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Reflection subgroups of finite and affine Weyl groups. (English) Zbl 1243.20051
The classification of the reflection subgroups of reflection groups is an interesting mathematical theme. The present paper is to give complete case-free classifications of the reflection subgroups of finite and affine Weyl groups. P. V. Tumarkin and A. A. Felikson, [in their paper Sb. Math. 196, No. 9, 1349-1369 (2005); translation from Mat. Sb. 196, No. 9, 103-124 (2005; Zbl 1138.20310)], gave a case-by-case description of the reflection subgroups of finite Coxeter groups up to isomorphism, the results in the case of Weyl groups \(W\) are based on the tables by E. B. Dynkin, [Am. Math. Soc., Transl., II. Ser. 6, 111-243 (1957; Zbl 0077.03404)], giving closed subsystems and maximal closed subsystems of the root system up to \(W\)-action. The arguments for the classification of reflection subgroups of finite Weyl groups provided by the present paper are closely analogous to those of Dynkin.
A classification (up to isomorphism) of the reflection subgroups of an affine Weyl group in terms of those of the corresponding finite Weyl group was conjectured and partly proved by H. S. M. Coxeter [in Proc. Camb. Philos. Soc. 30, 466-482 (1934; Zbl 0010.15403)] and completed by M. Dyer [in J. Algebra 135, No. 1, 57-73 (1990; Zbl 0712.20026)]. The paper by Tumarkin and Felikson [loc. cit.] includes explicit lists of the possible isomorphism types of maximal reflection subgroups and describes a procedure by which all reflection subgroups of a given affine Weyl group can be obtained. In contrast, the present paper provides two different bijective parametrisations of the reflection subgroups of affine Weyl groups. Both parametrisations are in terms of explicitly described combinatorial objects attached to finite root systems, and the proofs do not use the classification of finite or affine Weyl groups.

20F55 Reflection and Coxeter groups (group-theoretic aspects)
51F15 Reflection groups, reflection geometries
20E07 Subgroup theorems; subgroup growth
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