現代数学の系譜工学物理雑談古典ガロア理論も読む78

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377: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/02(土) 10:33:10.29 ID:ZLEqKHqI

>>376

つづき

・PSL(2, 5) < PGL(2, 5) → S6, of orders 60 and 120, which yields an embedding of S5 (respectively, A5) as a transitive subgroup of S6 (respectively, A6). This is an example of an exotic map S5 → S6,
and can be used to construct the exceptional outer automorphism of S6.[6] Note that the isomorphism PGL(2, 5) =~ S5 is not transparent from this presentation: there is no particularly natural set of 5 elements on which PGL(2, 5) acts.

Action on p points
While PSL(n, q) naturally acts on (qn?1)/(q?1) = 1+q+...+qn?1 points, non-trivial actions on fewer points are rarer. I
ndeed, for PSL(2, p) acts non-trivially on p points if and only if p = 2, 3, 5, 7, or 11; for 2 and 3 the group is not simple, while for 5, 7, and 11, the group is simple - further, it does not act non-trivially on fewer than p points.[note 5] This was first observed by Evariste Galois in his last letter to Chevalier, 1832.[7]

This can be analyzed as follows; note that for 2 and 3 the action is not faithful (it is a non-trivial quotient, and the PSL group is not simple), while for 5, 7, and 11 the action is faithful
(as the group is simple and the action is non-trivial), and yields an embedding into Sp. In all but the last case, PSL(2, 11), it corresponds to an exceptional isomorphism, where the right-most group has an obvious action on p points:
・ L2(2)=~ S3→ S2 via the sign map;
・ L2(3)=~ A_{4}→ A3=~ C3 via the quotient by the Klein 4-group;
・ L2(5)=~ A_{5}.To construct such an isomorphism, one needs to consider the group L2(5) as a Galois group of a Galois cover a5:
　X(5) → X(1) = P1, where X(N) is a modular curve of level N.
　This cover is ramified at 12 points. The modular curve X(5) has genus 0 and is isomorphic to a sphere over the field of complex numbers, and then the action of L2(5) on these 12 points becomes the symmetry group of an icosahedron.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/377

378: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/02(土) 10:34:37.27 ID:ZLEqKHqI

>>377

つづき

One then needs to consider the action of the symmetry group of icosahedron on the five associated tetrahedra.

・L2(7) =~ L3(2) which acts on the 1+2+4 = 7 points of the Fano plane (projective plane over F2); this can also be seen as the action on order 2 biplane, which is the complementary Fano plane.
・L2(11) is subtler, and elaborated below; it acts on the order 3 biplane.[8]
Further, L2(7) and L2(11) have two inequivalent actions on p points; geometrically this is realized by the action on a biplane,
which has p points and p blocks - the action on the points and the action on the blocks are both actions on p points,
but not conjugate (they have different point stabilizers);
they are instead related by an outer automorphism of the group.[9]

More recently, these last three exceptional actions have been interpreted as an example of the ADE classification:[10] these actions correspond to products (as sets, not as groups) of the groups as A4 × Z/5Z, S4 × Z/7Z, and A5 × Z/11Z,
where the groups A4, S4 and A5 are the isometry groups of the Platonic solids, and correspond to E6, E7, and E8 under the McKay correspondence.
These three exceptional cases are also realized as the geometries of polyhedra (equivalently, tilings of Riemann surfaces), respectively: the compound of five tetrahedra inside the icosahedron (sphere, genus 0),
the order 2 biplane (complementary Fano plane) inside the Klein quartic (genus 3), and the order 3 biplane (Paley biplane) inside the buckyball surface (genus 70).[11][12]

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/378

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