setis a�-orbit or that a certain other set is�-invariant. Nevertheless, in general we do not insiston such a complete notation, for the sake of simplicity. This will not be a problem, however,since we shall mostly assume that the group � is given and fixed so that we need not makethis dependence explicit, except for a few specific situations.

The set of real-valued linear functions defined over �� is denoted by �, and its typicalelements are written with Greek small letters, such as � , � etc.

Since �� is an inner-product vector space, for each � � � there is a unique associated� � �� such that

� �� (11)

The components of � are just the coefficients of the components of � when � is explicitlywritten as

� �� (12)

A linear function � � � is invariant if it is constant on each orbit. The set of suchinvariants is denoted by . Thus

� �� (13)

A consequence of (11) is that � is invariant if and only if the associated tensor � belongs to��.

Our main interest lies with elements of � which are not invariant but behave as such ona subspace of ��. To make this idea precise we need to introduce, for each � � �, theinvariance space �� of all elements of �� on whose orbit � is constant

�� (14)

Thus, �� is by definition the maximal subspace of �� on which � is invariant (the fact that�� is an invariant linear subspace is a straightforward consequence of the properties of theaction of� on��). Since the orbit of each� � �� is reduced to a single element, it followsthat �� . Thus

�� (15)

and, by definition, � � if and only if �� .We now spend a few words to show that, as anticipated before, there is no loss in

generality in the hypothesis that � be a subgroup of ��, rather than ��. Suppose,


indeed, that this restriction be removed and let � be a compact subgroup of ��. Then,build a larger group �� as the union of � itself and the set ��. Clearly, in view of definition(5), we can deduce that the orbits in�� under the actions of� or �� are the same (this followsbasically from the fact that � acts as ��). Now, since by construction �� , we knowthat

�� (16)

where �� (see e.g. [4, Chapter XIII, Section 9]). Again, by a similar argument,we deduce that the orbits in �� under the actions of �� or �� are exactly the same. Inconclusion, for each group � � �� the set of orbits in �� is the same as for the group�� . Thus, for each linear function � , �� is the invariance space with respect to thegroup � if and only if it is also the invariance space with respect to the group �� .This justifies the choice of restricting our discussion to groups of proper orthogonal tensors.

We call restricted invariants the elements of � for which the invariance space does notcoincide either with �� or ��. In other words, � � � is a restricted invariant if it behavesas an invariant on a subspace �� of��, and thus on a special class of elastic materials, whichis smaller than �� itself but larger than ��. As we noticed before, when �� thefunction � is an invariant. The opposite situation, when �� , occurs if � behaves asa (trivial) invariant only on the subspace of elements left fixed by the action of �. Thus, itseems appropriate to say that such a function is fully noninvariant, since it is impossible torestrict its domain in a non trivial way to make it behave as an invariant.

The full picture can be summarized through the following three definitional propertieswhich describe, respectively, the set of invariants, the set� of full noninvariants and theset� of restricted invariants:

� �� (17)

� � �� (18)

� � �� (19)

In view of (15) we notice that � is the union of three disjoint subsets

� � �� (20)

We are now ready to formulate more precisely the problem under discussion: For one� � ��, how do we know if a given � � � belongs to� or not? We show that, providedwe have at our disposal a direct sum decomposition of �� into irreducible subspaces, thereis an approach which makes it possible to answer systematically this question. Moreover, wegive more explicit results when � � �� or � � ��.


��

We present and comment on two examples of restricted invariants for� � ��, the wholegroup of rotations, and for � � ��, the group of rotations which leave the direction of ��fixed.

For � � ��, define � � � as

� �� (21)

This linear function is not invariant on �� under the action of � � ��. We will showthat there is a proper invariant subspace of �� larger than �� on which � is invariant, i.e. �is a restricted invariant. Consider the set � of anisotropic elastic materials, discussed by Tingin [1, 2], which require uniform pressure �when subjected to a uniform contraction describedby the strain tensor

�� (22)

where � � � is the volume change. Since the corresponding stress is given by

�� (23)

this class of materials can be defined by

� � �� (24)

The set � is then a proper invariant subspace of ��, larger than ��, on which the restrictionof � takes the constant value �. Thus, in our terminology, � is a restricted invariant and, aswe shall later prove, � � �� .

For � � ��, we present one of the restricted invariants found by Ting in [1, p. 515],where it is called a conditional invariant. Define for � � �� the linear function �

� �� (25)

Clearly, � is not invariant on �� for the action of �. Nevertheless, Ting proved in [1] that� is invariant if its domain is restricted to the invariant subspace � of �� defined as

� � �� (26)

Six additional examples of restricted (conditional) invariants can be found going throughTing’s paper, where, as mentioned before, they are derived through direct and explicitcomputations. We now might describe the motivating questions behind our research as: Canwe find more? Is there a rationale behind all this? What about groups other than �� and��?

We expected quick answers, easily obtained from readily available concepts and resultsin abstract algebra, but the entire issue turned out to be more complex.


The lengthy theoretical discussion developed below, however, will make finally clear, asshown in Sections 8 and 9, that the restricted invariants presented above and obtained in [1],far from being mere accidents, belong to a class which can be precisely and exhaustivelydescribed.

�� !��

In this section we wish to give an appropriate definition of the centroid of the orbit of anelasticity tensor under the group action. If the group is finite the average of a function �defined over it is defined as

�

��

� �� (27)

where �� is the order of �. By linearity this can be extended to the tensor valued function� �� , for any given � � ��. Thus, in this case, we let the centroid �� of the orbit of� � �� be

��

��

� �� (28)

It is a consequence of the linearity of the action of � that, for each �� ,

��

��

��

� � ��

��

��

��

�� (29)

thus showing that �� is left fixed by the action of �. A similar argument, moreover, showsthat �� does not really depend on the element of the orbit from which we compute it. Thus, if�� is on the same orbit of ��, �� .

