PhD Morgane Evin
email: evinmorgane@gmail.com

MD, PhD Alban Redheuil

MD Gilles Soulat

MD Ludivine Perdrix

MD Golmehr Ashrafpoor

PhD Alain Giron

MS Jérôme Lamy

MD Carine Defrance

MD Charles Roux

MD, PhD Stéphane N Hatem

MD, PhD Benoit Diebold

MD, PhD Elie Mousseaux

PhD Nadjia Kachenoura

Left atrial aging: a cardiac magnetic resonance feature-tracking study

Keywords: Left atrial function, aging, feature tracking, magnetic resonance imaging

Abstract

Aims

Importance of left atrial (LA) phasic function evaluation is increasingly recognized for its incremental value in terms of prognosis and risk stratification. LA phasic deformation in the pathway of normal aging have been characterized using echocardiographic speckle tracking. However no data are available regarding agerelated variations using feature tracking (FT) techniques from standard cine magnetic resonance imaging (MRI).

Methods and Results

We studied 94 healthy adults (41±14 years, 47 females.) who underwent MRI and Doppler echocardiography on the same day for left ventricular diastolic function evaluation. From cine MRI, longitudinal strain and strain rate, radial motion fraction and radial relative velocity, respectively corresponding to the reservoir, conduit and LA contraction phases were measured using dedicated FT software.

Longitudinal strain and radial motion fraction decreased gradually and significantly with aging for both reservoir (r>0.31, p<0.003) and conduit (r>0.54, p<0.001) phases whereas they remained unchanged during the LA contraction phase. Subsequently, the LA contraction to reservoir ratio increased significantly with age (r>0.44, p<0.001).

Longitudinal strain rate and radial relative velocity significantly decreased with age (reservoir: r=0.39, p<0.001, conduit: r>0.54, p<0.001) and these associations tended to be stronger in females than in males. Finally, associations of LA functional indices with age were stronger in individuals with lower E/A ratio and E' as well as higher E/E' ratio, highlighting the LV-LA interplay.

Introduction

Left atrial (LA) evaluation has been shown to be useful in various conditions such as valvular diseases, cardiomyopathies and heart failure [START_REF] Blume | Left atrial function: physiology, assessment, and clinical implications[END_REF]. In addition, LA dysfunction has been reported to be an important correlate of atrial fibrillation in the elderly and subsequently to stroke and poor outcome [START_REF] Heart | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC). Eur Eur Pacing Arrhythm Card Electrophysiol J Work Groups Card Pacing Arrhythm Card[END_REF][START_REF] January | AHA/ACC/HRS Guideline for the Management of Patients With Atrial Fibrillation A Report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF]. Accordingly, better characterization of LA function and geometry and their evolution with aging could be of major usefulness in predicting atrial fibrillation but also in differentiating physiological from pathological changes. Although age-specific changes in LA geometry and function have been described in echocardiography using speckle tracking or tissue Doppler techniques [START_REF] Sun | Left atrial regional phasic strain, strain rate and velocity by speckle-tracking echocardiography: normal values and effects of aging in a large group of normal subjects[END_REF], only LA geometry and ejection fraction indices have been described in cardiovascular magnetic resonance imaging (MRI) in the setting of aging [START_REF] Germans | Effects of Aging on Left Atrioventricular Coupling and Left Ventricular Filling Assessed Using Cardiac Magnetic Resonance Imaging in Healthy Subjects[END_REF]. However, the characterization of LA function in addition to its geometry and ejection fraction could be of interest since it enables a more comprehensive evaluation of the complex LA phasic behavior. Indeed, the assessment of the reservoir, conduit and LA contraction phases could enable a better evaluation of LA function as well as its coupling with the left ventricle (LV).

Regarding LA geometry, both echocardiography and MRI agreed on subtle LA dilation and changes in diameters with aging [START_REF] Maceira | Reference left atrial dimensions and volumes by steady state free precession cardiovascular magnetic resonance[END_REF]. Interestingly, echocardiographic characterization of LA function showed significant changes even without significant LA enlargement [START_REF] Thomas | Compensatory changes in atrial volumes with normal aging: is atrial enlargement inevitable?[END_REF]. More specifically, functional parameters such as LA strain, strain rate and LA active or passive emptying fractions were reported to be significantly altered with aging [START_REF] Boyd | Atrial strain rate is a sensitive measure of alterations in atrial phasic function in healthy ageing[END_REF][START_REF] Okamatsu | Effects of aging on left atrial function assessed by twodimensional speckle tracking echocardiography[END_REF].

Cardiac MRI has been reported as a reference modality for LA volumetric quantification ( 16) and more recently, few studies revealed the ability of MRI to characterize LA wall deformation using endocardial feature tracking [START_REF] Evin | Assessment of left atrial function by MRI myocardial feature tracking[END_REF][START_REF] Habibi | Feature Tracking Cardiac Magnetic Resonance Imaging in the Assessment of Left Atrial Function[END_REF][START_REF] Kowallick | Quantification of left atrial strain and strain rate using Cardiovascular Magnetic Resonance myocardial feature tracking: a feasibility study[END_REF], in patients. Nevertheless, to the best of our knowledge, age-related variations in LA myocardial strain and strain rate values have never been described in MRI literature. Accordingly, our main objective was to provide normal ranges of LA strains and strain rates for the three LA phases by applying feature tracking on MRI steady state free precession (SSFP) data of 94 healthy volunteers divided into 3 age groups, with equal distribution of males and females. Furthermore, associations between such LA functional indices and state of the art echocardiographic LV diastolic function indices were studied.

Glossary

Methods

Study population and MRI imaging

We studied 94 healthy volunteers (41±14years, 47 females) who underwent MRI exam on GE 1.5T Magnet, including steady state free precession (SSFP) acquisitions in short-axis views and in three long axis views (2, 3, and 4 chamber), during breath-holding. All subjects were asymptomatic and had no history of cardiovascular diseases. They all had normal electrocardiogram and a body mass index (BMI) <30kg/m². SSFP acquisitions were performed with the following average scan parameters: acquisition matrix of 260×192, TR of 3.7 ms, TE of 1.5 ms, flip angle of 50°, pixel size of 0.74 mm×0.74 mm, slice thickness of 8 mm, views per segment of 12 and a temporal resolution ranging between 20 and 30ms. The study protocol was approved by the institutional review board and informed consent was obtained from all participants.

LV volumes and ejection fraction as well as LV mass and LV myocardial wall thickness were computed after manual tracing of myocardial borders on MRI images using the Qmass software (Medis, v7.6, Leiden, the Netherlands). Blinded to strain analysis, the Qmass software was also used for LA volumes and ejection fraction evaluation using the biplane method (Simpson's method).

Doppler Echocardiography Diastolic Function

Standard diastolic function assessment was performed by Doppler echocardiography as described in the guidelines [START_REF] Nagueh | Recommendations for the evaluation of left ventricular diastolic function by echocardiography[END_REF] on the same day as MRI and included the estimation of : transmitral early (E, cm/s) and atrial (A, cm/s) maximal velocities, deceleration time (DT, ms) and mitral annulus maximal longitudinal velocity measured on the lateral wall (E', cm/s). The ratios conventionally used for diastolic function evaluation (E/E' and E/A) were calculated (Figure 1).

Cardiac MRI evaluation of LA functional indices

The LA endocardial feature tracking software used in this study was previously described in [START_REF] Evin | Assessment of left atrial function by MRI myocardial feature tracking[END_REF]. Briefly, it uses block matching which is based on spatial correlation constrained by active contours and edge detection, and enables to track over the cardiac cycle the endocardial contour drawn manually on a single phase corresponding to maximal LA dilation. After tracking, LA endocardial contours were used to estimate: 1) longitudinal strain, defined as the relative temporal change in LA border length, 2) radial motion fraction, defined as the relative radial displacement of the LA endocardial border towards LA center of mass (Figure 1). Then, quantitative indices corresponding to the three LA phases were estimated from both longitudinal strain (Sl R , Sl E , Sl A ) and radial motion fraction (Mr R , Mr E , Mr A ) curves and for each LA phase, respectively: reservoir phase longitudinal strain (Sl R ) and radial motion fraction (Mr R ), corresponding to the first maximum; LA contraction phase indices, longitudinal strain (Sl A ) and radial motion fraction (Mr A ), corresponding to the second maximum; and conduit phase indices, longitudinal strain (Sl E ) and radial motion fraction (Mr E ), corresponding to the difference between the first and the second maximum. Ratios between reservoir and LA contraction indices were then calculated for both longitudinal strain (Sl A /Sl R ) and radial motion fraction (Mr A / Mr R ).

Afterwards, longitudinal strain rate and relative radial velocity curves were, respectively, calculated as the time derivatives of the aforementioned longitudinal strain and radial motion fraction curves (Figure 1). Then, both longitudinal strain rates and radial relative velocity indices were estimated for each LA phase: reservoir phase indices, SRl S' and Vr S' , corresponding to the positive maximum; conduit phase indices, SRl E' and Vr E' , corresponding to the first negative maximum; and LA contraction phase indices, SRl A' and Vr A' , corresponding to the second negative maximum. Finally, ratios between conduit and LA contraction indices were provided for both longitudinal strain rate (SRl E' /SRl A' ) and relative radial velocity (Vr E' /Vr A' ).

Of note, functional LA indices were calculated as the average of values obtained on the three long axis views (2, 3, and 4 chamber).

