In RobinHankin/permutations: The Symmetric Group: Permutations of a Finite Set

knitr::include_graphics(system.file("help/figures/permutations.png", package = "permutations"))

knitr::opts_chunk$set(collapse = TRUE, comment = "#>", dev = "png", fig.width = 7, fig.height = 3.5, message = FALSE, warning = FALSE)
options(width = 80, tibble.width = Inf)

To cite the permutations package in publications, please use @hankin2020. Here we consider $S_9$, the symmetric group on 9 elements, and representations of its elements. First we will load the package and change the default print method:

library("permutations")
options("print_word_as_cycle" = FALSE)

Now we will generate a couple of permutations, a and b:

a <- as.word(char2cycle("(175296)(348)"))
b <- as.word(char2cycle("(27)(45)(89)"))
a
b

Now we will show a representation of $S_9$, using permutation matrices:

M <- diag(9)
rownames(M) <- 1:9
colnames(M) <- 1:9
M

We will use the map $\phi\colon S_9\longrightarrow\operatorname{Aut}\left(\mathbb{R}^9\right)$ given by $\phi(a)=$ M[a,]. Then

M[a,]

permutes the rows of $M$ with permutation a. Note how the row names are permuted as well as the elements of $M$. Verifying that $\phi$ is indeed a homomorphism---that is, $\phi(\iota)=I_9$ and $\phi(a)\phi(b)=\phi(a\circ b)$---is straightforward:

all(M[as.word(id,9),] == diag(9))
all(M[a*b,] == M[a,] %*% M[b,])

In the second line above, note that the left hand side of the equality is group composition, while the right hand side is matrix multiplication. We can further verify that $\phi\left(a^{-1}\right)=\phi(a)^{-1}$:

all(M[inverse(a),] == solve(M[a,]))

again with group inversion on the left and matrix inversion on the right.

The map $\phi(a)=$ M[,a] is not a homomorphism:

all(M[,a*b] == M[,a] %*% M[,b])

but we can "rescue" it by considering a group operation $\star$ defined by $a\star b=b\circ a$:

all(M[,b*a] == M[,a] %*% M[,b])

See how the operation has a and b in opposite order from the matrix multiplication; see vignette order_of_ops for a discussion of this phenomenon from a different perspective.