How is it possible to extend the definition of centroid to the orbits under the action ofany compact infinite group?

For this purpose we use Haar integration, a form of integration which can be defined overany compact Lie group and which, in particular, is both left and right invariant. For a quickoverview of the essential properties we refer the interested reader to the book by Golubitskyet al. [4, Chapter XII] and, for a deeper presentation and an explicit construction, to Hochshild[9] and Simon [10]. We write �

�

� ��

for the Haar integral of a real-valued continuous function � defined on �. Sometimes, forsimplicity, we omit the integration variable� and only write

�� . The properties of linearity

and positivity are both satisfied:��

��

�� , for each pair of

functions � � and � � and real constants �� and ��, and�� if � �� for each�.


The characterizing properties of the Haar integral, however, are left and right invariance��

� ��

��

� ��

��

� � �� (30)

The Haar integral can be proved to be unique, if we scale it so that�� . When � is

finite, moreover, Haar integration of a function � defined over � reduces to the expression(27).

Simon’s book [10, Theorem VII.3.2] contains a proof showing that, as a consequence ofuniqueness, Haar integration is also invariant under the map� �� . Thus�

�

� ��

��

� �� (31)

For the orbit of each elasticity tensor it is possible to define a centroid which, roughlyspeaking, is the center of the orbit and is left fixed by the action of �. The definition is basedon Haar integration extended by linearity to the tensor-valued function � �� . Thecentroid �� of the orbit of � is then defined as

��

��

� �� (32)

The linearity of the action of � defined by (5), together with invariance of the Haar integral,as written in (30), imply that, for each �� ,

��

��

��

��

� ��

��

� �� (33)

Thus, for each elasticity tensor �,

�� (34)

and

�� (35)

Since � � � is constant on the orbit of each � � �� we may also deduce that

� � �� (36)

The subspace of all elasticity tensors with the orbit centered at the origin is called � andis defined as

� � ��

� �� (37)

where� is the null tensor in ��. The first relevant consequence of this definition is that ��is the direct sum of orthogonal subspaces �� and �.


�� The space of elasticity tensors is the orthogonal direct sum of the subspace leftfixed by the action of � and the subspace of all tensors with the orbit centered at the origin.Thus

��

��"� It is obvious that if an elasticity tensor � is left fixed by the action of � and has theorbit centered at the origin then it must be the null tensor �. Thus, �� . Now,for any given �, let

��

Since we know that �� , in view of Definition (32) and property (35), we deduce that

��

� ��

��

� ��

�

��

� ��

� �� (38)

Thus, the orbit of �� is centered at the origin and �� . We conclude that each� � �� canbe written as the sum

� � ��

where �� , �� . Together with the fact that �� this implies that �� isthe direct sum of �� and �.

Let � � �� and � � �. Condition �� means that

� �

��

� ��

By use of (8) we deduce that

� � �

��

��

��

� ��

��

Property (31) implies that

� ��

��

� ��

but, since the orbit of � is centered at �,��

� ��


and we conclude that � � � �. Since � and � were arbitrarily chosen in �� and � thisproves that �� . �

We now define the subspace �� of all elasticity tensors with the orbit centered at theorigin and on which � is constant. Let

�� (39)

A trivial consequence of this definition is that

� � �� (40)

The next result shows that the invariance space �� is the direct sum of �� and �� .

�� The invariance space �� is the orthogonal direct sum of ��, the space leftfixed by the action of�, and�� , the space of tensors with the orbit centered at the origin andon which � is constant. Thus

��

��"� For � � �� let �� . The same argument used in (38) shows again that�� . For any� � � linearity of � now yields

� �� (41)

But, since � � �� , we know that � �� . Moreover, in view of (36),� �� . Then

� �� (42)

This, taken together with (41), shows that �� belongs both to �� and to �, and thus to �� .Each � � �� can now be written as the sum

� � ��

Since the same argument used in Proposition 2 shows that �� we concludethat �� . �

In view of definition (19), Proposition 2 and Proposition 3 tell us that � is a restrictedinvariant if and only if �� is a proper subspace of �.

�� .

In the next section we show how to obtain a more convenient formulation of this condition.


#� � �� $��

The isotypic decomposition for ��, the space of elasticity tensors, is the main tool on whichwe base the condition which characterizes restricted invariants. For simplicity, we only statethe theorem which defines the properties of the isotypic decomposition in the case needed forour discussion. For a wider perspective and a complete proof the interested reader is referredto [4]. We recall from Section 2 that two vector spaces � and� are copies of one another ifthere is an isomorphism which commutes with the actions of � on � and� .

�%&��&' �� There is a finite collection of vector spaces

��

such that

1. � acts irreducibly on each �� ;2. for � �� ,�� is not a copy of �� , i.e.�� ;3. �� , where each �� is defined as the union of all

subspaces of �� which are copies of �� .

For convenience, we let �� to be ��, the subspace of�� on which� acts as the identity,if �� . Notice that, in general, the action of � on �� need not be irreducible, even ifeach subspace �� is invariant.

The isotypic decomposition of �� should be compared with the result of Proposition 2,which shows that �� is the direct sum of orthogonal subspaces �� and �. Since, as wewrote above, �� , we are led to conjecture that � � �� .Indeed, the next proposition shows that this is the case.

�� #� � � �� .

