[seqfan] n-dimensional simplices
David Harden
oddleehr at alum.mit.edu
Fri Nov 7 19:26:41 CET 2008
In the course of work of mine, I came across a sequence I'd like to submit (preferably after having seen it extended):
The nth term is the number of permutation-isomorphism classes of possible symmetry groups of an n-dimensional simplex.
(notation: here the dihedral group of order 2n is denoted D_n)
Two n-simplices are congruent if and only if there is a bijection on their vertex sets which preserves all edge lengths (this is provable by using v_i - v_1, 2 <= i <= n+1, as a basis for R^n).
This gives a procedure for computing this sequence:
Take G to be a subgroup of S_(n+1). Take a generic n-simplex (one having all (n^2+n)/2 edge lengths distinct) and compute the orbits of G in its induced action on the unordered pairs {i,j}, 1 <= i < j <= n+1, of distinct indices.
Now compute the symmetry group of a 'generic G-symmetric simplex' (i.e., the group of permutations preserving (when they act via simultaneous row and column permutations, or, equivalently, via conjugation by the corresponding permutation matrix) the new distance table with a different distance for each orbit of the action of G on the unordered pairs). Call this group H. It can be larger than G: If G is doubly transitive (or just 2-homogeneous), the unordered pairs fall into a single orbit and H=S_(n+1).
There are, up to permutation isomorphism, finitely many choices for G. The number of H coming out of this is a_n.
I have the first 4 terms (with offset n=1):
n=1: 1
n=2: 3 (as a triangle can be scalene, isoceles, or equilateral)
n=3: 8 (the possible permutation groups are S_4, D_4, S_3, V, S_2xS_2, <(12)>, <(12)(34)>, and 1)
n=4: 11 (the possible permutation groups are S_5, S_4, S_3xS_2, D_5, D_4, S_3, V, S_2xS_2, <(12)>, <(12)(34)>, and 1)
It is clear that this sequence is not decreasing, because every possible symmetry group of an n-simplex is also a possible symmetry group of an (n+1)-simplex: Place the new vertex, v_(n+2), above (in the direction perpendicular to the hyperplane of the simplex formed by v_1,...,v_(n+1) ) the circumcenter of the simplex formed by vertices v_1 through v_(n+1). Then move it up far enough (e.g., so the distance from v_(n+2) to any other vertex is larger than any pre-existing edge length) so that no new symmetry is introduced.
(In fact, the sequence is strictly increasing because S_(n+2) is a simplicial symmetry group possible in n+1 dimensions but not in n dimensions.)
---- David
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