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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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

379: 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) 10:35:48.60 ID:ZLEqKHqI

>>378

つづき

The action of L2(11) can be seen algebraically as due to an exceptional inclusion L2(5)→ L2(11) - there are two conjugacy classes of subgroups of L2(11) that are isomorphic to L2(5),
each with 11 elements: the action of L2(11) by conjugation on these is an action on 11 points,
and, further, the two conjugacy classes are related by an outer automorphism of L2(11). (The same is true for subgroups of L2(7) isomorphic to S4, and this also has a biplane geometry.)

Geometrically, this action can be understood via a biplane geometry, which is defined as follows.
A biplane geometry is a symmetric design (a set of points and an equal number of "lines", or rather blocks) such that any set of two points is contained in two lines, while any two lines intersect in two points;
this is similar to a finite projective plane, except that rather than two points determining one line (and two lines determining one point),
they determine two lines (respectively, points). In this case (the Paley biplane, obtained from the Paley digraph of order 11),
the points are the affine line (the finite field) F11,
where the first line is defined to be the five non-zero quadratic residues (points which are squares: 1, 3, 4, 5, 9),
and the other lines are the affine translates of this (add a constant to all the points).
L2(11) is then isomorphic to the subgroup of S11 that preserve this geometry (sends lines to lines), giving a set of 11 points on which it acts - in fact two:
the points or the lines, which corresponds to the outer automorphism - while L2(5) is the stabilizer of a given line, or dually of a given point.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/379

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