��"� Fix �, with � � � � �, and choose � � �� . Since �� is invariant we deduce that�� . But we know that �� and then it must be �, since �� . Thus,�� , for each � � �� , and, in view of (37), this means that �� , which in turn impliesthat

�� (43)

Proposition 2 and Item 3 of Theorem 1 show that

�� (44)

where, as mentioned before, �� . Thus, in view of (43), we reach the conclusion

��

which completes the proof. �


The decomposition �� implies that we can identify an element � of ��with a pair �� , where �� and �� are orthogonal. Furthermore, the additionaldecomposition � � �� shows that we may also identify the element � witha list �� where �� and �� is orthogonal to each �� . Thus

� � �� (45)

and

� � �� (46)

Another concept we are planning to use is the idea of the space spanned by the orbit ofan element � of ��. We define such a space as

�� span �� (47)

Obviously, �� is invariant under the action of �.For a given function � � � we now prove that the space �� , as defined in (39), is the

orthogonal complement in� of the space spanned by ��, the component in � of �, as definedin (45). More precisely, we let

�� (48)

and we prove that �� .

�� (� The space �� is the orthogonal complement in � of ��,

��

��"� Let � � �� be given. This means that � � �� and

� ��

For � the element which represents � according to (11) this implies that

� � � � ��

But, since we may write � � ��with �� we deduce that, for each� � �,

��

Thus, � is orthogonal to the orbit of �� and, as a consequence, to the space spanned by it.We conclude that if � � �� then � � �� which implies that �� .

Now, let � be an element of �� . Then, in view of definition (48) we know that � � �and that


� � ��

Since � � �, the orbit of � is orthogonal to �� and thus to �� . Thus,

��

The last equality can be written as

� ��

which implies that � belongs to �� . Thus, �� . �

This result is important for us, because it allows for a better characterization of restrictedinvariants. In view of Propositions 3 and 6 we may write

�� (49)

Since � is a restricted invariant if and only if �� properly contains �� and is properlycontained in ��, in view of Proposition 4 we deduce that this condition is satisfied if andonly if �� is a non trivial subspace of �.

�� )� � � � � �� .

Recall that, as in (45), we may write for � the decomposition

� � �� (50)

Thus,

�� (51)

where each �� is uniquely determined by ��.We now show how the condition for � to be a restricted invariant, expressed in

Proposition 7, can be rewritten through the spaces spanned by the orbits of �� .The subspace ��, spanned in � by the orbit of ��, can be itself decomposed by use of

Theorem 1. In particular, we let �� be the union of all subspaces of �� (if any) which arecopies of the irreducible spaces �� , used earlier for the decomposition of the larger space��. Obviously, it may happen that for some value of the index � there are no subspaces of�� which are copies of�� . When this is the case, we simply let �� be ��, the trivial spaceformed by the null tensor alone. Since �� is a subspace of �� each �� is a subspace of �� .Thus, in view of Theorem 1, we write

�� with �� (52)


For convenience, we recall the content of Proposition 5:

� � ��

Then, �� is a proper subspace of � if and only if at least one �� is a proper subspaceof �� . Our next result shows that each �� is exactly the space spanned by the orbit of thecorrespondent �� so that we may conclude that � is a restricted invariant if and only if theorbit of at least one component �� spans a proper subspace of �� . The conditions under whichthis may happen will be the subject of the next section.

�%&��&' �� For � � ��

��

��"� First we show that, for each value of the index �, the space �� spanned by the orbit ofthe component �� of �� is a subspace of the isotypic component �� . For any � � �, in viewof (51), we write

� ��

� �� (53)

Since �� belongs to �� , which is an invariant subspace, we know that� �� . But, since� �� is an element of ��, in view of (52) it can also be written in a unique way as

� ��

� � � � �� (54)

The fact that each �� is a subspace of �� and the uniqueness of decomposition (53) imply that

� � � ��

which, in turn, shows that � �� belongs to �� . Thus, the whole orbit of �� is contained in�� and, as a consequence,

�� (55)

Next, we show that �� . Let � be any element of �� . Since �� we know that� also belongs to the space �� spanned by the orbit of �� in�. Thus, there are � constants ��and rotations�� (� � �� ) such that

� ��

��

In view of (51)


��

��

and

� ��

��

��

Linearity of the action of � on �� yields

� � ��

��

��

where��

��

Since we assumed � to be an element of �� uniqueness of decomposition (52) implies that

� ��

��

and

� ��

��

Thus, � is a linear combination of elements of the orbit of �� and, therefore � � �� . Since� was chosen as an arbitrary element of �� we deduce that �� . In view of the oppositeinclusion (55) this implies that �� . �

The conclusion which can be drawn from the above theorem is that �� is a propersubspace of � if and only if there is at least one value of the index � for which �� is aproper subspace of �� . Indeed, from (52) and Theorem 2 we deduce that

�� (56)

Let �� be the orthogonal complement of �� in �� ,

�� (57)

As a consequence of (56) and of (49) we write


�� (58)

In view of Proposition 4 we conclude that � is a restricted invariant if and only if (1) notall the components �� of the associated tensor � in the decomposition (51) are equal to� (inwhich case � would be an invariant), and (2) at least one space �� spanned by the orbit of�� is a proper subspace of �� , for � � �� , or, equivalently, that at least one space �� islarger than ��.

The first requirement guarantees that � is not an invariant and it is not hard to verify.The second condition, in contrast, is more subtle and shall be discussed in the next section.

(� �� *��

The results obtained in Section 4 show that, for a given function � � � which is not aninvariant, the question of it being a restricted invariant can be answered positively if andonly if for at least one value of the index � (� � �� ) the space spanned by the orbit of�� is a proper subspace of �� . We now try to better understand the meaning of this condition.