The inter-observer variability analysis for this method resulted in coefficients of variation of 4.3%, 7%, and 10.2% for the three phases for longitudinal strains, of 15.1%, 12.2%, and 9.8% for longitudinal strain rate parameters, of 2.9%, 8.3%, and 9% for the three phases of radial motion fraction and of 7.4%, 8.6%, and 10.2% for radial relative velocities [START_REF] Evin | Assessment of left atrial function by MRI myocardial feature tracking[END_REF].

Statistical analysis

To assess gradual variations of LA functional indices with age, subjects were divided into 3 subgroups according to age and with equal distribution of males and females in each sub-group: age group 1 (between 20 and 29 years), age group 2 (between 30 and 49 years), age group 3 (over 50 years). All continuous variables are given as means ± standard deviations. Differences across age groups were tested using one-way Anova test. Associations between continuous variables were studied using linear regression and Pearson correlation coefficients were provided for the whole group and separately for males and females. To study the association of LA functional indices with LV diastolic function indices, distributions of LA indices according to age groups and to diastolic LV function indices (E/A, E/E', E') were represented while stratifying LV diastolic function indices according to their median values. For this latter analysis, the non-parametric Wilcoxon test was used to assess the significance of differences between the elderly subjects group and the remaining groups. Statistical analysis was performed using the R software and statistical significance was indicated by p-value< 0.05. There was a significant increase in BMI and a gradual and significant increase in systolic and diastolic blood pressures (p<0.001) with age. Regarding MRI LV systolic function indices, no significant changes were observed between age groups in terms of LV mass, ejection fraction and volumes, although these measures followed the expected trends such as an increase in LV mass along with a decrease in LV volumes in the elderly. Finally, there was a slight and significant increase in LV wall thickness with age.

Results

Patient characteristics and LV function

All diastolic function parameters varied significantly across age groups. Furthermore, all diastolic variables followed the expected trend with age. Indeed, there was a decrease in E wave (p<0.001) along with an increase in A wave (p<0.001) resulting in a decrease in E/A ratio (p<0.001). Moreover, deceleration time increased with age (p=0.01) while E' decreased with age (p<0.001) resulting in a gradual increase in E/E' ratio (p<0.001). Strain (Sl A /Sl R ) and radial motion fraction (Mr A /Mr R ) ratios combining LA reservoir and contraction phases increased significantly with age. Additionally, a significant decreasing trend was observed for strain rate (SRl E' /SRl A' ) and relative velocity (Vr E' /Vr A' ) ratios combining LA conduit and contraction phases. Regarding gender differences, the strongest associations between LA functional indices and age were found in women group, regardless of the LA phase (Figure 2, supplementary table). LA volumes indexed to BSA were equivalent between males and females except for individuals ≥ 50 years old, where LA volumes were higher in males (supplementary table). Concerning LA functional parameters, magnitudes of longitudinal strain and strain rate as well as radial motion fraction and relative velocity, were higher in females <30 years old, regardless of the LA phase (supplementary table). Conversely, such magnitudes were slightly higher in males ≥

Effect of aging on LA parameters

Association between atrial and LV diastolic functions indices

Distributions of LA conduit phase longitudinal strain and radial motion fraction indices according to age groups and to diastolic LV function indices (E/A, E/E', E') are shown in Figure 3. Independent of diastolic function indices, a decreasing trend with aging was observed for LA strain and motion fraction values corresponding to conduit phases. However, such decrease in longitudinal strain and radial motion fraction with age was statistically significant only for low values of E/A (p<0.014), of E' (p<0.004) and high values of E/E' (p<0.031).

Discussion

Changes in LA function and dimensions have been mostly studied as a reflection of LV diastolic dysfunction. Although a part of LA functional alterations could be attributed to LA/LV coupling which are intimately related to variations in pressure gradients between the two chambers, another part could be associated to intrinsic LA tissue alterations which may increase LA stiffness and alter its contraction and relaxation, as a result of electrophysiological alterations [START_REF] Boyd | Atrial dilation and altered function are mediated by age and diastolic function but not before the eighth decade[END_REF]. Indeed, associations between LA tissue and its functional alteration are supported by recent studies which indicated significant relation between LA function evaluated by echocardiographic speckle tracking and the amount and myocardial fibrosis provided by histological analysis (4).

Studying LA through the sole evaluation of volumes and ejection fraction is limited since LA function is complex and comprises three main phases whose mechanisms and variations with age may be explained by both LV diastolic function alterations with aging, intrinsic LA aging and their interplay. Accordingly, in this study, an insight into LA phasic function modifications in the pathway of healthy aging was provided and its association with state of the art evaluation of LV diastolic function using Doppler echocardiography was studied.

First, the expected age-related changes in diastolic function indices were found in our population [START_REF] Okura | Age-and gender-specific changes in the left ventricular relaxation: a Doppler echocardiographic study in healthy individuals[END_REF][START_REF] Yamada | Effect of aging on diastolic left ventricular myocardial velocities measured by pulsed tissue Doppler imaging in healthy subjects[END_REF]. Indeed, there was a significant decrease in E wave, E/A and E' as well as a significant increase in A wave, E/E' and deceleration time with aging. Furthermore, our results regarding conventional LA volumetric indices were in line with previous MRI findings (16) despite slight discrepancies with echocardiographic results [START_REF] Kou | Echocardiographic reference ranges for normal cardiac chamber size: results from the NORRE study[END_REF]. LA volumes indexed to BSA increased slightly through age groups and their linear association with age resulted in significant correlations. Moreover, as previously shown [START_REF] Germans | Effects of Aging on Left Atrioventricular Coupling and Left Ventricular Filling Assessed Using Cardiac Magnetic Resonance Imaging in Healthy Subjects[END_REF], LA ejection fraction remained unchanged across age groups. This might be explained by the relatively parallel augmentation of both LA enddiastolic and end-systolic volumes with age.

Regarding the newly proposed MRI LA functional indices, the linear regression of their association with age resulted in higher correlation coefficient than those found for conventional LA volumetric indices although lower than those found for LV diastolic function indices. Specifically, Tables 1 and2 indicated that the main significant variations of LA functional indices with age were: 1) a decrease in amplitude of functional indices related to reservoir and conduit phases for both longitudinal strain and radial motion fraction, 2) an increase in LA contraction over reservoir ratio for both longitudinal strain and radial motion fraction, 3) a decrease in longitudinal strain rates and radial relative velocities corresponding to LA reservoir (S') and conduit phases (E'), 4) a decrease in longitudinal strain rates and radial relative velocities ratios combining the LA conduit and contraction phases (E'/A'). Of note, the strongest associations with age were found for longitudinal strain rate and relative velocity corresponding to the conduit phase. This might be due to the LV longitudinal relaxation which affects LA wall. When studied separately in males and females, age related variations in LA function indices were found to be stronger in the females group. While, magnitudes of strain indices were higher in young females (<30 years), as compared to males, they were equivalent or even slightly inferior in elderly females (≥50 years), resulting in an accentuated decreasing trend in females.

These findings are in line with those shown on the LV aging according to gender [START_REF] Cheng | Age-related left ventricular remodeling and associated risk for cardiovascular outcomes: the Multi-Ethnic Study of Atherosclerosis[END_REF][START_REF] Cheng | Correlates of echocardiographic indices of cardiac remodeling over the adult life course: longitudinal observations from the Framingham Heart Study[END_REF] and might be explained by the important role played by estrogens in heart remodeling.

In addition, associations of LA function indices with LV diastolic function parameters revealed significant variations of LA longitudinal strain with age in the subgroup of individuals corresponding to higher E/E' as well as lower E/A and E', especially during the conduit phase. This finding is concomitant with the previous knowledge regarding the important role of LV-LA coupling [START_REF] Germans | Effects of Aging on Left Atrioventricular Coupling and Left Ventricular Filling Assessed Using Cardiac Magnetic Resonance Imaging in Healthy Subjects[END_REF] in LA aging.

Such age-related variations in LA functional parameters are in line with physiological knowledge. Briefly, during reservoir phase, the pulmonary veins inflows enter the LA cavity while the mitral valve is closed and the mitral annulus is moving towards the apex inducing LA dilation. Age-related changes in this phase can be explained by alteration in relaxation capacities of the LA associated with tissue changes (i.e. myocardial fibrosis) and the underlying electromechanical alterations [START_REF] Spadaccio | The role of extracellular matrix in age-related conduction disorders: a forgotten player?[END_REF] as well as reduced mitral valve annulus longitudinal motion. After reaching its maximal dilation, the mitral valve opens resulting in passive LA emptying due to pressure gradient between LA and LV. Such phase is named LA conduit phase and follows the significant drop in LV pressures which occurs just before opening of the mitral valve and can disappear in case of severe LV alterations, such as dilated cardiomyopathy.

Effect of age on this phase combines the decrease in LA wall elastic recoil capacity with LV aging which is known to be associated with altered myocardial relaxation capacities [START_REF] Lakatta | Arterial and cardiac aging: major shareholders in cardiovascular disease enterprises: Part II: the aging heart in health: links to heart disease[END_REF]. Finally, LA emptying is completed by its active contraction whose contribution to LV filling increases with age as depicted by the increase in LA contraction to reservoir strain ratio. In our data, despite a significant increase in A wave with age, LA contraction strain and radial motion fraction remain unchanged probably because of the slightly dilated LA in the elderly which might result in higher late LV filling volume without a substantial change in LA contraction. Overall changes in LA functional indices with age are concomitant with the knowledge regarding LV ageing which consists in an impaired relaxation while its contraction remains preserved. Furthermore, these findings are consistent with the known importance of a preserved LA contraction in different conditions of impaired LV relaxation such as aging and potentially other causes of LV hypertrophy (hypertension, hypertrophic cardiomyopathy) as they highlight the increased importance of atrial contraction relative to the reservoir function.

