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| author | dos-reis <gdr@axiomatics.org> | 2010-04-22 01:40:53 +0000 |
|---|---|---|
| committer | dos-reis <gdr@axiomatics.org> | 2010-04-22 01:40:53 +0000 |
| commit | 9fac9abefef7f727a59b7861317d148352a43d88 (patch) | |
| tree | 6cf281301d4609d71aae0540b6cd098abd41de2b /src/algebra/irsn.spad.pamphlet | |
| parent | dca6da4bba9a14e544345d4b54f623450d47f283 (diff) | |
| download | open-axiom-9fac9abefef7f727a59b7861317d148352a43d88.tar.gz | |
* algebra/irsn.spad.pamphlet (IrrRepSymNatPackage): Tidy.
* algebra/partperm.spad.pamphlet (PartitionsAndPermutations): Likewise.
* algebra/cycles.spad.pamphlet (complete$CycleIndicators): Now
take only positive integers.
(powerSum$CycleIndicators): Likewise.
(elementary$CycleIndicators): Likewise.
(alternating$CycleIndicators): Likewise.
(cyclic$CycleIndicators): Likewise.
(dihedral$CycleIndicators): Likewise.
(graphs$CycleIndicators): Likewise.
Diffstat (limited to 'src/algebra/irsn.spad.pamphlet')
| -rw-r--r-- | src/algebra/irsn.spad.pamphlet | 43 |
1 files changed, 22 insertions, 21 deletions
diff --git a/src/algebra/irsn.spad.pamphlet b/src/algebra/irsn.spad.pamphlet index 4cde0675..95d9a257 100644 --- a/src/algebra/irsn.spad.pamphlet +++ b/src/algebra/irsn.spad.pamphlet @@ -55,27 +55,28 @@ IrrRepSymNatPackage(): public == private where ICF ==> IntegerCombinatoricFunctions Integer PP ==> PartitionsAndPermutations PERM ==> Permutation + macro PI == PositiveInteger public ==> with - dimensionOfIrreducibleRepresentation : L I -> NNI + dimensionOfIrreducibleRepresentation : L PI -> NNI ++ dimensionOfIrreducibleRepresentation(lambda) is the dimension ++ of the ordinary irreducible representation of the symmetric group ++ corresponding to {\em lambda}. ++ Note: the Robinson-Thrall hook formula is implemented. - irreducibleRepresentation : (L I, PERM I) -> M I + irreducibleRepresentation : (L PI, PERM I) -> M I ++ irreducibleRepresentation(lambda,pi) is the irreducible representation ++ corresponding to partition {\em lambda} in Young's natural form of the ++ permutation {\em pi} in the symmetric group, whose elements permute ++ {\em {1,2,...,n}}. - irreducibleRepresentation : L I -> L M I + irreducibleRepresentation : L PI -> L M I ++ irreducibleRepresentation(lambda) is the list of the two ++ irreducible representations corresponding to the partition {\em lambda} ++ in Young's natural form for the following two generators ++ of the symmetric group, whose elements permute ++ {\em {1,2,...,n}}, namely {\em (1 2)} (2-cycle) and ++ {\em (1 2 ... n)} (n-cycle). - irreducibleRepresentation : (L I, L PERM I) -> L M I + irreducibleRepresentation : (L PI, L PERM I) -> L M I ++ irreducibleRepresentation(lambda,listOfPerm) is the list of the ++ irreducible representations corresponding to {\em lambda} ++ in Young's natural form for the list of permutations @@ -84,10 +85,10 @@ IrrRepSymNatPackage(): public == private where private ==> add -- local variables - oldlambda : L I := nil$(L I) + oldlambda : L PI := nil$L(PI) flambda : NNI := 0 -- dimension of the irreducible repr. younglist : L M I := nil$(L M I) -- list of all standard tableaus - lprime : L I := nil$(L I) -- conjugated partition of lambda + lprime : L PI := nil$L(PI) -- conjugated partition of lambda n : NNI := 0 -- concerning symmetric group S_n rows : NNI := 0 -- # of rows of standard tableau columns : NNI := 0 -- # of columns of standard tableau @@ -100,7 +101,7 @@ IrrRepSymNatPackage(): public == private where -- (signum(k,l,id))_1 <= k,l <= flambda, where id -- denotes the identity permutation - alreadyComputed? : L I -> Void + alreadyComputed? : L PI -> Void -- test if the last calling of an exported function concerns -- the same partition lambda as the previous call @@ -119,9 +120,9 @@ IrrRepSymNatPackage(): public == private where -- otherwise -- signum(k,l,pi) = 0. - sumPartition : L I -> NNI - -- checks if lambda is a proper partition and results in - -- the sum of the entries + -- checks if lambda is a proper partition and results in + -- the sum of the entries + sumPartition : L PI -> NNI testPermutation : L I -> NNI -- testPermutation(pi) checks if pi is an element of S_n, @@ -157,12 +158,12 @@ IrrRepSymNatPackage(): public == private where alreadyComputed?(lambda) == - if not(lambda = oldlambda) then + if lambda ~= oldlambda then oldlambda := lambda lprime := conjugate(lambda)$PP - rows := (first(lprime)$(L I))::NNI - columns := (first(lambda)$(L I))::NNI - n := (+/lambda)::NNI + rows := first(lprime)$L(PI) + columns := first(lambda)$L(PI) + n := +/lambda younglist := listYoungTableaus(lambda)$SGCF flambda := #younglist aIdInverse() -- side effect: creates actual aId @@ -240,14 +241,14 @@ IrrRepSymNatPackage(): public == private where sumPartition(lambda) == ok : B := true prev : I := first lambda - sum : I := 0 + sum : NNI := 0 for x in lambda repeat sum := sum + x ok := ok and (prev >= x) prev := x if not ok then error("No proper partition ") - sum::NNI + sum testPermutation(pi : L I) : NNI == @@ -273,7 +274,7 @@ IrrRepSymNatPackage(): public == private where dimensionOfIrreducibleRepresentation(lambda) == nn : I := sumPartition(lambda)::I --also checks whether lambda dd : I := 1 --is a partition - lambdaprime : L I := conjugate(lambda)$PP + lambdaprime := conjugate(lambda)$PP -- run through all rows of the Youngtableau corr. to lambda for i in 1..lambdaprime.1 repeat -- run through all nodes in row i of the Youngtableau @@ -284,7 +285,7 @@ IrrRepSymNatPackage(): public == private where (factorial(nn)$ICF quo dd)::NNI - irreducibleRepresentation(lambda:(L I),pi:(PERM I)) == + irreducibleRepresentation(lambda: L PI,pi: PERM I) == nn : NNI := sumPartition(lambda) alreadyComputed?(lambda) piList : L I := listPermutation pi @@ -298,8 +299,8 @@ IrrRepSymNatPackage(): public == private where irreducibleRepresentation(lambda) == - listperm : L PERM I := nil$(L PERM I) - li : L I := nil$(L I) + listperm : L PERM I := nil$L(PERM I) + li : L I := nil$L(I) sumPartition(lambda) alreadyComputed?(lambda) listperm := @@ -314,7 +315,7 @@ IrrRepSymNatPackage(): public == private where irreducibleRepresentation(lambda,listperm) - irreducibleRepresentation(lambda:(L I),listperm:(L PERM I)) == + irreducibleRepresentation(lambda: L PI,listperm: L PERM I) == sumPartition(lambda) alreadyComputed?(lambda) [irreducibleRepresentation(lambda, pi) for pi in listperm] |