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H 2 Invariants of ternary forms under the orthogonal group

Auteurs : Hubert, Evelyne (Auteur de la Conférence)
CIRM (Editeur )

 Loading the player... diffusion tensor quartic model action of orthogonal group invariants slice lemma $B_3$ equivariant basis minimal generating set of invariants evaluation of invariants rewriting of invariants questions from the audience

Résumé : Classical invariant theory has essentially addressed the action of the general linear group on homogeneous polynomials. Yet the orthogonal group arises in applications as the relevant group of transformations, especially in 3 dimensional space. Having a complete set of invariants for its action on ternary quartics, i.e. degree 4 homogeneous polynomials in 3 variables, is, for instance, relevant in determining biomarkers for white matter from diffusion MRI.
We characterize a generating set of rational invariants of the orthogonal group acting on even degree forms by their restriction on a slice. These restrictions are invariant under the octahedral group and their explicit formulae are given compactly in terms of equivariant maps. The invariants of the orthogonal group can then be obtained in an explicit way, but their numerical evaluation can be achieved more robustly using their restrictions. The exhibited set of generators futhermore allows us to solve the inverse problem and the rewriting.
Central in obtaining the invariants for higher degree forms is the preliminary construction, with explicit formulae, for a basis of harmonic polynomials with octahedral symmetry, dif- ferent, though related, to cubic harmonics.
This is joint work with Paul Görlach (now at MPI Leipzig), in a joint project with Téo Papadopoulo (Inria Méditerranée).

Codes MSC :
05E05 - Symmetric functions and generalizations
13A50 - Actions of groups on commutative rings; invariant theory
13P10 - Gröbner bases; other bases for ideals and modules
68W30 - Symbolic computation and algebraic computation
92C55 - Biomedical imaging and signal processing, tomography

 Informations sur la Vidéo Langue : Anglais Date de publication : 09/04/2018 Date de captation : 05/04/2018 Collection : Research talks ; Computer Science Format : MP4 (.mp4) - HD Durée : 00:41:27 Domaine : Numerical Analysis & Scientific Computing ; Computer Science Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2018-04-05_Hubert.mp4 Informations sur la rencontre Nom de la rencontre : Symmetry and computation / Symétries dans les méthodes de calculOrganisateurs de la rencontre : Hubert, Evelyne ; Mansfield, Elizabeth ; Szanto, AgnesDates : 03/04/2018 - 07/04/2018 Année de la rencontre : 2018 URL Congrès : https://conferences.cirm-math.fr/1772.html Citation Data DOI : 10.24350/CIRM.V.19387203 Cite this video as: Hubert, Evelyne (2018). Invariants of ternary forms under the orthogonal group. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19387203 URI : http://dx.doi.org/10.24350/CIRM.V.19387203

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