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We present a polynomial description of this set for the Weyl groups associated to root systems of types [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. Our description is given through a polynomial matrix inequality. The novelty lies in an approach via Hermite quadratic forms and a closed form formula for the matrix entries. The orbit space of the multiplicative Weyl group action is the orthogonality region of generalized Chebyshev polynomials. In this polynomial basis, we show that the matrices obtained for the five types follow the same, surprisingly simple pattern. This is applied to the optimization of trigonometric polynomials with crystallographic symmetries.<\/jats:p>","DOI":"10.1137\/23m158173x","type":"journal-article","created":{"date-parts":[[2024,8,19]],"date-time":"2024-08-19T04:21:47Z","timestamp":1724041307000},"page":"612-649","source":"Crossref","is-referenced-by-count":2,"title":["Orbit Spaces of Weyl Groups Acting on Compact Tori: A Unified and Explicit Polynomial Description"],"prefix":"10.1137","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1456-9524","authenticated-orcid":true,"given":"Evelyne","family":"Hubert","sequence":"first","affiliation":[{"name":"Centre Inria d\u2019Universit\u00e9 C\u00f4te d\u2019Azur, 06902 Sophia Antipolis, France."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0688-7074","authenticated-orcid":true,"given":"Tobias","family":"Metzlaff","sequence":"additional","affiliation":[{"name":"Centre Inria d\u2019Universit\u00e9 C\u00f4te d\u2019Azur, 06902 Sophia Antipolis, France."},{"name":"Department of Mathematics, RPTU Kaiserslautern\u2013Landau, 67663 Kaiserslautern, Germany. Former address: Centre Inria d\u2019Universit\u00e9 C\u00f4te d\u2019Azur, 06902 Sophia Antipolis, France."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1192-3500","authenticated-orcid":true,"given":"Cordian","family":"Riener","sequence":"additional","affiliation":[{"name":"Department of Mathematics, UiT The Arctic University, 9037 Troms\u00f8, Norway."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,8,19]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-014-1070-7"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1991-1019520-3"},{"key":"ref3","author":"Bourbaki N.","year":"1968","journal-title":"Actualit\u00e9s Scientifiques et Industrielles 1337"},{"key":"ref4","series-title":"MOS\/SIAM Ser. 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