Outer Automorphisms of S6
TL;DR: In this paper, the outer automorphisms of S6 were studied and discussed in the context of the American Mathematical Monthly: Vol. 89, No. 6, pp. 407-410.
Abstract: (1982). Outer Automorphisms of S6. The American Mathematical Monthly: Vol. 89, No. 6, pp. 407-410.
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TL;DR: In this paper, it was shown that Nichols algebras over alternating groups (m ≥ 5) are infinite-dimensional, except for the transpositions considered in Fomin and Kirillov.
Abstract: It is shown that Nichols algebras over alternating groups \({\mathbb A_m}\) (m ≥ 5) are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to \({\mathbb A_m}\) is isomorphic to the group algebra. In a similar fashion, it is shown that the Nichols algebras over the symmetric groups \({\mathbb S_m}\) are all infinite-dimensional, except maybe those related to the transpositions considered in Fomin and Kirillov (Progr Math 172:146–182, 1999), and the class of type (2, 3) in \({\mathbb S_5}\). We also show that any simple rack X arising from a symmetric group, with the exception of a small list, collapse, in the sense that the Nichols algebra \({\mathfrak B(X, \bf q)}\) is infinite dimensional, q an arbitrary cocycle.
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Cites background from "Outer Automorphisms of S6"
... −→χ 0. (iii) (23) in S 6, ρ 2 = −→χ 1 ⊗ǫor −→χ 1 ⊗sgn. Actually, the rack O 2in S 6 is isomorphic to O 3, since any map in the class of the outer automorphism of S 6 applies (1 2) in (1 2)(3 4)(5 6) [JR]. Thus, case (iii) is contained in (i). The remaining cases can not be treated by consideration of Nichols algebras of subracks, see Remark 4.2. 1.4. The main results in this paper are negative, in th...
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01 Jan 1995
TL;DR: In this paper, a dictionary of translation into Dirac groups is provided, with a discussion of the application of finite geometry results to Dirac algebras and some interesting finite geometry symmetry methods, the latter being given a rather full treatment in appendices.
Abstract: Associated with the real Clifford algebra Cl(p, q), p + q = n, is the finite Dirac group G(p, q) of order 2n+1. The quotient group V n = G(p, q)/ *#x007B;± 1*#x007D;, viewed additively, is an ndimensional vector space over GF(2) = *#x007B;0, 1*#x007D; which comes equipped with a quadratic form Q and associated alternating bilinear form B. Properties of the finite geometry over GF(2) of V n B, Q — in part familiar, in part less so — are given a rather full description, and a dictionary of translation into their Dirac group counterparts is provided. The knowledge gained is used, in conjunction with facts concerning representations of G(p, q), to give a pleasantly clean derivation of the well-known table of Porteous (1969/1981) of the algebras Cl(p, q). In particular the finite geometry highlights the “antisymmetry” of the table about the column p - q = -1. Several low-dimensional illustrations are given of the application of finite geometry results to Dirac groups. Particular emphasis is laid on certain interesting finite geometry symmetry methods, the latter being given a rather full treatment in the appendices. Finite geometry is also used to study the automorphisms of the Dirac groups, and the splitting of certain exact sequences.
31 citations
TL;DR: In this article, the problem of determining the symmetry of the automorphism group of octahedral complexes has been investigated and the group is confirmed to be S6 and representative symmetry permutations (one for each of the 11 classes of S6) are shown.
Abstract: Isomerization of octahedral complexes XY6, by a mechanism known as diagonal twist (in which two ligands in cis positions, one to another, exchange sites) is considered. Construction of the corresponding isomerization graph is outlined (reported before by Balaban) and the problem of determining its symmetry is considered. Alternative routes for deducing the order of the automorphism group are described. The group is confirmed to be S6 and representative symmetry permutations (one for each of the 11 classes of S6) are shown. Alternative pictorial representations of the 15-vertex graph are also shown.
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TL;DR: In this paper, it was shown that very few pairs of conjugacy classes might give rise to finite-dimensional Nichols algebras, where O is the class of transpositions and ρ is the sign representation.
Abstract: It is an important open problem whether the dimension of the Nichols algebra B(O,\rho) is finite when O is the class of the transpositions and \rho is the sign representation, with m>= 6. In the present paper, we discard most of the other conjugacy classes showing that very few pairs (O,\rho) might give rise to finite-dimensional Nichols algebras.
17 citations
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TL;DR: In this article, it was shown that every quasisimple sporadic group possesses an unmixed strongly real Beauville structure aside from the Mathieu groups M11 and M23 (and possibly 2B and M).
Abstract: We first show that every quasisimple sporadic group possesses an unmixed strongly real Beauville structure aside from the Mathieu groups M11 and M23 (and possibly 2B and M). We go on to show that no almost simple sporadic group possesses a mixed Beauville structure. We then go on to use the exceptional nature of the alternating group A6 to give a strongly real Beauville structure for this group explicitly correcting an earlier error of Fuertes and Gonzalez-Diez. In doing so we complete the classification of alternating groups that possess strongly real Beauville structures. We conclude by discussing mixed Beauville structures of the groups A6:2 and A6:2^2.
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