Limitations

The first limitation of our study is the low number of subjects > 70 years. However, finding healthy subjects without any cardiovascular comorbidities in such age range was difficult. The second limitation is due to the fact that feature tracking techniques might suffer from out of plane motion and the temporal resolution of our MRI data. However, feature tracking offers a good alternative to the tagging technique whose application is still restricted to LV myocardium because of the wall thickness issues.

In addition, our temporal resolution was equivalent to those of previous feature tracking studies and a study performed on phantom showed that up to 30 phases per cardiac cycle the error on strain measurements remains reasonable [START_REF] Evin | Assessment of left atrial function by MRI myocardial feature tracking[END_REF]. Finally, the comparison with ultrasound speckle tracking was not performed since our primary goal in the present study was to provide additional LA functional parameters to the already established reference evaluation of LA volumes by MRI and to show their variations with age. A future study, where echocardiographic acquisitions are specifically designed for LA function evaluation should be performed to enable translation of physiological knowledge between the two modalities.

Conclusions

This MRI feature tracking study on healthy volunteers highlighted a reduction of LA functional parameters associated to reservoir and conduit phases with aging along with an increase in the ratio of LA contraction to reservoir phase indices. Linear association with age resulted in higher correlation coefficients for the newly proposed indices than those obtained for conventional LA volumetric indices. The association with LV diastolic function indicated the intricate role of the LV in age-related LA dysfunction and further studies focusing on LA myocardial tissue characterization in addition to its function are needed to fully assess the possible role of intrinsic agerelated alteration on LA compliance. Such studies made possible by MRI tissue mappings [START_REF] Imai | Multi-Ethnic Study of Atherosclerosis: Association between Left Atrial Function Using Tissue Tracking from Cine MR Imaging and Myocardial Fibrosis[END_REF] and strain measurements could help providing a better insight into LA and LV-related functional alterations.

Figure captions

Figure 1 -

 1 Figure 1 -Example of echocardiographic diastolic function graphs, and MRI

Figure 2 -

 2 Figure 2 -Linear regressions between LA functional parameters and age in

Figure 3 -

 3 Figure 3 -Distribution of LA functional parameters according to age groups and

Table 1

 1 shows subjects description by age groups including basic characteristics, brachial pressures, MRI indices of LV systolic function and echocardiographic indices of LV diastolic function. Moreover, correlation coefficients corresponding to the linear regressions of the described indices with age are summarized in table 1.

Table 2 summarizes

 2 LA volumetric indices as well as mean values of LA functional parameters across age groups. Similar to table 1, correlation coefficients corresponding to the linear regression between LA indices and age are provided. Regarding LA volumes, whether indexed or not to BSA, they increased slightly across age groups while EF remained stable. Linear associations of LA volumes with age were significant (p<0.01) although not as strong as those obtained for LV diastolic The strongest associations with age (Table 2) were found for indices characterizing LA conduit phase (Sl E , SRl E' , Mr E , Vr E' ). Such associations were nearly as strong as those found for LV diastolic function indices.

	function indices.
	While indices of longitudinal strain (Sl R , Sl E ) and strain rate (SRl S' , SRl E' ), as well as
	radial motion fraction (Mr R , Mr E ) and relative velocity (Vr S' , Vr E' ) associated with LA
	reservoir and conduit phases significantly decreased in terms of magnitude across age
	groups (p<0.01), those corresponding to LA contraction remained unchanged across
	age groups.

Acknowledgements

We thank the ICAN Institute of Cardiometabolism and Nutrition, Paris, France for the funding of a Post-Doctoral Grant for ME on this research program.

Grants

This work was supporting by the ICAN Institute of Cardiometabolism and Nutrition, Paris.

Disclosure

None declared. LV for left ventricular, EDV for end diastolic volume, ESV for end systolic volume, EF for ejection fraction, SBP and DBP: systolic and diastolic blood pressures.

Text tables

Differences across age groups are tested using one-way Anova test and the p values are provided for significant differences, while the non-significant ones are noted "ns".


MSME N Auffray

B Kolev

M Olive

Handbook of bidimensional tensors: Part I: Harmonic decomposition and symmetry classes

Keywords: Anisotropy, Symmetry classes, Constitutive laws, Higher-order tensors

To investigate complex physical phenomena, bi-dimensional models are often an interesting option. It allows spatial couplings to produce while keeping them as simple as possible. For linear physical laws, constitutive equations involve the use of tensor spaces. As a consequence the different types of anisotropy that can be described are encoded in tensor spaces involved in the model. In the present paper, we solve the general problem of computing symmetry classes of constitutive tensors in R 2 using mathematical tools coming from representation theory. The power of this method is illustrated through the tensor spaces of Mindlin strain-gradient elasticity.

Introduction

To investigate complex physical phenomena such as anisotropic damage mechanics, elastic couplings within composite structures, or wave propagation in architected materials, bi-dimensional models constitute an interesting step before studying the complete 3D situation. It allows spatial couplings while keeping the process as simple as possible.

For linear physical laws, constitutive equations are described by tensors. A sound analysis of these tensors, and of the vectorial spaces they belong to, gives precious information on the physics that can be described by their mean. This way of exploring models through their tensorial description is well-developed in condensed matter physics [START_REF] Jerphagnon | The description of the physical properties of condensed matter using irreducible tensors[END_REF], but not so much known in continuum mechanics. Nevertheless, these last years a renew of interest in tensor methods in the field of continuum mechanics has been observed [START_REF] Desmorat | An alternative to the Kelvin decomposition for plane anisotropic elasticity[END_REF][START_REF] De Saxcé | Invariant measures of the lack of symmetry with respect to the symmetry groups of 2"d" elasticity tensors[END_REF][START_REF] Forte | A unified approach to invariants of plane elasticity tensors[END_REF]. This fact is related to the wish to properly design materials' architecture to optimize overall specific features. The spectrum of tensor methods is relatively broad and includes, among other, the optimal conception of composite materials [START_REF] Vincenti | Optimization of laminated composites by using genetic algorithm and the polar description of plane anisotropy[END_REF], damage mechanics [START_REF] François | A damage model based on Kelvin eigentensors and Curie principle[END_REF][START_REF] Desmorat | An alternative to the Kelvin decomposition for plane anisotropic elasticity[END_REF], full-field identification method [START_REF] Auffray | Invariant-based reconstruction of bidimensionnal elasticity tensors[END_REF], and wave control [START_REF] Bacigalupo | Homogenization of periodic hexa-and tetrachiral cellular solids[END_REF][START_REF] Chen | Micropolar continuum modelling of bi-dimensional tetrachiral lattices[END_REF][START_REF] Auffray | A complete description of bi-dimensional anisotropic strain-gradient elasticity[END_REF][START_REF] Rosi | Anisotropic and dispersive wave propagation within strain-gradient framework[END_REF]. These methods are all based on the invariants of the elasticity tensor. These quantities allow to determine an elastic material uniquely [START_REF] Boehler | On the polynomial invariants of the elasticity tensor[END_REF], and the polynomial relations between them indicate the symmetry classes of this material [START_REF] Auffray | On anisotropic polynomial relations for the elasticity tensor[END_REF].

Tensor methods are not restricted to elasticity and can naturally be applied to other linear laws. For examples, the piezoelectricity effect is modelled by a third order tensor [START_REF] Geymonat | Symmetry classes of piezoelectric solids[END_REF][START_REF] Vannucci | The polar analysis of a third order piezoelectricity-like plane tensor[END_REF], and extended models of elasticity involve tensors of order greater than 4 [START_REF] Forest | Mechanics of generalized continua: construction by homogenizaton[END_REF][START_REF] Trinh | Evaluation of generalized continuum substitution models for heterogeneous materials[END_REF]. Hence, instead of treating each specific case separately, we found interesting to formulate the problem in its full generality to obtain general results.

In the following, tensor spaces used to model physical laws will be called Constitutive Tensor Spaces, a precise construction of those spaces will be given latter on. Considering a constitutive tensor space, from a physical perspective at least three questions naturally arise:

1. What are the different types of anisotropic behaviour that can be described ? 2. How can different materials (with respect to a chosen physics) be uniquely labelled ? 3. How to identify the anisotropic class of a specific tensor from its components in an arbitrary basis ?

In this paper, attention will only be devoted to the first of these three questions and only in the 2D situation. The two other questions, which have been solved for elasticity in 2D by Vianello [START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF], will be the objects of forthcoming papers. Albeit being different in essence, those questions are closely related by the use of the same mathematical tools. Before getting into the core of the matter, let us give few words on the historical background of our problem.

