Irreducible Coxeter Groups
Irreducible Coxeter Groups
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不可约考克塞特群
DOI：
10.1142/s0218196707003779
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
L. Paris
中科院分区：
文献类型：
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作者：
L. Paris
证明了非球面不可约Coxeter群是（直接）不可分解的，而不定不可约Coxeter群是强不可分解的，因为它的所有有限指数子群是（直接）不可分解的.设W是Coxeter群。记为W = WX1 × WXb × WZ3，其中WX1，...，WXb是非球面不可约Coxeter群，WZ3是有限群。根据Krull-Remak-施密特定理，群WZ 3有一个分解WZ 3 = H1 × H1 × Hq作为不可分解群的直积，它在中心自同构和因子置换下是唯一的。现在，W = WX1 × WXb × H1 × WXb × Hq是W作为不可分解子群的直积的分解。我们证明了这样的分解是唯一的中心自同构和置换的因素。记W = WX1 × WX2 × WX3，其中WX1，...，WX3是无限不可约Coxeter群，WZ2是不可约分支都是无限的仿射Coxeter群，WZ3是有限Coxeter群。群WZ 2包含同构于d的有限指数子群R，其中d =| Z2|- B + a和B-a是WZ 2的不可约分支的数目。选择d个副本R1，...，Rd，使得R = R1 × Rd × Rd。则G = WX1 × R2 × WXa × R1 × R2 × Rd是W作为强不可分解子群直积的一个虚分解.我们证明了这样的虚拟分解是唯一的可并行性和置换的因素。
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 = WX1 × ⋯ × WXb × WZ3, where WX1, … , WXb are non-spherical irreducible Coxeter groups and WZ3 is a finite one. By a classical result, known as the Krull–Remak–Schmidt theorem, the group WZ3 has a decomposition WZ3 = H1 × ⋯ × Hq as a direct product of indecomposable groups, which is unique up to a central automorphism and a permutation of the factors. Now, W = WX1 × ⋯ × WXb × H1 × ⋯ × Hq 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 = WX1 × ⋯ × WXa × WZ2 × WZ3, where WX1, … , WXa are indefinite irreducible Coxeter groups, WZ2 is an affine Coxeter group whose irreducible components are all infinite, and WZ3 is a finite Coxeter group. The group WZ2 contains a finite index subgroup R isomorphic to ℤd, where d = |Z2| - b + a and b - a is the number of irreducible components of WZ2. Choose d copies R1, … , Rd of ℤ such that R = R1 × ⋯ × Rd. Then G = WX1 × ⋯ × WXa × R1 × ⋯ × Rd 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.
无限不可约Coxeter群的直接不可分解性及Coxeter群的同构问题
DOI：
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发表时间：
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期刊：
Communications in Algebra (印刷中)
影响因子：
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作者：
T.Nakano;M.Sugiyama;Y.Nakano;Y.Shimogaki;Koji Nuida
通讯作者：
Koji Nuida
DOI：
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发表时间：
2005
期刊：
影响因子：
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作者：
T. Kobayashi;T. Oshima
通讯作者：
T. Oshima