For the rest of this section we let the value of the index � be fixed so that, for simplicityof notation, it can be omitted from spaces and tensors �� , �� , �� , for which we shall onlywrite �,� , �. We recall that all subspaces of � which are irreducible under � are copies of� and formulate our problem as follows: Find a necessary and sufficient condition for ��,spanned by the orbit of � � �, to be a �� subspace of �.

In order to state our main result we need some preliminary considerations. The completereducibility theorem (see e.g. [4, Chapter XII, Corollary 2.2]) states that any space on whicha compact group� acts orthogonally may be written as a direct sum of irreducible orthogonalsubspaces. In general, this decomposition is not unique.

Let

� � �� (59)

be a decomposition of � into a direct sum of orthogonal subspaces �� (� � �� ) whichare irreducible under the action of �, and hence are all copies of� (i.e., �� ). Sincethis decomposition is not unique the subspaces �� can be chosen in different ways.

A given element � of � can be written as the sum of its components with respect to (59):

� � �� (60)

In particular, we shall use the decomposition of �:

� � �� (61)


Both these decompositions should not be confused with (46), which refers to different spaces.Indeed, as noted before, in this section the space � is one of the spaces �� which appear inProposition 5 and � is one of the components �� in (51).

The complete reducibility theorem can also be applied to �� , the orthogonal complementof the space spanned by the orbit of � � �. Thus

�� (62)

where, again, each � ( � � �� ! ) is a copy of� (notice the use of superscripts, ratherthan subscripts). Thus, the space � can be decomposed as

� � � � � � � � � � � �� (63)

Of course, �� is a proper subspace of � if and only if the collection of orthogonal subspaces � is not empty, i.e. ! � �.

The decomposition of � � � with respect to (63) is written as

� � �� (64)

(notice the use of superscripts rather than subscripts, to avoid confusion with (60)).Let�� be an invariant isomorphism which makes�� a copy of� and let � � be a similar

isomorphism which makes � a copy of� . Thus,

�� (65)

We call� the orthogonal projection of � onto �� and let� � be the orthogonal projectionof � onto � . Then, for each � � �,

��

��

� � �� (66)

and

� � �� !� (67)

Since the action of � on � is orthogonal and all spaces mentioned above are invariant itfollows that the orthogonal projections � and � � are invariant.

Finally, we let � �� be the restriction of � � from � to the subspace �� . Thus, for

� � �� and � � �� !,

� ��

��

�� (68)

Then, since in view of (60) we know that

� �

��

��


from (68) it follows that

��

��

��

� �� (69)

and

� � ��

� �� !� (70)

A simple diagram is useful to make the situation clear. For each value of � and � we have

��

��

� �(71)

We recall that the maps �� and � � are invariant isomorphisms of �� and � onto � .Moreover, since �� and � are both copies of� from Proposition 1 we deduce that eachmap � �

� is either an isomorphism or a null map.The images in � of the components �� under the invariant isomorphisms

�� have an important role in formulating a necessary and sufficient conditionfor �� to be a proper subspace of �. We write {��} for the set of vectors

�� (72)

Before stating our main result we need to define a concept which we choose to call �-dependence.

�&"�� A set �� of vectors in a space� on which � acts irreduciblyis �-dependent if there is a subset of " �� " � �� elements

�� "

and a collection of " invariant isomorphisms �� of� onto itself such that

��

�� (73)

This definition recalls the well known concept of linear dependence and, indeed, canbe formulated in a more familiar way if we first make the observation that, in view ofProposition 1, each invariant linear map � of� into itself is either a null map or an invariantisomorphism. The following proposition gives an equivalent definition of �-dependence.

�� +� A set �� of vectors in a space� on which� acts irreduciblyis �-dependent if and only if there are � invariant linear maps of� into itself

��


not all of them null, such that

��

Notice that when the action is absolutely irreducible the only linear maps on� which areinvariant are scalar multiples of the identity (by definition), and, since one such map is anisomorphism if and only if the scalar itself is different from zero, in this case �-dependenceis clearly equivalent to linear dependence.

In general, however, �-dependence is weaker than linear dependence, since the latterimplies the former, but not vice versa. The fact that �-dependence does not imply lineardependence can be easily seen through a counterexample, which we prefer to omit, andintuitively justified by the observation that multiplication by a constant is only one of themany invariant isomorphisms which might exist on�.

�-independence is defined as the opposite of �-dependence, as expected. For the sakeof clarity we write a diagram showing the relevant implications.

Linear dependence �� -dependence�

�-independence �� Linear independence� (74)

Before stating our final result we recall that vectors �� are defined through (72) and(61) as the images in� of elasticity tensors �� through the invariant isomorphisms �� .

�%&��&' �� A necessary and sufficient condition for �� to be a proper subspace of � is thatthe set �� be �-dependent.

��"� We first show that�-dependence is a sufficient condition for �� to be a proper subspaceof �. Thus, assume there are " (� � " � �) vectors �� and " invariant isomorphisms �� of� onto itself such that (73) holds. For � � �� " let �� be the isomorphism of �� ontoitself which makes the following diagram commutative:

��

��

��

� � �� "� (75)

Then

�� (76)

and condition of �-dependence (73) can be rewritten as

��

��

�� (77)

which, in view of (72), is equivalent to


��

��

�� (78)

Linearity of the group action and �-invariance of each �� and �� imply that thiscondition is satisfied not only by the components �� of � but also by the components ��of any � � ��. Thus,

��

��

�� (79)

We now prove that �� is a proper subspace of � by showing that there is at least one � � �which does not satisfy (79). Indeed, we choose one value for the index �, say ��, and let�� for all � � �� except for �� , which we take arbitrary but different from �.For this choice of � condition (79) reduces to the single term

�� (80)

But this condition cannot be satisfied by one �� different from �, since �� and �� areboth isomorphisms. Thus, since there is at least one � � � which does not satisfy (79) weconclude that �-dependence is a sufficient condition for �� to be a proper subspace of �.