State of the arts

The determination of the different type of anisotropy associated to a tensor space consists in computing the isotropy (or symmetry) classes of an SO(d)-or O(d)-representation on that space. This problem has been properly set up in mechanics for the first time by Forte and Vianello in 1996 [START_REF] Forte | Symmetry classes for elasticity tensors[END_REF]. In their paper, they provided the number and type of symmetry classes for 3D linear elasticity. Since then, other specific studies in 3D have been conducted, and answers have been provided for photo-elasticity [START_REF] Forte | Symmetry classes and harmonic decomposition for photoelasticity tensors[END_REF], piezo-electricity [START_REF] Geymonat | Symmetry classes of piezoelectric solids[END_REF], or flexo-elasticity [START_REF] Quang | The number and types of all possible rotational symmetries for flexoelectric tensors[END_REF]. In two recent publications [START_REF] Olive | Symmetry classes for even-order tensors[END_REF][START_REF] Olive | Symmetry classes for odd-order tensors[END_REF] a general and complete answer has been provided for any 3D constitutive tensor space. Aside from the 3D situation, some studies have been conducted in 2D. The 2D elasticity tensor has been studied in numerous publications [START_REF] Blinowski | Two-dimensional Hooke's tensors -isotropic decomposition, effective symmetry criteria[END_REF][START_REF] He | On the symmetries of 2"d" elastic and hyperelastic tensors[END_REF][START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF][START_REF] De Saxcé | Invariant measures of the lack of symmetry with respect to the symmetry groups of 2"d" elasticity tensors[END_REF][START_REF] Forte | A unified approach to invariants of plane elasticity tensors[END_REF]. The last cited reference provides a clear and comprehensive review of most of the work that have been done on plane elasticity tensors. This synthesis is important since a lot of different denominations for the same technique can be found in the literature [START_REF] Forte | A unified approach to invariants of plane elasticity tensors[END_REF]. The other plane physical behaviours investigated have been: piezo-electricity [START_REF] Vannucci | The polar analysis of a third order piezoelectricity-like plane tensor[END_REF], Mindlin strain gradient elasticity [START_REF] Auffray | Derivation of anisotropic matrix for bi-dimensional strain-gradient elasticity behavior[END_REF][START_REF] Auffray | A complete description of bi-dimensional anisotropic strain-gradient elasticity[END_REF]. Here, instead of considering each case separately, we propose to set the problem in its full generality and to provide a general answer from which specific cases can be deduced. This paper intents to be as self-contained as possible and many examples are provided along the lines.

Organisation

The article is organised as follows: section 2 is devoted to the formulation of the problematic, and by its end our main results are given and some examples provided. The next two sections are devoted to the proofs of these results. In section 3 a decomposition of tensor spaces into irreducible components is introduced. This decomposition, also refereed to as the harmonic decomposition, is the cornerstone of our analysis of the geometrical properties of tensor spaces. In section 4, the link is established between harmonic decomposition and symmetry classes through the use of a tool named the clips operator [START_REF] Olive | Symmetry classes for even-order tensors[END_REF][START_REF] Olive | Symmetry classes for odd-order tensors[END_REF].

Notations

Let us introduce the main notations that will be used throughout the paper. The following matrix groups will be considered:

• GL(2): the group of all linear invertible transformations of R 2 , i.e. F ∈ GL(2) iff det(F) ̸ = 0;

Prepared using sagej.cls • O(2): the orthogonal group, that is the group of all isometries of R 2 i.e. Q ∈ O(2) iff Q ∈ GL(2)
and Q -1 = Q T , where the superscript T denotes the transposition. As a matrix group O(2) can be generated by:

r θ = ( cos θ -sin θ sin θ cos θ ) , 0 ≤ θ < 2π, and σ x = ( 1 0 0 -1 ) ,
in which r θ is a rotation by angle θ and σ x is the reflection across the x axis; • SO(2): the special orthogonal group, i.e. the subgroup of O(2) consisting of transformations satisfying det(Q) = 1. It corresponds to the group of plane rotations generated by r θ ;

Classical notations will be used to denote tensor products:

• ⊗ stands for the classical product, and ⊗ n indicates its n-th power;

• S 2 denotes its symmetrized version, and S n its extension to product of n elements;

• ∧ 2 indicates the antisymmetric tensor product.

Vector spaces will be denoted using blackboard fonts. The subsequent tensor spaces will be considered:

• G n := ⊗ n R 2
is the space of general nth-order tensors, that is tensors with no particular index symmetry; • S n := S n (R 2 ) is the subspace of totally symmetric tensors of G n ; • T n is any subspace of G n which contains S n ; it is generated by taking repeated tensorial products ⊗ p or S q of R 2 ; • K n is the space of n-th order completely symmetric and traceless tensors on R 2 , called harmonic tensors (this space is described in subsection 3.1).

When needed, index symmetries of both spaces and their elements are expressed as follows: (..) indicates invariance under permutations of the indices in parentheses, .. .. and .. .. indicate, respectively, symmetry and antisymmetry with respect to permutations of the underlined blocks.

Bidimensional tensors

In the following, the physical space will be described by E 2 , the 2-dimensional Euclidean space. The mathematical framework used in this section is introduced in Appendix A.

Material and physical symmetry groups

We represent a physical body as a compact 2-dimensional manifold M embedded in the Euclidean space E 2 and having a microstructure attached to each of its points P ∈ M. This microstructure describes the local organisation of matter at a scale below the one used for the continuous description. For crystalline materials this microstructure corresponds to the crystal lattice, for polymers to the organisation of polymeric chains, . . . . As for crystals, microstructures can possess invariance properties with respect to orthogonal transformations Q ∈ O(2). Hence, at each material point P , the set of such transformations forms a point group G M (P ) ⊂ O(2) which describes local material symmetries. At the continuous macroscopic scale detailed description of the microstructure is lost, and information on the microstructure is contained in G M (P ). In the case of an homogeneous medium the point dependence vanishes and G M (P ) = G M . In this paper our attention will be restricted to linear constitutive laws. In this perspective, physical properties are encoded by tensors: we attach a tensor T(P ) to each point P of M, resulting in a tensor field T that needs not be continuous in general. If the material is homogeneous this tensor field is constant, and the material is thus described by a single tensor. In general, the physical model imposes some index symmetries and T(P ) belongs to T n ⊂ G n . The action of O(2) on an element of T n is defined by:

(Q • T) i1i2...in := Q i1j1 . . . Q injn T j1...jn .

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Hence, within our hypotheses, the symmetry group of a physical property described by T is defined as the set of operations that leave this tensor invariant. The symmetry group of T(P ) is:

G T(P ) := {Q ∈ O(2)|Q • T(P ) = T(P )}.
Since constitutive tensors are supposed to describe physical properties of the matter, their symmetry groups must be connected. This connection is stated by the Curie-Neumann's Principle [START_REF] Zheng | The description, classification, and reality of material and physical symmetries[END_REF]: Principle 2.1. Curie-Neumann's Principle. At each material point P ∈ M with a physical property P described by T(P ) ∈ T n , every material symmetry is a physical symmetry, i.e.:

G M (P ) ⊂ G T(P ) .
It is important to emphasize that the notion of point group naturally contains information on the orientation of either the "microstructure" or the "physical property" attached to this point with respect to a reference frame.

Physical symmetry classes

The intrinsic characterization of the symmetry of a physical object has to be independent of its orientation in space. Hence, as proposed by [START_REF] Forte | Symmetry classes for elasticity tensors[END_REF], from both a physical and mathematical point of view, the relevant notion to characterize the physical symmetry of an object is the symmetry class of its related tensor. Recall that for a tensor T ∈ T n , having G T as symmetry group, its symmetry class is defined as the set of all the subgroups of O(2) conjugate to G T :

[G T ] = { QG T Q T ⊂ O(2), Q ∈ O(2) } . ( 2.1)
In other words, the symmetry class of T corresponds to its symmetry group modulo its orientation in O(2). Furthermore, it is known that there is only a finite number of different symmetry classes [START_REF] Bredon | Introduction to compact transformation groups[END_REF], and that each symmetry group is a closed subgroup of O(2). Conjugacy classes of closed subgroup of O(2) are known [START_REF] Armstrong | Groups and symmetry[END_REF] and can be described as follows:

{[1], [Z σx 2 ], [Z k ], [D k ], [SO(2)], [O(2)]} k≥2 ,
in which 1 stands for the trivial subgroup, SO(2) is the continuous rotation group and the other discrete groups are described below:

• Z σx 2 is the group generated by σ x , which denotes a mirror through the x-axis; • Z k is the cyclic group 1 with k ≥ 2 elements generated by the rotation r 2π/k ; • D k (k ≥ 2), the dihedral group with 2k elements generated by r 2π/k and σ x .

In the previous designation k indicates the order of the generating rotation of angle 2π

k . Obviously applying k times this generating rotation will lead to identity. As it will be important latter on, this number can either be odd or even, and only in the first case the group contains the inversion -1. By convention we would say that a group has an infinite order of rotation if it contains SO(2) as a subgroup.

Constitutive tensor spaces

Following [START_REF] Olive | Symmetry classes for even-order tensors[END_REF][START_REF] Olive | Symmetry classes for odd-order tensors[END_REF], constitutive tensor spaces are constructed according to the following procedure:

• Given two physical objects described by tensor spaces T m1 and T m2 , for which we have

S mi ⊂ T mi ⊂ G mi , i = 1, 2,
a linear constitutive law is given by an element of the tensor space

T n = L(T m1 , T m2 ) ≃ T m1 ⊗ T m2 , which, by construction, satisfies S n ⊂ T n ⊂ G n .
• These inclusions can be refined with additional hypotheses:

for self-adjoint constitutive laws, we have m 1 = m 2 = m and tensor spaces T n are constructed from the symmetric product:

T n := S 2 (T m ) .
The symmetric tensor product is a consequence of the existence of a potential. The space of elasticity tensors is an example of such a situation. for coupling laws, tensor spaces T n are constructed from the standard product:

T n := T m1 ⊗ T m2 .
This situation happens for coupling tensors that appear in multiphysical models (for instance, for the piezoelectric tensor).

Finally, we have to consider 3 different cases:

• for self-adjoint constitutive laws:

S 2 (S m ) ⊂ S 2 (T m ) ⊂ S 2 (G m ) ;
(2.2)

• for coupling laws:

S m1 ⊗ S m2 ⊂ T m1 ⊗ T m2 ⊂ G m1+m2 , ( 2.3)
with 2 cases, whether m 1 and m 2 have the same parity or not. The order of resulting tensor space is even in the first case, and odd in the second.

General results

Motivated by Herman's theorem [START_REF] Herman | Some theorems of the theory of anisotropic media[END_REF], which limits the possibilities for the symmetry class of a tensor of given order, we introduce the following definition.

