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Groups of Exceptional Type, Coxeter Groups and Related Geometries pp 217–238Cite as

On the Isomorphism Problem for Coxeter Groups and Related Topics

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 82))

Abstract

The isomorphism problem, which is a problem of deciding whether or not given two elements in a class of some kind of mathematical objects are isomorphic to each other, is a ubiquitous problem in mathematics. This paper gives a survey of recent developments for the isomorphism problem for (finitely generated or general) Coxeter groups and some related topics.

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Notes

  1. 1.

    It is not obvious but well-known that the Coxeter matrix is uniquely determined by the pair \((W,S)\); namely, \(m(s,t)\) is precisely the order of the element \(st\) in \(W\).

  2. 2.

    Usually, when \(m = 3\), the type of the group is classified as \(A_2\) rather than \(I_2(3)\). Similarly, when \(m = 4\), the type of the group is classified as \(B_2\) (also called \(C_2\)) rather than \(I_2(4)\).

  3. 3.

    The author does not know whether or not there exists a standard name for the type; the same also holds for type \(A_{\infty ,\infty }\) below.

  4. 4.

    For simplicity, we identify a matrix in \({\mathrm {GL}}(2,{\mathbb {Z}} )\) with its equivalence class in \({\mathrm {PGL}}(2,{\mathbb {Z}} )\).

  5. 5.

    To the author’s best knowledge, the term “rigid” for Coxeter groups was first introduced in the title of Radcliffe’s preprint [43], though the term does not appear in its text.

  6. 6.

    For example, \(W\) is strongly rigid if and only if \(W\) is rigid and \({\mathrm {Out}}(W)\) is trivial.

  7. 7.

    A “free product decomposition” version of Theorem 3.1 was also shown in a preprint version of the paper [30] by Mihalik, Ratcliffe and Tschantz.

  8. 8.

    We note that, in the present case, the paper [42] generalizes the result on direct indecomposability in the finite rank case to a wider class consisting of subgroups of finite index of those Coxeter groups, therefore the result of [42] does not follow from the results in [37].

  9. 9.

    We note that such a counterexample cannot be found in finite Coxeter groups, since finite Coxeter groups are known to be reflection rigid; see Theorem 3.10 of [5].

  10. 10.

    In the same paper, Radcliffe mentioned (without proof) that the condition for the finiteness of ranks can be removed.

  11. 11.

    This condition means that the dimension of the Davis complex \(\varSigma (W,S)\) for \((W,S)\) is at most two.

  12. 12.

    The latter condition is equivalent to that all elements of \({\mathrm {Ref}}_S(W)\) are conjugate in \(W\).

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Acknowledgments

This survey paper is based on a two-hour talk by the author at the Workshop and Conference on Groups and Geometries, Indian Statistical Institute Bangalore, India, December 10–21, 2012. The author would like to thank Professor N. S. Narasimha Sastry for inviting him to the conference and giving him an opportunity to give a talk. The author would also like to thank the participants to the conference, especially Professor Luis Paris, who gave precious comments to the talk.

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  1. Innovative Security Research Group, Research Institute for Secure Systems, National Institute of Advanced Industrial Science and Technology, Ibaraki, Japan

    Koji Nuida

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  1. Stat Math Unit, Indian Statistical Institute, Bangalore, Karnataka, India

    N.S. Narasimha Sastry

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Nuida, K. (2014). On the Isomorphism Problem for Coxeter Groups and Related Topics. In: Sastry, N. (eds) Groups of Exceptional Type, Coxeter Groups and Related Geometries. Springer Proceedings in Mathematics & Statistics, vol 82. Springer, New Delhi. https://doi.org/10.1007/978-81-322-1814-2_12

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