Next, we prove that �-dependence is also necessary for �� to be a proper subspace of �.Thus, assume that � can be decomposed as

� � � � � � � � � � � �� (81)

with ! � �. Consider the set of orthogonal projections �� of �� on �. Not all of these

projections can be null maps, since, otherwise, all elements of � would be orthogonal to �,which, by hypothesis, is a non trivial subspace of �.

In view of Proposition 1 we know that any projection �� which is not a null map is

an isomorphism. Let ��

be the collection of " (� � " � �) such projections which areisomorphisms of �� onto �.

Since, trivially, � � ��, Equation (70), written for � � �, yields

��

�� (82)

In this sum we can neglect all projections �� which are null maps and, thus, we are left

with

��

�� (83)

We now let �� be the invariant map which makes the following diagram commutative:


��

��

� ��

�� "� (84)

Thus,

�� (85)

and, since each ��

is an invariant isomorphism of �� onto �, we deduce that each �� isan invariant isomorphism of� onto itself.

Since �� is an isomorphism of � onto� we know that

��

��

��

��

��

�� (86)

Finally, recall that, in view of (72),

��

so that, by substitution in (86), we obtain

��

��

��

��

and, in view of the definition of �� given by (85), we conclude that

��

�� (87)

This shows that�-dependence is necessary for �� to be a proper subspace of �, and the proofis complete. �

A straightforward consequence of this result is that if the number � of copies of� inthe orthogonal direct sum decomposition (59) is larger than the dimension of �, then thespace �� is a proper subspace of �. In view of Theorem 3, this follows at once from the trivialobservation that a set of vectors larger in number than the dimension of the space to whichthey belong is always linear dependent and, hence,�-dependent. We make this result formal,for later reference.

��,,-�. �� If the number � in the orthogonal direct sum decomposition (59) is larger thanthe dimension of� then �� is a proper subspace of �.

The result obtained in Theorem 3 can be expressed in a different form. Let �� be theinvariant isomorphism between�� and� obtained through composition of�� and�� when the index � takes a value �� in the finite subset mentioned in Theorem 3,and let otherwise �� simply be �� , when � takes a value which is not in that subset. Formally,


��

�� if � � �� if � #� ��

(88)

If we define

�� (89)

condition (73) can be reformulated as stating that �� has a proper subset �� of " elementssuch that

�� (90)

If we recall that linear dependence implies �-dependence, and that multiplication by ascalar different from zero is an invariant isomorphism of � into itself, we readily reach theconclusion that Theorem 3 may be restated as follows.

�%&��&' �� A necessary and sufficient condition for �� to be a proper subspace of � is thatthere are invariant isomorphisms �� such that the set ��, where �� ,is linearly dependent.

Notice that, in this formulation, the invariant isomorphisms �� are chosen in anappropriate way, while the invariant isomorphisms �� were arbitrarily given.

)� ��

Once a methodology is given for deciding if a given linear function � on �� is a restrictedinvariant we would like to show how it is possible to describe and construct the space ��on which this function behaves as an invariant. The content of Propositions 3 and 49 andTheorem 2 lead to Equation (58), which we rewrite here for convenience as

��

�� (91)

Thus, we now wish to investigate the structure of �� and �� , for any given value of theindex �. For simplicity of notation we fix a value for � and, as in the previous section, weomit the corresponding indication in �� , �� . Rather, we reserve the letter � for describing theorthogonal decomposition (59) of � into a direct sum of � invariant subspaces�� . Henceforth,we shall again use the notation of the previous section, such as (60) and (61).

For � � �� let �� be the image of �� under the invariant isomorphism�� , as described by (72).

In view of Theorem 3 we know that �� is a proper subspace of � if and only if the set�� is �-dependent. Our goal is now the following: describe the structure of the subspace��, and find a necessary and sufficient condition for � � � to belong to it.


First of all, we need to further explore the concept of �-independence. In view ofProposition 8, �-independence can be phrased as follows: a set of � vectors �� in � is�-independent if the only invariant linear maps of� into itself such that

�� (92)

are null maps.Without loss of generality we may assume that the first $ (� � $ � �) vectors in ��

form a�-independent set which is maximal, in the sense that if we add any other vector �� $� to the collection �� we obtain a�-dependent set. This means that,for any value of the index �, there are $ � � invariant linear maps, not all of them null, suchthat

�� (93)

�-invariance of the set formed by the first $ vectors implies that �� is certainly non null,and thus an isomorphism of�. Since an isomorphism is invertible, by composition on theleft with the inverse of �� we are finally able to express each of the last � � $ vectors�� as a combination of the first $ vectors �� through appropriateinvariant maps ��

��

�� $� (94)

Now, since each �� is the image of �� under the invariant isomorphism �� shown in diagram(71), by right composition with�� and by left composition with the inverse of�� we define

�� (95)

the invariant linear map which makes the following diagram commutative:

��

��

��

(96)

(Notice that the � are invariant isomorphisms, while the � and the � are either invariantisomorphisms or null maps.)

Invariance of all the maps involved shows that condition (94) is equivalent to

��

�� $� (97)

Our next result shows that ��, the space spanned by the orbit of �, can be describedthrough the invariant maps �� . More precisely, an elasticity tensor in � belongs to �� if andonly if the last $�� components�� can be written as combinations of the first $ components�� through the invariant linear maps �� obtained from �� by (95) and diagram (96).