Definition 2.2. For any integer p ≥ 1, we define

{ I 2p max := {[Z 2k ] , [D 2k ]} 1≤k≤p ∪ {[SO(2)] , [O(2)]} I 2p+1 max := {[Z 2k+1 ] , [D 2k+1 ]} 0≤k≤p ∪ {[O(2)]} .
These two sets correspond respectively to the maximal theoretical set of symmetry classes that a tensor space of order 2p (resp. 2p + 1) can possess.

Remark 2.3. Let us consider a physical property described by an odd-order tensor in 2D. If the material symmetry group contains an even-order or an infinite-order rotation, then the related tensor is the null tensor. Indeed, classes related to finite subgroups in I 2p+1 max contain only odd-order rotations, and the only odd-order tensor having [O(2)] as symmetry class is the null tensor. We can also note that classes related to finite subgroups in I 2p max only contain even-order rotations. The main results of the present paper are summed up in the following tables and their proof are postponed in subsection 4.4.

G 2p I 2p max G 2p+1 I 2p+1 max \{[Z 2p+1 ]}
S 2 (T n ) , S n T n I 2n max S 2 (S n ) , n > 2 I 2n max \{[Z 2n ] , [Z 2n-2 ]} T m1 ⊗ T m2 , m i > 0 { I m1+m2 max , if same parity I m1+m2 max \{[Z m1+m2 ]}, if different parity Table 2
with the two degenerate cases

I ( S 2 ( S 1 )) = {[D 2 ] , [O(2)]} I ( S 2 ( S 2 )) = {[Z 2 ] , [D 2 ] , [D 4 ] , [O(2)]}.
The following lemma can be obtained by induction on p.

Lemma 2.5. For each integer p ≥ 1, we have

I ( S 2p ) = {[Z 2k ] , [D 2k ]} 1≤k≤⌊ p 2 ⌋ ∪ {[D 2k ]} ⌊ p 2 ⌋+1≤k≤p ∪ {[O(2)]} I ( S 2p+1 ) = {[1] , [Z σx 2 ] , [O(2)]} ∪ { [ Z O(⌊ 2k+1 3 ⌋) ] , [D 2k+1 ]} 1≤k≤p
where ⌊x⌋ is the integer part of x and the function O (.) is defined as

O (k) := { k, if k is odd; 0, if k is even. .
Let us now consider some explicit examples.

Examples

Classical elasticity tensors The most famous example in the literature is the space of elasticity tensors [START_REF] Blinowski | Two-dimensional Hooke's tensors -isotropic decomposition, effective symmetry criteria[END_REF][START_REF] He | On the symmetries of 2"d" elastic and hyperelastic tensors[END_REF][START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF][START_REF] De Saxcé | Invariant measures of the lack of symmetry with respect to the symmetry groups of 2"d" elasticity tensors[END_REF][START_REF] Forte | A unified approach to invariants of plane elasticity tensors[END_REF], namely

Ela := S 2 ( S 2 ) = { E ijkl ∈ ⊗ 4 (R 2 )|E (ij) (kl)
} .

The symmetry classes of Ela are summed-up in the following table 2 :

Name Monoclinic, Diagonal Orthotropic Tetragonal Isotropic [G E ] [Z 2 ] [D 2 ] [D 4 ] [O(2)] # indep (E) 6 (5) 4 3 2 Table 3
The in-parenthesis number indicates the minimal number of independent components of the matrix in an adapted basis. Some observations should be made here:

• there is no odd-order rotational invariant symmetry class;

• each elasticity tensor is at least in [Z 2 ]; • there is no tetrachiral class [Z 4 ].
Higher order elasticity tensors Other examples involve higher order tensors and are borrowed from Mindlin strain gradient elasticity [START_REF] Mindlin | Micro-structure in linear elasticity[END_REF][START_REF] Mindlin | On first strain-gradient theories in linear elasticity[END_REF]. Some works on its bi-dimensional anisotropic formulation have recently been conducted in [START_REF] Auffray | Derivation of anisotropic matrix for bi-dimensional strain-gradient elasticity behavior[END_REF][START_REF] Auffray | A complete description of bi-dimensional anisotropic strain-gradient elasticity[END_REF]. In this model, aside from the classical fourth-order elasticity tensor, a fifth-and a sixth-order tensor are added. Those tensors belong to:

• the space of coupling elasticity tensors:

Cou = { M ijklm ∈ ⊗ 5 (R 2 )|M (ij)(kl)m } ;
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• the space of second-order elasticity tensors:

Str = { A ijklmn ∈ ⊗ 6 (R 2 )|A (ij)k (lm)n } .
Hence the first space is the coupling space S 2 ⊗ ( S 2 ⊗ K 1 ) , while the second space is S 2 ( S 2 ⊗ K 1 ) . For the fifth-order tensor space Cou, we get:

Name Biclinic Rectangular Trichiral Trigonal Pentagonal Isotropic [G M ] [1] [Z σx 2 ] [Z 3 ] [D 3 ] [D 5 ] [O(2)] # indep (M) 18 (17) 9 6 (5) 3 1 0 Table 4
In the previous table, some observations can be made:

• there is no even-order invariant rotational symmetry class;

• there is no pentachiral class [Z 5 ].

Now for Str, we have:

Name Monoclinic, Diagonal Orthotropic Tetrachiral Tetragonal [G A ] [Z 2 ] [D 2 ] [Z 4 ] [D 4 ] # indep (A) 21 (20) 12 9 (8) 6 Name Hexachiral Hexagonal Hemitropic Isotropic [G A ] [Z 6 ] [D 6 ] [SO(2)] [O(2)] # indep (A) 7 (6) 5 5 4
Table 5 and we notice that:

• there is no odd-order rotational invariant symmetry class;

• each Str-tensor is at least in [Z 2 ] ;

• a "surprising" hemitropic class [SO(2)] appears.

The proofs of these results, as well as the calculation of the number of independent components, for each symmetry class rely on the harmonic decomposition of tensor spaces, which is introduced in the next section. An algorithm to compute this decomposition will be provided, together with a formula which gives the dimension of the fixed point spaces (elements that are fixed by the action of a given subgroup H ⊂ O(2)). From a mechanical point of view, this dimension is the number of physical parameters required to describe a tensor in each symmetry class. This information was provided in the last line of the former tables.

Harmonic decomposition

In order to properly set up the problem, tensor spaces should be decomposed into a collection of elementary spaces called irreducible representations of O(2). We refer the reader to subsection A.2 for an overview of the general mathematical framework, and focus here on the particular case of O(2). Example 3.1. Let us illustrate this decomposition in the special case of second-order symmetric tensors. Any element T ∈ S 2 (R 2 ) admits the following decomposition

T = K 2 + 1 2 K 0 q = ϕ(K 2 , K 0 ) (3.1)
where K 2 ∈ K 2 and K 0 ∈ K 0 are, respectively, the 2D deviatoric and 1D spheric part of T and q is the identity tensor. They are defined by the following formulas:

K 0 := T ii ; K 2 ij := T ij - 1 2 K 0 δ ij .
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The linear mapping ϕ, defined by expression (3.1), is an isomorphism between the direct sum

K 2 ⊕ K 0 and S 2 (R 2 ) K 2 ⊕ K 0 ≃ S 2 (R 2 ).
This decomposition is moreover O(2)-equivariant, or expressed in other way: the components (K 0 , K 2 ) are covariant with T under the O(2)-action, i.e.

QTQ T = ϕ(QK 2 Q T , K 0 ), ∀Q ∈ O(2), ∀T ∈ S 2 (R 2 ).
This decomposition is also irreducible, which means that those tensors cannot be split into smaller tensors satisfying this property. Let us now extend this procedure to higher-order tensors. The first step to achieve is to generalize the notion of deviator.

O(2)-irreducible representations

For each integer n ≥ 1, there is a natural O(2)-representation ρ n on the two dimensional space R 2 given by

ρ n (r θ ) := ( cos nθ -sin nθ sin nθ cos nθ ) , ρ n (σ x ) := ( 1 0 0 -1 ) . (3.2)
There are also two natural O(2)-representations on the one dimensional space R: one of them ρ 0 is the trivial representation, and the other one ρ -1 is the multiplication by det(Q). All these representations are irreducible and each irreducible representation of O( 2) is equivalent to one of them.

Given n ≥ 1, let H n be the space of homogeneous polynomials of degree n in two variables. There is a natural action ρn of the orthogonal group O(2) on H n :

(ρ n (Q)(P))(x) := P(ρ n (Q) -1 (x)), P ∈ H n , x ∈ R 2 , Q ∈ O(2).
The subspace K n ⊂ H n of harmonic polynomials (with vanishing Laplacian) is invariant under this action. Since any (real) harmonic function on R 2 is the real or imaginary part of a holomorphic function, we conclude that K n is generated by Re(z n ) and Im(z n ). In this basis, the action of O( 2)

is precisely (R 2 , ρ n ) introduced in (3.

2).

A third model for this irreducible representation is given by K n , the space of harmonic tensors; i.e n-th order totally symmetric and traceless tensors on R 2 (with its natural O(2)-action), where the trace of an n-th order totally symmetric tensor T is defined as the (n -2)-th order tensor:

(tr T)(x 1 , . . . , x n-2 ) := 2 ∑ i=1 T(x 1 , . . . , x n-2 , e i , e i ).
Remark 3.2. The link between the space of harmonic tensors K n and the space of homogeneous harmonic polynomials K n is described as follows. First, note that to each totally symmetric tensor T corresponds a homogeneous polynomials P, given by P(x) := T(x, . . . , x). This correspondence can be shown to induce an O(2)-equivariant isomorphism Φ between the spaces S n and H n and its inverse can be obtained explicitly by polarization [START_REF] Olver | Classical invariant theory[END_REF][START_REF] Auffray | A complete integrity basis for the elasticity tensor[END_REF]. When restricted to the subspace of traceless tensors, the image of Φ corresponds to the space of harmonic polynomials, since (△P)(x) = n(n -1)(tr T)(x, . . . , x).

