commutator | R Documentation |
In the permutations package, the dot is defined as the
Group-theoretic commutator:
\mjeqn[x,y]=x^-1y^-1xy[x,y]=x^(-1)y^(-1)xy. This is a bit of an
exception to the usual definition of xy-yx
(along with the
freegroup package). Package idiom is commutator(x,y)
or
.[x,y]
.
The Jacobi identity does not make sense in the context of the permutations package, but the Hall-Witt identity is obeyed.
The “dot” object is defined and discussed in inst/dot.Rmd
,
which creates file data/dot.rda
.
commutator(x, y)
x,y |
Permutation objects, coerced to word |
Robin K. S. Hankin
group_action
.[as.cycle("123456789"),as.cycle("12")] x <- rperm(10,7) y <- rperm(10,8) z <- rperm(10,9) uu <- commutator(commutator(x,y),z^x) * commutator(commutator(z,x),y^z) * commutator(commutator(y,z),x^y) stopifnot(all(is.id(uu))) # this is the Hall-Witt identity .[x,y] is.id(.[.[x,y],z^x] * .[.[z,x],y^z] * .[.[y,z],x^y]) is.id(.[.[x,-y],z]^y * .[.[y,-z],x]^z * .[.[z,-x],y]^x)
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