�%&��&' #� An elasticity tensor � � � belongs to the space ��, spanned by the orbit of �,if and only if

��

�� (98)

for each � � �� $, where the invariant linear maps

�� $� % � �� $�

are defined by (95).

��"� Assume that � belongs to ��, the space spanned by the orbit of �. This means thatthere are � constants �� and rotations�� such that

� ��

�� (99)

Since � � �� , upon substitution in the last expression we obtain

� ��

��

��

��

��

��

�� (100)

Since each �� is invariant we know that

��

��

But, since the decomposition of � into a sum of elements of spaces �� is unique we readilyconclude that

��

�� (101)

In particular,

��

�� $�

where we can substitute the expression for �� from (97). Thus,

��

��

��

��

��

which, due to the invariance of each �� under the action of� � �, can be rewritten as


��

��

��

��

which, in view of (101), yields Equation (98):

��

�� $� (102)

and this proves the ‘‘only if ’’ part of this theorem.Now, suppose that condition (98) is satisfied by an element� � �. We want to show that

� belongs to the space spanned by the orbit of �.Consider the space

��

the direct sum of the first $ spaces �� in the decomposition (59). Since the set of vectors�� is assumed to be�-independent, Theorem 3 (applied to ��) shows that the spacespanned by the orbit of �� is the whole space ��. This means that, forany possible choice of the first $ components of � in the decomposition (60), there are �constants �� and rotations�� such that

��

�� % � �� $� (103)

By substitution of this expression for �� in Equation (98) we deduce that

��

��

��

�� $� (104)

which, in view of linearity and invariance of each �� , can be rewritten as

��

��

�� $� (105)

The expression for �� of Equation (97) finally shows that

��

�� $� (106)

This relation, taken together with (103), proves that � is in the space spanned by the orbitof �. �

A few words of comment on condition (98), to be satisfied by all and only the elasticitytensors which belong to ��, seems to be appropriate here. Indeed, the natural interpretationis given by thinking of the first $ components as free variables in the description of the linearspace ��. This picture, however, and our result are made non-trivial by the fact that, here, thelinear combination is made through invariant linear maps rather than, simply, real scalars.


We now give a precise recipe for constructing �� , the space orthogonal to �� in �. Toavoid a possible source of confusion we use the letter � for a generic element of this space.Furthermore, we let

�� $� % � �� $�

be the transpose of the invariant linear maps defined by (95) and used in Equation (98).

�%&��&' (� A tensor � � � belongs to �� if and only if

��

�� % � �� $� (107)

��"� We compute the inner product between elasticity tensors � and � through de-composition (60)

� � �

��

��

��

��

Since all � � �� satisfy condition (98) we substitute the appropriate sum for each �� andwrite

� � ��

��

��

��

which, after some rearrangements and in view of the property of the transpose of a linear mapwith respect to the inner product, yields

� � ��

��

��

��

��

��

��

��

��

This inner product is zero for an arbitrary choice of tensors �� (% � �� $) if and only ifcondition (107) is satisfied. �

+� � ��

The results of the previous sections show how to find whether a linear function defined on�� is a restricted invariant for a given group � and how to construct its invariance space.Now we are ready to apply the procedure to the isotropic case, in which G is the whole groupof rotations.

We must first introduce the notion of harmonic tensor. A tensor of any order is totallysymmetric if its components are not changed under any index permutation. A tensor isharmonic if it is totally symmetric and traceless, which means that the trace with respectsto every pair of indexes is zero. In particular we shall deal with � �, the five-dimensionalspace of second-order harmonic tensors


� � � �� !" �� (108)

and with �", the nine-dimensional space of fourth-order harmonic tensors

�" � � � �� (109)

A relevant property of � � and �" is that they are both irreducible under the action of��, which means that they have no proper invariant subspaces [4]. Moreover, it followsfrom Chapter XIII of [4], Theorem 7.3, that the representation of �� on� � is absolutelyirreducible. The same holds for the representation of �� on �". Notice that we couldeasily prove that � � is absolutely irreducible for the action of ��, applying results oflinear continuum mechanics. Indeed, it is well known that the constitutive equation of a linearisotropic material is

�� &�#�� (110)

where � !" is the Cauchy stress tensor and � !" is the infinitesimal deformationtensor. Therefore, the elasticity tensor of an isotropic material is

� � �� &�� (111)

The representation theorem for isotropic linear functions [6] assures that (111) is the onlyclass of isotropic tensors in ��, i.e. tensors which satisfy the identity

��

�� !"� � � ��

The elements of� � are the traceless elements of !". The restriction of linear map (110)to � � is a multiple of the identity. Therefore, the linear transformations of � � into itselfwhich commute with the action of �� are only multiple of the identity and hence� � isabsolutely irreducible.

The reason of our interest in the spaces of second- and fourth-order harmonic tensors isexplained by the decomposition of �� into a direct sum of such spaces.

�%&��&' )� There is an ��-invariant isomorphism between �la and the direct sum� � � � � � � � � � rm. Indeed, for � � �la, there is a unique choice of and � �, � and � � � �, � rm, such that

�� #��

� � �� (112)

The values of , , �, � and are determined from � as


� �� #�$� (113)

� �� #%�� (114)

�� %�� #�� (115)

�� %�� #�� (116)

�� #�

� ��

� ��

� �� #��

� �� #��&� (117)

The proof can be given through simple but lengthy computations. The above decompositionis also known in the literature as the harmonic decomposition of the space of elasticity tensors.Notice that different expressions of harmonic decomposition of �� can be found in theliterature (see e.g. [11, 12, 13, 14, 15]). It suffices to use invertible linear combinations of�and � and, likewise, invertible linear combinations of and . The expression presentedabove differs slightly from what can be found in [14] and [7], since, here, the pairs ofsubspaces of �� which are copies, respectively, of � and � � are chosen to be orthogonalto each other, as can be checked by direct computations.

