0
$\begingroup$

As far as I know the group of symmetries of the euclidean n-cube is generated by reversions ${r_i}$ and transpositions ${s_i}$ for i=1,...,n-1. Transpositions generate the subgroup of permutations of coordinates, which is isomorphic to $S_n$, the symmetric group, acting by swapping two coordinates and leaving the others unchanged:

$s_i(a_1,...,a_n)=(a_1,...,a_{i-1},a_{i+1},a_i,a_{i+2},...,a_n)$

I need to know if there is a known set of transpositions which considers also (perhaps starting from n=3) the transposition

$s_n(a_1,...,a_n)=(a_n,a_2,...,a_{n-1},a_1)$

and which group does it generate.

$\endgroup$
1
$\begingroup$

This is not a transposition, but in any case it is a permutation and hence is an element of $S_n$. The group of symmetries of a hypercube may be combinatorially realized as the hyperoctahedral group, or the group of signed permutations. These are permutations $f$ of $\{-n,-n+1,\ldots,-1,1,2,\ldots,n\}$ such that $f(-i)=-f(i)$ for all $i$ and are usually identified by the sequence of values $(f(1),f(2),\ldots,f(n))$. This group is generated as a Coxeter group by the adjacent transpositions together with the element $(-1,2,3,\ldots,n)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.