Finally, we set K 0 = R with the identity O(2)-action and K -1 = R with the determinant O(2)action. We can interpret these representations as follows:

K -1 is the space of pseudo-scalars, or hemitropic coefficients; K 0 is the space of scalars, that is isotropic coefficients; K 1 is the space of vectors; K 2 is the space of deviators;

K n (n ≥ 3) is the space of n-th order deviators. 3 Prepared using sagej.cls

Decomposition into irreducible representations

We can decompose any representation T n ⊂ G n of the group O(2) or SO(2) into irreducible pieces, that is into harmonic tensor spaces K k [START_REF] Forte | Symmetry classes for elasticity tensors[END_REF]. We shall write

T n ≃ q ⊕ i=1 K ki . (3.3)
Since several pieces may be isomorphic to the same harmonic tensor space K k , it is preferable to group all these pieces into one space

α k K k := α k ⊕ l=1 K k , called an isotypic component (of type K k ) of the representation T n . The integer α k indicates the multiplicity of K k into the decomposition (3.
3) (it is zero if the representation K k does not appear in (3.3)). We shall write therefore

T n ≃ ⊕ k α k K k , ( 3.4)
where it is understood that only a finite number of α k do not vanish. Remark 3.3. It is worth noting that the decomposition (3.3) is uniquely defined only if each k i appears once (c.f. [START_REF] Golubitsky | Singularities and groups in bifurcation theory[END_REF]). At the opposite the global structure of the decomposition (3.4) is uniquely defined.

To determine the symmetry classes of a tensor space, the computation of an explicit isomorphism that realizes the harmonic decomposition (3.3) is not mandatory, only the knowledge of the sequence (α k ) in the isotypic decomposition (3.4) is required. In the next section, an easy way to determine this sequence is presented.

Clebsch-Gordan formula

The structure of the isotypic decomposition can easily be determined using the Clesbch-Gordan formula, which is a direct consequence of the character formula for O(2) representations, as detailed in the Appendix A. Lemma 3.4. For every integers m > 0 and n > 0, we have the following isotypic decompositions

⊗ K n K 0 K -1 K m { K 2n ⊕ K 0 ⊕ K -1 , n = m; K n+m ⊕ K |n-m| , n ̸ = m; K m K m K 0 K n K 0 K -1 K -1 K n K -1 K 0 Table 6
Note that the tensor product between two identical spaces can be decomposed into a symmetric (S 2 ) and an antisymmetric product (Λ 2 ):

∀n ≥ 1, K n ⊗ K n = S 2 (K n ) ⊕ Λ 2 (K n ) .
Therefore Table 6 can be completed by the following ones: Lemma 3.5. For every integer n ≥ 1, we have the following isotypic decompositions, in which meaningless products are indicated by ×

S 2 K n K 0 K -1 K n K 2n ⊕ K 0 × × K 0 × K 0 × K -1 × × K 0 Λ 2 K n K 0 K -1 K n K -1 × × K 0 × 0 × K -1 × × 0
Prepared using sagej.cls Example 3.6. Consider the space of symmetric second-order tensors S 2 (R 2 ) = S 2 (K 1 ). From Lemma 3.5, we get:

S 2 (K 1 ) ≃ K 2 ⊕ K 0 .
The classical decomposition between a deviatoric part and a spherical one is retrieved. For a more complex example, we consider Ela = T (ij) (kl) . This space can be viewed as S 2 (S 2 (R 2 )) [START_REF] Ansemil | The symmetric tensor product of a direct sum of locally convex spaces[END_REF]:

Ela ≃ S 2 (S 2 (R 2 )) ≃ S 2 (K 2 ⊕ K 0 ) ≃ S 2 (K 2 ) ⊕ S 2 (K 0 ) ⊕ (K 2 ⊗ K 0 ) ≃ K 4 ⊕ K 0 ⊕ K 0 ⊕ K 2 ≃ K 4 ⊕ K 2 ⊕ 2K 0 .
Hence we recover the structure of the well-known harmonic decomposition of plane elasticity [START_REF] Blinowski | Two-dimensional Hooke's tensors -isotropic decomposition, effective symmetry criteria[END_REF][START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF][START_REF] De Saxcé | Invariant measures of the lack of symmetry with respect to the symmetry groups of 2"d" elasticity tensors[END_REF][START_REF] Forte | A unified approach to invariants of plane elasticity tensors[END_REF] without any computation.

Other harmonic structures can be found in the following table :   Symmetries Tensor space Isotypic dec. Physical meaning

T i R 2 K 1 Displacement T (ij) S 2 (R 2 ) K 2 ⊕ K 0 Conduction T (ij)k S 2 (R 2 ) ⊗ R 2 K 3 ⊕ 2K 1 Piezoelectricity [36] T (ij) (lm) S 2 (S 2 (R 2 )) K 4 ⊕ K 2 ⊕ 2K 0
Elasticity [START_REF] Blinowski | Two-dimensional Hooke's tensors -isotropic decomposition, effective symmetry criteria[END_REF][START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF][START_REF] De Saxcé | Invariant measures of the lack of symmetry with respect to the symmetry groups of 2"d" elasticity tensors[END_REF][START_REF] Forte | A unified approach to invariants of plane elasticity tensors[END_REF]]

T (ij)(kl)m S 2 (S 2 (R 2 )) ⊗ R 2 K 5 ⊕ 3K 3 ⊕ 5K 1 Strain-gradient coupling [4] T (ij) (kl)m Λ 2 (S 2 (R 2 )) ⊗ R 2 K 3 ⊕ 2K 1 Gyrotropicity [4] T (ij)k (lm)n S 2 (S 2 (R 2 ) ⊗ R 2 ) K 6 ⊕ 2K 4 ⊕ 5K 2 ⊕ 4K 0 ⊕ K -1
Strain-gradient elasticity [START_REF] Auffray | Derivation of anisotropic matrix for bi-dimensional strain-gradient elasticity behavior[END_REF] Table 7 As can be observed in previous examples, the decomposition of even-order tensor spaces contains only even-order harmonic spaces, and the same occurs for odd-order tensors. This fact is indeed a general fact and can be proven using the Clebch-Gordan decomposition (c.f. Lemma 3.9).

Isotypic decomposition of usual tensor spaces

Following observations made in subsection 2.3 to demonstrate our main theorem, the harmonic decomposition of S n , S 2 (S n ) and G n have to be determined, which will be achieved in this section. Lemma 3.7. Let S n = S n (R 2 ) be the space of n-th order completely symmetric tensors. Then:

S 2q ≃ q ⊕ k=0 K 2k , S 2q+1 ≃ q ⊕ k=0 K 2k+1
Proof. It is a classical result (see [START_REF] Sternberg | Group theory and physics[END_REF] for instance) that any homogeneous polynomial F of degree n can be decomposed uniquely as

F = P 0 + qP 1 + • • • + q r P r ,
where q(x, y) = x 2 + y 2 , P k is an harmonic polynomial of degree n -2k and r = [n/2]. The conclusion follows using the isomorphism between the space of totally symmetric tensors S n and the space H n of homogeneous polynomials of degree n, described in remark 3.2.

Prepared using sagej.cls Lemma 3.8. The isotypic decompositions of S 2 ( S 2p ) (p ≥ 1) and S 2 ( S 2q+1 ) (q ≥ 0) are given by:

S 2 ( S 2p ) ≃ (p + 1)K 0 ⊕ ( p ⊕ k=1 (p -k + 1) ( K 4k ⊕ K 4k-2 ) ) , S 2 ( S 2q+1 ) ≃ q ⊕ k=0 (q -k + 1) ( K 4k+2 ⊕ K 4k ) .
Proof. It suffices to note that

S 2 ( S 2p ) = p ⊕ k=0 S 4k
and to apply Lemma 3.7.

Lemma 3.9. For any integer n ≥ 1, the isotypic decomposition of G n = ⊗ n R 2 is given by:

G n ≃            ( 2p -1 p ) ( K -1 ⊕ K 0 ) ⊕ ( p ⊕ k=1 ( 2p p + k ) K 2k ) , if n = 2p, p > 0, q ⊕ k=0 ( 2q + 1 q + 1 + k ) K 2k+1 , if n = 2q + 1, q ≥ 0, where ( k n ) is the binomial coefficient.
Proof. It is an application of the character formula given in subsection A.3, using the development of the character χ G n for the O(2)-representation

G n = ⊗ n R 2 χ G n (r θ ) = (2 cos θ) n = ( e iθ + e -iθ ) n , χ G n (σ x r θ ) = 0.
We end this section with some results about dimensions of fixed point spaces. As it will be illustrated below, this knowledge is not sufficient to determine the symmetry classes. Nevertheless, once the symmetry classes is determined, these formula are needed to establish Table 3, Table 4 and Table 5.

Fixed-point spaces

We introduce here formulas which allow to easily determine the number of parameters of a given H-invariant tensor. Recall that given a tensorial representation T n of n-th order tensors and a subgroup H ⊂ O(2), the fixed-point space (T n )

H associated to H is defined as

(T n ) H := {T ∈ T n , Q • T = T, ∀Q ∈ H} .
The dimension of (T n ) H can easily be computed.

