diff options
Diffstat (limited to 'src/algebra/perm.spad.pamphlet')
| -rw-r--r-- | src/algebra/perm.spad.pamphlet | 36 |
1 files changed, 15 insertions, 21 deletions
diff --git a/src/algebra/perm.spad.pamphlet b/src/algebra/perm.spad.pamphlet index 69f33f83..2dd9da5b 100644 --- a/src/algebra/perm.spad.pamphlet +++ b/src/algebra/perm.spad.pamphlet @@ -15,7 +15,7 @@ ++ Authors: Holger Gollan, Johannes Grabmeier, Gerhard Schneider ++ Date Created: 27 July 1989 ++ Date Last Updated: 29 March 1990 -++ Basic Operations: cycle, cycles, eval, orbit +++ Basic Operations: cycle, cycles, orbit ++ Related Constructors: PermutationGroup, PermutationGroupExamples ++ Also See: RepresentationTheoryPackage1 ++ AMS Classifications: @@ -24,7 +24,8 @@ ++ Description: PermutationCategory provides a categorial environment ++ for subgroups of bijections of a set (i.e. permutations) -PermutationCategory(S:SetCategory): Category == Group with +PermutationCategory(S:SetCategory): Category == + Join(Group,Eltable(S,S)) with cycle : List S -> % ++ cycle(ls) coerces a cycle {\em ls}, i.e. a list with not ++ repetitions to a permutation, which maps {\em ls.i} to @@ -37,12 +38,9 @@ PermutationCategory(S:SetCategory): Category == Group with ++ {\em ls.i} to {\em ls.i+1}, indices modulo the length of the list, ++ then these permutations are mutiplied. ++ Error: if repetitions occur in one cycle. - eval : (%,S) -> S - ++ eval(p, el) returns the image of {\em el} under the - ++ permutation p. - elt : (%,S) -> S - ++ elt(p, el) returns the image of {\em el} under the - ++ permutation p. + support: % -> Set S + ++ \spad{support p} returns the set of points not fixed by the + ++ permutation \spad{p}. orbit : (%,S) -> Set S ++ orbit(p, el) returns the orbit of {\em el} under the ++ permutation p, i.e. the set which is given by applications of @@ -61,7 +59,7 @@ PermutationCategory(S:SetCategory): Category == Group with ++ Authors: Johannes Grabmeier, Holger Gollan ++ Date Created: 19 May 1989 ++ Date Last Updated: 2 June 2006 -++ Basic Operations: _*, degree, movedPoints, cyclePartition, order, +++ Basic Operations: _*, degree, support, cyclePartition, order, ++ numberOfCycles, sign, even?, odd? ++ Related Constructors: PermutationGroup, PermutationGroupExamples ++ Also See: RepresentationTheoryPackage1 @@ -126,8 +124,6 @@ Permutation(S:SetCategory): public == private where degree : % -> NonNegativeInteger ++ degree(p) retuns the number of points moved by the ++ permutation p. - movedPoints : % -> Set S - ++ movedPoints(p) returns the set of points moved by the permutation p. cyclePartition : % -> Partition ++ cyclePartition(p) returns the cycle structure of a permutation ++ p including cycles of length 1 only if S is finite. @@ -300,7 +296,7 @@ Note that before [[patch--50]] this read coercePreimagesImages preImageAndImage == p : % := [preImageAndImage.1,preImageAndImage.2] \end{verbatim} -causing bugs when computing [[movedPoints]], [[fixedPoints]], [[even?]], +causing bugs when computing [[support]], [[fixedPoints]], [[even?]], [[odd?]], etc., as reported in Issue~\#295. The other coercion facilities check for fixed points. It also seems that [[*]] @@ -308,7 +304,7 @@ removes fixed points from its result. <<TEST PERM>>= p := coercePreimagesImages([[1,2,3],[1,2,3]]) - movedPoints p -- should return {} + support p -- should return {} even? p -- should return true p := coercePreimagesImages([[0,1,2,3],[3,0,2,1]])$PERM ZMOD 4 fixedPoints p -- should return {2} @@ -319,9 +315,9 @@ removes fixed points from its result. <<domain PERM Permutation>>= - movedPoints p == construct p.1 + support p == construct p.1 - degree p == #movedPoints p + degree p == #support p p = q == #(preimp := p.1) ~= #(preimq := q.1) => false @@ -334,12 +330,12 @@ removes fixed points from its result. orbit(p ,el) == -- start with a 1-element list: out : Set S := brace list el - el2 := eval(p, el) + el2 := p(el) while el2 ~= el repeat -- be carefull: insert adds one element -- as side effect to out insert!(el2, out) - el2 := eval(p, el2) + el2 := p(el2) out cyclePartition p == @@ -450,13 +446,11 @@ removes fixed points from its result. inv p == [p.2, p.1] - eval(p, el) == + elt(p,el) == pos := position(el, p.1) pos = 0 => el (p.2).pos - elt(p, el) == eval(p, el) - numberOfCycles p == #coerceToCycle(p, false) @@ -480,7 +474,7 @@ Up to [[patch--50]] we did not check for duplicates. Up to [[patch--50]] we did not check for duplicates. <<domain PERM Permutation>>= - fixedPoints ( p ) == complement movedPoints p + fixedPoints ( p ) == complement support p cyclePartition p == pt := partition([#c::PI for c in coerceToCycle(p, false)])$Partition |