As a straightforward consequence of the harmonic decomposition of the space ofelasticity tensors, we obtain the isotypic decomposition of �� under the action of ��through three vector spaces �� (� � �� ), which are the spaces of harmonic tensors oforder, respectively, �, � and �.

�%&��&' +� Through the isotypic decomposition, the space of elasticity tensors is the directsum of three orthogonal subspaces, invariant under the action of ��

��

where

1. ��, the space left fixed by the group action, union of all subspaces of �� copies of �, istwo-dimensional;

2. ��, the union of all subspaces of �� which are copies of � �, is the direct sum of twosuch spaces, �� ;

3. �� rm.


Notice that the different expressions of harmonic decomposition of ��, shown in theliterature, can be considered as a consequence of different choices of the irreducible spacesinto which �� and �� are decomposed.

In view of Theorem 7 we can define �� and �� as

�� (118)

and

�� (119)

The spaces�� (� � �� ) are orthogonal and both copies of� �. Indeed, Equations (115)and (116) define two invariant isomorphisms �� of �� onto� � for � � �� , as in (65).

Let � be a linear function defined on ��. The problem is to decide whether � is arestricted invariant. The associated tensor � � ��, according to the isotypic decompositionof the space of elasticity tensors, can be written as

� � �� (120)

where �� , (� � �� ).Let �� and �� denote the components of �� in�� and�� respectively. We know that��

and�� are both�-isomorphic to� � through the invariant maps�� and�� defined in (115)and in (116), respectively. Let ��

�� and ��

�� be the images in � � of

the components of ��. Hence

�� (121)

The following proposition holds.

�� /� A linear function � defined on �� is a restricted invariant under the actionof the group of rotations if and only if either �� and �� , or there are real constants�� and ��, not both of them zero, such that

��

��"� If the tensor �, associated to the � � � has components �� (� � �� ), then� is an invariant. If �� and �� , then � is a restricted invariant and its space ofinvariance contains �". In view of Theorem 3 we know that �� is a proper subspace of�� if and only if�� and�� are �-dependent. This means that there are two invariant linearmaps of� � into itself, such that

�� (122)

Since the action of �� on � � is absolutely irreducible, �� and �� are �-dependent ifand only if they are linearly dependent.


Therefore we may conclude that �� is a proper subspace of �� if and only if there arereal constants �� and ��, not both of them zero, such that

�� (123)

We can rewrite the above condition in terms of the components �� of the tensor �, bysubstitution of�� and�� as given in (115) and in (116)

�� (124)

where �� and �� .Following Voigt’s notation we have in compact form the five equations which express

the linear dependence between�� and��

��

��

��

�� (125)

��

��

��

Now we would like to construct the space �� on which a given function � � � behavesas an invariant. According to the isotypic decomposition of the space of elasticity tensors,� � �� can be written as� � ��, where�� ,�� and�� . Let��and �� denote the components of �� in �� and �� respectively. The following Propositionis a consequence of Theorem 6 and Proposition 9.

�� 0� Let � be a restricted isotropic invariant and � be the associated tensor. Then,

1. If �� and �� , for any �� , not both of them zero, the spaceof invariance is �� rm.

2. If �� and the condition (123) holds for some ��, �� , the space of invariance is�� ! �rm, where ! � �� .

3. If �� and the condition (123) holds for some ��, �� , the space of invariance is�� ! , where ! � �� .

We conclude the discussion on isotropic restricted invariants by examining the example(21) presented in Section 2, where the linear function � was defined as


� �� (126)

for � � ��.The tensor � associated with � has components

� ��

��

�

�� (127)

the other components being zero. It is easy to prove that �� and that �� has nullcomponent in ��, while �� . Therefore, we come to the expected conclusion that � isa restricted invariant. Its invariance space, moreover, is the direct sum �� . InSection 2 we noticed that this function is invariant on the space

� � �� (128)

For a given � � �� , by taking the trace on the second couple of indicesin the relation (112), we can easily prove that � � �, while , , �, are arbitrary. Thus,� � �� .

/� �� $ � ��

We turn now our discussion to the transversely isotropic case, in which the group � is ��,the group of rotations which leave a given direction, say the axis of ��, fixed. �� isgenerated by ��, the group of rotations which leave ��, fixed and by ��, which is a rotationof ' about ��.

The first step of the procedure shown in the previous sections is to find the isotypicdecomposition for �� under the action of ��. In view of Theorem 7 there is an ��invariant isomorphism between �� and the direct sum � � � � � � � � � � �".Notice that this decomposition is invariant, but no longer irreducible for the action of �.By a technique named after Cartan, we now decompose the spaces of harmonic tensors oforder two (� �) and of order four (�"), which are present in the decomposition of ��,into irreducible subspaces invariant under the action of ��. For a detailed description ofthe technique see, for example, the book of Golubitsky et al. [4] and for further reference andapplication to continuum mechanics see [7, 16, 17].

