Français
Invariants of ternary forms under the orthogonal group
Post-edited
Authors : Hubert, Evelyne (Author of the conference)
CIRM (Publisher )
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
CIRM (Publisher )
Abstract :
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).
MSC Codes : 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).
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
|
Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 09/04/2018 Conference Date : 05/04/2018 Subseries : Research talks arXiv category : Symbolic Computation Mathematical Area(s) : Numerical Analysis & Scientific Computing ; Computer Science Format : MP4 (.mp4) - HD Video Time : 00:41:27 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2018-04-05_Hubert.mp4 
|
Information on the Event
Event Title : Symmetry and computation / Symétries dans les méthodes de calculEvent Organizers : Hubert, Evelyne ; Mansfield, Elizabeth ; Szanto, Agnes Dates : 03/04/2018 - 07/04/2018 Event Year : 2018 Event URL : https://conferences.cirm-math.fr/1772.html
Citation Data
DOI : 10.24350/CIRM.V.19387203Cite 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 |
See Also
- [Multi angle] Hybrid finite element methods preserving local symmetries and conservation laws / Author of the conference Stern, Ari.
- [Multi angle] Symmetry and computation: a few of my favourite things / Author of the conference Munthe-Kaas, Hans Z..
Bibliography
- Görlach, P., Hubert, E., & Papadopoulo, T. (2017). Rational invariants of ternary forms under the orthogonal group.
