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McKay’s E 6 Observation on the Largest Fischer Group

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Abstract

In this paper, we study McKay’s E 6-observation on the largest Fischer 3-transposition group Fi24. We investigate a vertex operator algebra \({VF^{\natural}}\) of central charge \({23\frac{1}{5}}\) on which the Fischer group Fi24 naturally acts. We show that there is a natural correspondence between dihedral subgroups of Fi24 and certain vertex operator subalgebras constructed by the nodes of the affine E 6 diagram by investigating so-called derived Virasoro vectors of central charge 6/7. This allows us to reinterpret McKay’s E 6-observation via the theory of vertex operator algebras.

It is also shown that the product of two non-commuting Miyamoto involutions of σ-type associated to derived c = 6/7 Virasoro vectors is an element of order 3, under certain general hypotheses on the vertex operator algebra. For the case of \({VF^{\natural}}\) , we identify these involutions with the 3-transpositions of the Fischer group Fi24.

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Authors and Affiliations

  1. Department of Mathematics, Kansas State University, 138 Cardwell Hall, Manhattan, KS, 66506-2602, USA

    Gerald Höhn

  2. Institute of Mathematics, Academia Sinica, No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, Taiwan

    Ching Hung Lam

  3. Department of Mathematics, Tokyo Woman’s Christian University, 2-6-1 Zempukuji, Suginami-ku, Tokyo, 167-8585, Japan

    Hiroshi Yamauchi

Authors
  1. Gerald Höhn
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  2. Ching Hung Lam
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  3. Hiroshi Yamauchi
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Corresponding author

Correspondence to Hiroshi Yamauchi.

Additional information

Communicated by Y. Kawahigashi

Partially supported by Kansas NSF EPSCoR grant NSF32239KAN32240.

Partially supported by NSC grant 97-2115-M-006-015-MY3 and National Center for Theoretical Sciences, Taiwan.

Partially supported by JSPS Grant-in-Aid for Young Scientists (Start-up) No. 19840025 and (B) No. 21740011.

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Höhn, G., Lam, C.H. & Yamauchi, H. McKay’s E 6 Observation on the Largest Fischer Group. Commun. Math. Phys. 310, 329–365 (2012). https://doi.org/10.1007/s00220-011-1413-8

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  • DOI: https://doi.org/10.1007/s00220-011-1413-8

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