Lemma 3.10. Let T n ≃ ⊕ k α k K k
where α k is the multiplicity of K k in the decomposition. Then

dim (T n ) Zp = α -1 + α 0 + 2 ∑ k≥1 p|k α k ; dim (T n ) Dp = α 0 + ∑ k≥1 p|k α k
where the notation p | k means p divides k and with the obvious limit cases:

dim (T n ) SO(2) = α -1 + α 0 ; dim (T n ) O(2) = α 0 .
Remark 3.11. It is important to note that the knowledge of the different fixed-point spaces dimensions does not allow to conclude on the number and type of symmetry classes. For example, by using these formulas we establish that:

• The dimension of (Str) D6 and (Str) SO(2) are identical, but the symmetry classes [D 6 ] and [SO [START_REF] Armstrong | Groups and symmetry[END_REF]] are distinct; • The dimension of (Ela)

Z4 and (Ela) D4 are different, but both elements belongs to the same class [D 4 ].

In the next section, a rigorous method to determine the symmetry classes of a tensor representation will be presented.

Symmetry classes and clips operation

The purpose of this section is to present some general results about symmetry classes of constitutive tensor spaces under O( 2

Symmetry classes of irreducible O(2)-representation

Before computing isotropy classes of a reducible O(2)-representation, we list the isotropy classes of all the irreducible O(2)-representations. These classes are given by the following theorem.

I(K n ) =      {[SO(2)] , [O(2)]} ; if n = -1 {[O(2)]} ; if n = 0 {[D n ] , [O(2)]} ; if n ≥ 1 with the convention D 1 = Z σx 2 .
Proof. For n ≥ 1, there are two possibilities.

If the vector v ∈ R 2 is null its symmetry class is [O(2)].
In the other case, there is an element

Q ∈ O(2) such that Q • v = (r, 0) (with r > 0), hence v is Z σx 2 -invariant

Clips operator

Every O(2)-representation V can be decomposed into a direct sum of irreducible O(2)representations V i (i = 1, . . . , n). To obtain the symmetry class of V, we need therefore to compute the symmetry classes of a direct sum V 1 ⊕ V 2 , knowing independently the symmetry classes of each space V i . This can be achieved using the clips operator (which was first introduced in [29, 30] for SO(3) and O(3) representations).

Lemma 4.2. Let V be a representation of a group G that split into a direct sum of two G-invariant

subspaces V = V 1 ⊕ V 2 where g • V 1 ⊂ V 1 and g • V 2 ⊂ V 2 , ∀g ∈ G.
If I denotes the set of all isotropy classes associated to V, I i the set of all isotropy classes associated to

V i (i = 1, 2),
[1] [H] = {[1]} and [G] [H] = {[H]}.
We also have the following result which is a corollary of Lemma 4.2.

Corollary 4.6. Given two

G-representations V 1 and V 2 , if I k denotes the set of isotropy classes of V k for k = 1, 2, then I 1 I 2 := ∪ [H1]∈I1,[H2]∈I2 [H 1 ] [H 2 ].
is the set of all the isotropy classes of V 1 ⊕ V 2 .

Symmetry classes of a reducible O(2)-representation

The symmetry classes of a reducible O(2)-representation can now be computed using iterated clips operations and the following table :   Theorem 4.7. Clips operations between O(2)-closed subgroups are given in the following table: where Z 1 := 1, D 1 := Z σx 2 and d(n, m) := gcd(n, m). Due to the commutativity of the clips operator the table is symmetric and, accordingly, only its lower half part has been filled.

[1] [Z σx 2 ] [Z n ] [D n ] [SO(2)] [O(2)] [1] [1] [Z σx 2 ] [1] [1] , [Z σx 2 ] [Z m ] [1] [1] [ Z d(n,m) ] [D m ] [1] [1] , [Z σx 2 ] [ Z d(n,m) ] [ Z d(n,m) ] , [ D d(n,m) ] [SO(2)] [1] [1] [Z n ] [Z n ] [SO(2)] [O(2)] [1] [Z σx 2 ] [Z n ] [D n ] [SO(2)] [O(2)]

Proof. First we get

[Z n ] [Z m ] := { [Z n ∩ QZ m Q -1 ], Q ∈ O(2) } = {[Z n ∩ Z m ]} = {[ Z d(n,m) ]} .
Now, D n is generated by Z n and the mirror symmetry σ x along the axis (Ox). Therefore

D n ∩ QZ m Q -1 = D n ∩ Z m = Z n ∩ Z m Prepared using sagej.cls and thus [D n ] [Z m ] = {[ Z d(n,m) ]} .
To compute the clips [D n ] [D m ], note that given Q ∈ O(2), the subgroup QD m Q -1 is generated by Z m and the mirror symmetry along the Q(Ox) axis. We get therefore

[D n ] [D m ] = {[ Z d(n,m) ] , [ D d(n,m) ]} .
The other clips operations are obvious.

Using iterated clips operations, we get the following lemma.

Lemma 4.8. For n ≥ 1:

∀k ≥ 1, k I (K n ) := I (K n ) ( k-1 I (K n )) = {[Z n ] , [D n ] , [O(2)]} with 0 (K n ) := K n .
We have moreover:

Proposition 4.9. For p ≥ 2:

I ( K 2p ⊕ K 2p-2 ) = {[Z 2 ] , [D 2 ] , [D 2p-2 ] , [D 2p ] , [O(2)]}
Proof. From corollary 4.6 we have

I ( K 2p ⊕ K 2p-2 ) = I ( K 2p ) I ( K 2p-2 ) . Now from theorem 4.1, I (K n ) = {[D n ] , [O (2)
]} (n ≥ 0) and thus from table 8 we get:

[D 2p ] [O(2)] [D 2p-2 ] [ Z d(2p,2p-2) ] , [ D d(2p,2p-2) ] [D 2p-2 ] [O(2)] [D 2p ] [O(2)]
But d(2p, 2p -2) = 2 for p ≥ 1, which achieved the proof.

Proof of the main result

Recall first that given a constitutive tensor space T n , we necessarily have α n = 1 and

α k = 0 for k > n, if T n = ⊕ k α k K k , because S n ⊂ T n ⊂ G n .
We will now determine sufficient conditions on the isotypic decomposition of T n to have I (T n ) = I n max . We will split the study according to n is even or odd. Proposition 4.10. Let T 2p (p > 0) be an even order tensor space which contains the irreducible representation K -1 in its harmonic decomposition. Then

I ( T 2p ) = I 2p max .
Proof. We can write

T 2p ≃ α -1 K -1 ⊕ p ⊕ k=0 α 2k K 2k with α -1 ̸ = 0. Since S 2p ⊂ T 2p we have I ( S 2p ) ⊂ I ( T 2p
) . As a consequence of Lemma 2.5, we get

{[D 2k ]} 1≤k≤p ∪ {[O(2)]} ⊂ I ( T 2p
) .

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But since α -1 ̸ = 0, we have

{[SO(2)] , [O(2)]} ({[D 2k ]} 1≤k≤p ∪ {[O(2)]}) ⊂ I ( T 2p ) Now {[SO(2)] , [O(2)]} ({[D 2k ]} 1≤k≤p ∪ {[O(2)]}) = {[Z 2k ] , [D 2k ]} 1≤k≤p ∪ {[SO(2)] , [O(2)]} = I 2p max . Thus I ( S 2p ) = I 2p max ⊂ I ( T 2p
) ⊂ I 2p max , which achieves the proof.

In fact there is no other way to obtain a complete set of classes. Nevertheless, in the odd order case we have: Proposition 4.11. Let T 2q+1 be an odd-order tensor space such that:

T 2q+1 ≃ q ⊕ k=0 α 2k+1 K 2k+1 where α 2k+1 ≥ 2, for k < q. Then I ( T 2q+1 ) = I 2q+1 max \{[Z 2q+1 ]}. Proof. Since S 2q+1 ⊂ T 2q+1 , we have I ( S 2q+1 ) ⊂ I ( T 2q+1
) and thus by Lemma 2.5, we get

{[D 2k+1 ]} 0≤k≤q ∪ {[O(2)]} ⊂ I ( T 2q+1
) .

For k < q we have α 2k+1 ≥ 2 and α 2q+1 = 1. Now

I ( α 2k+1 K 2k+1 ) = α 2k+1 ( K 2k+1 ) = {[Z 2k+1 ] , [D 2k+1 ] , [O(2)]} .
and for the last irreducible representation

I ( K 2q+1 ) = {[D 2q+1 ] , [O(2)]} .
which show all isotropy classes in I 2q+1 max appear but Z 2q+1 , hence

I ( T 2q+1 ) = I 2q+1 max \{[Z 2q+1 ]}.
which concludes the proof.

These first two results and the harmonic decompositions of G 2p and G 2p+1 given by Lemma 3.9 directly lead to the following lemma. Lemma 4.12. We have:

{ I ( G 2p ) = I 2p max I ( G 2q+1 ) = I 2q+1 max \{[Z 2q+1 ]} .
Let us now observe that given two integers m 1 , m 2 > 0, we get the following:

• If m 1 and m 2 have the same parity, the harmonic decomposition of S m1 ⊗ S m2 involves a tensor product of two spaces of same order. As a consequence of the Clebsch-Gordan formula (Lemma 3.4), there is a term K -1 in the decomposition. Thus, we can apply Proposition 4.10 on S m1 ⊗ S m2 to achieve the proof; • If m 1 and m 2 have different parity (i.e. m 1 = 2p and m 2 = 2q + 1) a direct induction on p + q leads to the harmonic decomposition

S 2p ⊗ S 2q+1 = p+q ⊕ k=0 α 2k+1 K 2k+1 with α 2k+1 ≥ 2, for k < p + q.
Thus we can apply Proposition 4.11 on S m1 ⊗ S m2 .

