{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T06:21:40Z","timestamp":1775802100177,"version":"3.50.1"},"reference-count":23,"publisher":"World Scientific Pub Co Pte Ltd","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2007,5]]},"abstract":"<jats:p>We prove that a non-spherical irreducible Coxeter group is (directly) indecomposable and that an indefinite irreducible Coxeter group is strongly indecomposable in the sense that all its finite index subgroups are (directly) indecomposable. Let W be a Coxeter group. Write W = W<jats:sub>X<jats:sub>1<\/jats:sub><\/jats:sub>\u00d7 \u22ef \u00d7 W<jats:sub>X<jats:sub>b<\/jats:sub><\/jats:sub>\u00d7 W<jats:sub>Z<jats:sub>3<\/jats:sub><\/jats:sub>, where W<jats:sub>X<jats:sub>1<\/jats:sub><\/jats:sub>, \u2026 , W<jats:sub>X<jats:sub>b<\/jats:sub><\/jats:sub>are non-spherical irreducible Coxeter groups and W<jats:sub>Z<jats:sub>3<\/jats:sub><\/jats:sub>is a finite one. By a classical result, known as the Krull\u2013Remak\u2013Schmidt theorem, the group W<jats:sub>Z<jats:sub>3<\/jats:sub><\/jats:sub>has a decomposition W<jats:sub>Z<jats:sub>3<\/jats:sub><\/jats:sub>= H<jats:sub>1<\/jats:sub>\u00d7 \u22ef \u00d7 H<jats:sub>q<\/jats:sub>as a direct product of indecomposable groups, which is unique up to a central automorphism and a permutation of the factors. Now, W = W<jats:sub>X<jats:sub>1<\/jats:sub><\/jats:sub>\u00d7 \u22ef \u00d7 W<jats:sub>X<jats:sub>b<\/jats:sub><\/jats:sub>\u00d7 H<jats:sub>1<\/jats:sub>\u00d7 \u22ef \u00d7 H<jats:sub>q<\/jats:sub>is a decomposition of W as a direct product of indecomposable subgroups. We prove that such a decomposition is unique up to a central automorphism and a permutation of the factors. Write W = W<jats:sub>X<jats:sub>1<\/jats:sub><\/jats:sub>\u00d7 \u22ef \u00d7 W<jats:sub>X<jats:sub>a<\/jats:sub><\/jats:sub>\u00d7 W<jats:sub>Z<jats:sub>2<\/jats:sub><\/jats:sub>\u00d7 W<jats:sub>Z<jats:sub>3<\/jats:sub><\/jats:sub>, where W<jats:sub>X<jats:sub>1<\/jats:sub><\/jats:sub>, \u2026 , W<jats:sub>X<jats:sub>a<\/jats:sub><\/jats:sub>are indefinite irreducible Coxeter groups, W<jats:sub>Z<jats:sub>2<\/jats:sub><\/jats:sub>is an affine Coxeter group whose irreducible components are all infinite, and W<jats:sub>Z<jats:sub>3<\/jats:sub><\/jats:sub>is a finite Coxeter group. The group W<jats:sub>Z<jats:sub>2<\/jats:sub><\/jats:sub>contains a finite index subgroup R isomorphic to \u2124<jats:sup>d<\/jats:sup>, where d = |Z<jats:sub>2<\/jats:sub>| - b + a and b - a is the number of irreducible components of W<jats:sub>Z<jats:sub>2<\/jats:sub><\/jats:sub>. Choose d copies R<jats:sub>1<\/jats:sub>, \u2026 , R<jats:sub>d<\/jats:sub>of \u2124 such that R = R<jats:sub>1<\/jats:sub>\u00d7 \u22ef \u00d7 R<jats:sub>d<\/jats:sub>. Then G = W<jats:sub>X<jats:sub>1<\/jats:sub><\/jats:sub>\u00d7 \u22ef \u00d7 W<jats:sub>X<jats:sub>a<\/jats:sub><\/jats:sub>\u00d7 R<jats:sub>1<\/jats:sub>\u00d7 \u22ef \u00d7 R<jats:sub>d<\/jats:sub>is a virtual decomposition of W as a direct product of strongly indecomposable subgroups. We prove that such a virtual decomposition is unique up to commensurability and a permutation of the factors.<\/jats:p>","DOI":"10.1142\/s0218196707003779","type":"journal-article","created":{"date-parts":[[2007,5,30]],"date-time":"2007-05-30T11:38:28Z","timestamp":1180525108000},"page":"427-447","source":"Crossref","is-referenced-by-count":18,"title":["IRREDUCIBLE COXETER GROUPS"],"prefix":"10.1142","volume":"17","author":[{"given":"LUIS","family":"PARIS","sequence":"first","affiliation":[{"name":"Institut de Math\u00e9matiques de Bourgogne, UMR 5584 du CNRS, Universit\u00e9 de Bourgogne, B. P. 47870, 21078 Dijon Cedex, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","first-page":"1011","volume":"19","author":"Borcherds R. E.","journal-title":"Int. Math. Res. 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Helv."},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9167-8"},{"key":"rf14","first-page":"91","volume":"5","author":"de la Harpe P.","journal-title":"Expos. Math."},{"key":"rf15","first-page":"62","volume":"21","author":"Howlett R. B.","journal-title":"J. London Math. Soc. (2)"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1007\/BF02677488"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-6398-2"},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511623646"},{"key":"rf20","doi-asserted-by":"publisher","DOI":"10.1007\/s102400200001"},{"key":"rf21","unstructured":"B.\u00a0M\u00fchlherr, The Coxeter Legacy (Amer. Math. 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