According to Cartan decomposition the space " of harmonic tensors of order � in thethree-dimensional Euclidean space can be decomposed into the direct sum of �� subspaces"� (� � �� ), which are irreducible for the action of ��. The dimension of "�is � for � � � and is � for � � �. We can view the two-dimensional spaces "� (� � �) as‘‘planes’’. A right-handed rotation � of ( about �� acts on each ‘‘plane’’ "� as a rotationthrough �( . Moreover, �� acts on "� as either the identity, if the order � is even, or as areflection, if � is odd, while for � � � it acts as a reflection about either a ‘‘horizontal’’, or‘‘vertical’’ axis in"� , depending on evenness of � and �.

Applying Cartan decomposition to� �, the space of second-order harmonic tensors, weobtain a direct orthogonal sum of three irreducible spaces, invariant under the action of��:


� � � #� �#� �#��

where #� is one-dimensional and #�, #� are two-dimensional. We notice that, since theorder � is even, the group leaves#� fixed. Moreover, the different action of the group on#� ,with � � �, leads us to conclude that the planes #� are not copies of one another.

By use of ‘‘Cartan decomposition’’, we can similarly decompose�", the space of fourthorder harmonic tensors, into the direct sum of five irreducible spaces, invariant under theaction of ��:

�" � $� �$� �$� �$� �$��

where $� is one-dimensional and $�, $�, $�, $� are two-dimensional. Since the order � iseven, the group leaves $� fixed. Moreover, examining the different action of the group on$� , with � � �, we may conclude that the ’’planes’’ $� are not copies of one another.

Let us consider now, for a given �, (� � �� ), the ‘‘planes’’#� and$� . The group has thesame action on them, hence, there is a trivial �-invariant isomorphism between #� and $� ,and we may conclude that the two subspaces are copies of one another.

Now we recall that �� has an invariant decomposition under the action of �� intothe direct sum of �� , the two-dimensional space left fixed by the action of thegroup, �� and ��,�-isomorphic to� �, and �". By ‘‘Cartan decomposition’’ each space�� , � � �� has an ��-invariant decomposition into the direct sum of three vector spaces,�-isomorphic to#� (with � � �� ), while�" has an��-invariant decomposition intofive spaces $� (with � � � � �).

The geometric view of the group action on each subspace of the decomposition makesstraightforward the identification of the five vector spaces which are the isotypic componentsof �� under the action of ��:

1. ��, which is �-isomorphic to reals, and on which the group � acts as the identity.We recognize, in the harmonic decomposition of �� and in the subsequent Cartan de-composition, five vector spaces, which are copies of��: namely the two one-dimensionalspaces left fixed by the action of ��, the one-dimensional space �-isomorphic to #�in the decomposition of �� with (� � �� ) and finally $�.

2. ��, a ‘‘plane’’ rotated through ( by the action of a right-handed rotation� of ( about ��and reflected about the ‘‘vertical’’ axis by the action of ��. There are three two-dimensionalvector spaces in the irreducible decomposition of ��, which are �-isomorphic to ��,namely the ‘‘planes’’ �-isomorphic to #� in the decomposition of �� with (� � �� ) and$�.

3. ��, a ‘‘plane’’ rotated through �( by the action of a right-handed rotation� of ( about�� and reflected about the ‘‘horizontal’’ axis by the action of ��. There are three two-dimensional vector spaces in the irreducible decomposition of ��, which are �-isomor-phic to��, namely the ‘‘planes’’ �-isomorphic to #� in the decomposition of �� with(� � �� ) and the ‘‘plane’’ $� in the decomposition of �".

4. �� $�, a ‘‘plane’’ rotated through �( by the action of a right-handed rotation� of (about ��.


5. �� $�, a ‘‘plane’’ rotated through �( by the action of a right-handed rotation� of (about ��.

We summarize the results in the following theorem.

�%&��&' /� Through the isotypic decomposition, the space of elasticity tensors is the directsum of five orthogonal subspaces, invariant under the action of ��:

�� (129)

where ��, the space left fixed by the group action, is five-dimensional, �� and �� are boththe direct sum of three irreducible two-dimensional subspaces, and finally �� and �� are bothtwo-dimensional irreducible subspaces.

The following proposition, which is an immediate consequence of the isotypic de-composition of �� under the action of ��, shows that every linear function is a restrictedinvariant under the action of ��, which is quite surprising. One might conjecture that thesame property holds if � is an arbitrary subgroup of ��.

�� Every linear function on�� is a restricted invariant under the action of��.

��"� Consider the subspace ��,

��

� � � �� (130)

where the subspaces � �� (� � �� ) are �-isomorphic to ��. We notice that the number

of irreducible subspaces, copies of��, in the orthogonal direct sum decomposition (130) islarger than the dimension of��. The result follows from Corollary 1. Obviously, we obtainthe same result if we consider the subspace ��. �

We conclude this survey on transversely isotropic restricted invariants briefly discussingthe example (25) presented in Section 2, where � � � was defined as

� �� (131)

The tensor � associated with � has components (in Voigt’s notation)

� ��

��

�

�� (132)

the other components being zero.In order to find the space of invariance �� we must know the components �� (� � � � �)

of the associated tensor in the isotypic decomposition of ��. For an explicit description ofthe orthonormal basis for the component spaces �� , with � � � � �, we refer to [17]. It iseasy to verify that the components of � in �� and in �� are different from zero, while � hasnull components in �� , with � � � � �. Therefore, we may conclude that the invariancespace is the direct sum �� , i.e., �� . From


the knowledge of a basis for ��,(see [17]), one can deduce that (in Voigt’s notation and inagreement with Ting [1])

�� (133)

Acknowledgments. We wish to express our gratitude to Professor B. Webster and Professor W. C. Waterhouse for use-ful suggestions concerning the proof of Theorem 3 through an interesting discussion over the Internet, in the newsgroupsci.math.research.

��

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Restricted Invariants on the Space of Elasticity Tensors

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