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Finally we have: Lemma 4.13. Let m 1 , m 2 > 0. We have:

I(S m1 ⊗ S m2 ) = { I m1+m2 max , if m 1 , m 2 have the same parity, I m1+m2 max \{[Z m1+m2 ]}, if m 1 , m 2 have different parity.
.

In the case of the symmetric tensor space S 2 (S n ):

Lemma 4.14. For any integer n ≥ 2, we have:

I ( S 2 (S n ) ) = I 2n max \ {[Z 2n ] , [Z 2n-2 ]} . Proof.
Using the harmonic decomposition of S 2 (S n ) given in Lemma 3.7, all irreducible representations K 2k (k < n -1) come with multiplicity α 2k ≥ 2 and thus by Lemma 4.8 we have

I ( α 2k K 2k ) = α 2k ( K 2k ) = {[Z 2k ] , [D 2k ] , [O(2)]} . Now K 2n ⊕ K 2n-2
is also present in the isotypic decomposition of S 2 (S n ), and by Proposition 4.9

I ( K 2n ⊕ K 2n-2 ) = {[Z 2 ] , [D 2 ] , [D 2n-2 ] , [D 2n ] , [O(2)]} .
Applying the clips operator to these two sets, we get finally

I ( S 2 (S n ) ) = I 2n max \ {[Z 2n ] , [Z 2n-2 ]
} . Proof of Theorem 2.4. When T n = G n or T n = S 2 (S n ), the theorem is a direct consequence of Lemma 4.12 and Lemma 4.14. We still need to consider the case S 2 (T n ) where T n S n and the case T m1 ⊗ T m2 (coupling constitutive laws).

If T n S n , the harmonic decomposition of T n contains necessarily a factor K k ⊕ K k and thus S 2 (T n ) contains a factor

S 2 (K k ⊕ K k ) ≃ S 2 (K k ) ⊕ S 2 (K k ) ⊕ (K k ⊗ K k ).
Thus, the factor K -1 is present in the harmonic decomposition of S 2 (T n ) and due to proposition 4.10, we have then I ( S 2 (T n ) ) = I 2n max . In the case T m1 ⊗ T m2 , we always have S m1 ⊗ S m2 ⊂ T m1 ⊗ T m2 .

• if m 1 and m 2 have the same parity, the conclusion follows from Lemma 4.13;

• if m 1 and m 2 don't have the same parity we have

I m1+m2 max \{[Z m1+m2 ]} ⊂ I (T m1 ⊗ T m2 ) ,
by Lemma 4.13. Besides, the factor K m1+m2 occurs in the harmonic decomposition of T m1 ⊗ T m2 with multiplicity one and since

I ( K m1+m2 ) = {[D m1+m2 ], [O (

Conclusion

In the present paper, the general problem of finding the symmetry classes of 2D constitutive tensor spaces has been solved. To tackle this question, numerous tools from group representation theory have been used. We believe that these methods could find applications to other problems. In a forthcoming paper, the harmonic decomposition will be used to generate bases of polynomial invariants, i.e. integrity bases, for general constitutive tensor spaces in 2D. The knowledge of an integrity basis has many practical interests ranging from the labelling of materials to the formulation of non-linear constitutive laws.

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A.3 Character's formula

The characters χ ρi of the irreducible representations (V i , ρ i ) constitute an orthonormal system for the inner product

⟨f 1 , f 2 ⟩ := ∫ G f 1 (g)f 2 (g) dµ(g). (A.3)
where µ is the Haar mesure on G. Thus, using the rule (A.1), the coefficient α i of the irreducible representation (V i , ρ i ) in (A.2) is given by:

α i = ⟨χ ρ , χ ρi ⟩.
The characters of the irreducible representations (K n , ρ n ) (n = -1, 0, . . .) of O(2) has the following explicit expressions χ ρn (r θ ) = 2 cos(nθ), χ ρn (σ x r θ ) = 0, (n ≥ 1) (A.4) and χ ρ0 ≡ 1, χ ρ-1 (r θ ) = 1, χ ρ-1 (σ x r θ ) = -1.

(A.5)

For O(2), the inner product (A.3) is explicitly given by ⟨f 1 , f 2 ⟩ := 1 4π

∫ 2π 0 f 1 (r θ )f 2 (r θ ) dθ + 1 4π ∫ 2π 0 f 1 (σ x r θ )f 2 (σ x r θ ) dθ.

Notes

1. It has to be noted that Z σx 2 and Z2 are isomorphic as groups but are not conjugate. 2. The name for the different bi-dimensional symmetry systems are borrowed from [START_REF] Zheng | The description, classification, and reality of material and physical symmetries[END_REF]. 3. There is some ambiguity in the literature on what is the proper generalisation of the classical notion of deviator to higher-order tensors. For some authors [START_REF] Toupin | Elastic materials with couple-stresses[END_REF][START_REF] Mindlin | Micro-structure in linear elasticity[END_REF] it means tensors that contain no scalar part, for others [START_REF] Jerphagnon | The description of the physical properties of condensed matter using irreducible tensors[END_REF] it means traceless symmetric higher-order tensors. These definitions coincide for secondorder tensors, but differ for higher-order tensors. We prefer the second definition since then deviators correspond to irreducible representations of the orthogonal group.

Theorem 2 . 4 .

 24 Symmetry classes. For every integers p > 0 and n > 0 Type of tensor space Symmetry classes S n c.f. Lemma 2.5

  )-group action . In subsection 4.1, we describe the symmetry classes irreducible O(2)-representations. In subsection 4.2, we introduce the clips operator, a tool to compute the symmetry classes of the direct sum of two representations. Finally, in subsection 4.3, we present an algorithm to compute the symmetry classes of every reducible O(2)-representation. The proof of Theorem 2.4 and subsequent results for tensor spaces are provided in subsection 4.4.

Theorem 4 . 1 .

 41 The symmetry classes of the O(2)-irreducible space K n are:

  . Furthermore, since v belongs to K n , a rotation of angle θ turns v by an angle nθ. Hence, v is invariant by the rotation of angle 2π n , i.e. v is Z n -invariant. Thus, v belongs to the symmetry class [D n ]. The other cases are obvious.

  2)]} by Theorem 4.1, we getI (T m1 ⊗ T m2 ) ⊂ I m1+m2 max \{[Z m1+m2 ]}which achieves the proof.

Table 1

 1 Furthermore we have

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  then [H] ∈ I if and only if there exist [H 1 ] ∈ I 1 and [H 2 ] ∈ I 2 such that H = H 1 ∩ H 2 . Lemma 4.2 shows that the isotropy classes of a direct sum are related to intersections of isotropy subgroups. But as intersection of classes is meaningless, the result cannot be directly extended. The clips operator solves this problem. Clips Operator. For every conjugacy classes [H 1 ] and [H 2 ], we define the clips operator of [H 1 ] and [H 2 ], noted [H 1 ] [H 2 ], to be [H 1 ] [H 2 ] := {[H 1 ∩ gH 2 g -1 ] for all g ∈ G} which is a set of conjugacy classes. The clips operation is commutative. For every conjugacy class [H] of a group G, we have

	Definition 4.3. One can observe some immediate properties:
	Proposition 4.4. Proposition 4.5.
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Table 8 .

 8 Clips operation on O(2) closed subgroups

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Acknowledgements

N. Auffray and B. Kolev would like to gratefully acknowledge the Région Pays de la Loire and the pole EMC2 for their financial support.

Declaration of conflicting interests

None declared.

A Elements of representation theory

A.1 Group representations

Let G denote a compact group and V a finite dimensional R-vector space. A linear representation (V, ρ) of G is a group morphism from G to the group GL(V) of linear invertible endomorphisms of V:

Thus, for each element g ∈ G, we have an invertible linear mapping ρ(g) of V and for each v ∈ V we write

Two representations (ρ 1 , V 1 ) and (ρ 2 , V 2 ) of G are said to be equivalent if there exists an isomorphism φ such that the following diagram is commutative

The character of a representation (V, ρ) of G is the real value function χ ρ (g) := tr(ρ(g)).

Note that two equivalent representations (ρ 1 , V 1 ) and (ρ 2 , V 2 ) of G have the same character. Furthermore, for two representations (V, ρ) and (V ′ , ρ ′ ) of G, the characters of the induced representations (V ⊕ V ′ , ρ ⊕ ρ ′ ) and (V ⊗ V ′ , ρ ⊗ ρ ′ ) are given by:

A.2 Decomposition into irreducible representations

A subspace W of a linear representation (V, ρ) of a group G is stable if

The linear representation (V, ρ) is irreducible if {0} and V are the only stable subspaces of V. Due to Peter-Weyl's theorem [START_REF] Sternberg | Group theory and physics[END_REF], we know that every finite dimensional representation V of a compact group G can be split into a direct sum of finite dimensional irreducible representations

where, for each k i , (V ki , ρ ki ) is an irreducible representation of G, and ≃ indicates a G-equivariant isomorphism. We consider also the isotypic decomposition, in which equivalent factors are group together

When applied to the special case of a finite dimensional representation of the orthogonal group O(2) the isotypic decomposition is often referred to as the harmonic decomposition [START_REF] Backus | A geometrical picture of anisotropic elastic tensors[END_REF][START_REF] Jerphagnon | The description of the physical properties of condensed matter using irreducible tensors[END_REF][START_REF] Forte | Symmetry classes for elasticity tensors[END_REF][START_REF] Forte | A unified approach to invariants of plane elasticity tensors[END_REF].

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