Arithmetic Ops Group Methods for permutations
Allows arithmetic operators to be used for manipulation of permutation objects such as addition, multiplication, division, integer powers, etc.
## S3 method for class 'permutation' Ops(e1, e2) cycle_power(x,pow) cycle_power_single(x,pow) cycle_sum(e1,e2) cycle_sum_single(c1,c2) group_action(e1,e2) word_equal(e1,e2) word_prod(e1,e2) word_prod_single(e1,e2) permprod(x) vps(vec,pow) ccps(n,pow) helper(e1,e2)
x,e1,e2 |
Objects of class “ |
c1,c2 |
Objects of class |
pow |
Integer vector of powers |
vec |
In function |
n |
In function |
The function Ops.permutation() passes binary arithmetic
operators (“+”, “*”, “/”,
“^”, and “==”) to the appropriate
specialist function.
Multiplication, as in a*b, is effectively
word_prod(a,b); it coerces its arguments to word form (because
a*b = b[a]).
Raising permutations to integer powers, as in a^n, is
cycle_power(a,n); it coerces a to cycle form and returns
a cycle. Negative and zero values of n operate as expected.
Function cycle_power() is vectorized; it calls
cycle_power_single(), which is not. This calls vps()
(“Vector Power Single”), which checks for simple cases such as
pow=0 or the identity permutation; and function vps()
calls function ccps() which performs the actual
number-theoretic manipulation to raise a cycle to a power.
Raising a permutation to the power of another permutation, as in
a^b, is idiom for inverse(b)*a*b, sometimes known as
group action; the notation is motivated by the identities
x^(yz)=(x^y)^z and (xy)^z=x^z*y^z.
Permutation addition, as in a+b, is defined if the cycle
representations of the addends are disjoint. The sum is defined as
the permutation given by juxtaposing the cycles of a with those
of b. Note that this operation is commutative. If a
and b do not have disjoint cycle representations, an error is
returned. This is useful if you want to guarantee that two
permutations commute (NB: permutation a commutes with
a^i for i any integer, and in particular a
commutes with itself. But a+a returns an error: the operation
checks for disjointness, not commutativity).
Permutation “division”, as in a/b, is
a*inverse(b). Note that a/b*c is evaluated left to
right so is equivalent to a*inverse(b)*c. See note.
Function helper() sorts out recycling for binary functions, the
behaviour of which is inherited from cbind(), which also
handles the names of the returned permutation.
None of these functions are really intended for the end user: use the ops as shown in the examples section.
The class of the returned object is the appropriate one.
It would be nice to define a unary operator which inverted a
permutation. I do not like “id/x” to represent a
permutation inverse: the idiom introduces an utterly redundant object
(“id”), and forces the use of a binary operator where a
unary operator is needed.
The natural unary operator would be the exclamation mark, !x.
However, redefining the exclamation mark to give permutation inverses,
while possible, is not desirable because its precedence is too low.
One would like !x*y to return inverse(x)*y but instead
standard precendence rules means that it returns inverse(x*y).
This caused such severe cognitive dissonance that I removed it.
There does not appear to be a way to define a new unary operator due to the construction of the parser.
Robin K. S. Hankin
x <- rperm(20,9) # word form y <- rperm(20,9) # word form x*y # word form x^5 # coerced to cycle form x^as.cycle(1:5) # group action; coerced to word. x*inverse(x) == id # all TRUE # the 'sum' of two permutations is defined if their cycles are disjoint: as.cycle(1:4) + as.cycle(7:9) data(megaminx) megaminx[1] + megaminx[7:12]
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