![](`r system.file("help/figures/permutations.png", package = "permutations")`){width=10%} ![](`r system.file("help/figures/freealg.png", package = "freealg")`){width=10%}
set.seed(0) knitr::opts_chunk$set(echo = TRUE) library("freealg") library("permutations")
This short document introduces the dot object and shows how it can be
used to work with commutators and verify the Hall-Wittidentity. The dot
object is a (trivial) S4
object of class dot
,
`.` <- new("dot") save(".", file="dot.rda") # copy dot.rda to the data/ directory.
The dot uses the dot
class of the freealg
package. The point of
the dot (!) is that it allows one to calculate the bracket
$[x,y]=x^{-1}y^{-1}xy$ using R idiom .[x,y]
. Thus:
x <- as.cycle(1:3) y <- as.cycle(2:5) x^-1*y^-1*x*y .[x,y]
It would have been nice to use \code{[x,y]} (that is, without the dot) but although this is syntactically consistent, it cannot be done in R AFAICS.
It is possible to apply the dot construction .[x,y]
to more
complicated examples. Note that the Jacobi identity does not really
make sense in this context, but the Hall-Witt identity does:
x <- rperm(10,7) y <- rperm(10,8) z <- rperm(10,9) .[.[x,y],z^x] * .[.[z,x],y^z] * .[.[y,z],x^y]
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