From: input/repa6.input fricas -- This file demonstrates Representation Theory in Scratchpad -- using the packages REP1,
Type: List(Permutation(Integer))fricas -- pRA6 is the permutation representation over the Integers...
Type: List(Matrix(Integer))fricas -- ... and pRA6m2 is the permutation representation over PrimeField 2:
Type: List(Matrix(PrimeField?(2)))fricas -- Now try to split pRA6m2:
Type: List(List(Matrix(PrimeField?(2))))fricas -- We have found the trivial module as a factormodule -- and a 5-dimensional submodule.
Type: List(Matrix(PrimeField?(2)))fricas -- Try to split again...
Type: List(List(Matrix(PrimeField?(2))))fricas -- ... and find a 4-dimensional submodule,
Type: List(Matrix(PrimeField?(2)))fricas -- Now we want to test,
Type: Booleanfricas -- ...and see: dA6d4a is absolutely irreducible. -- So we have found a second irreducible representation.
Type: List(PositiveInteger)fricas dimensionOfIrreducibleRepresentation lambda
Type: PositiveIntegerfricas -- now create the restriction to A6:
Type: List(Matrix(Integer))fricas -- ... and d2211m2 is the representation over PrimeField 2:
Type: List(Matrix(PrimeField?(2)))fricas -- and split it: -- )set output tex off -- )set output algebra on sp2 := meatAxe d2211m2
Type: List(List(Matrix(PrimeField?(2))))fricas -- A 5 and a 4-dimensional one.
Type: List(Matrix(PrimeField?(2)))fricas -- This is absolutely irreducible,
Type: Booleanfricas -- ... and dA6d4a and dA6d4b are not equivalent:
Type: Matrix(PrimeField?(2))fricas -- So the third irreducible representation is found. fricas )set output tex off fricas )set output algebra on Type: List(Matrix(PrimeField?(2)))fricas -- and try to split it... Type: List(List(Matrix(PrimeField?(2))))fricas -- The representation is irreducible, Type: Booleanfricas -- So let's try the same over the field with 4 elements: Type: Typefricas dA6d16gf4 : List Matrix gf4 := dA6d16 fricas sp4 := meatAxe dA6d16gf4 fricas -- Now we find two 8-dimensional ones, fricas dA6d8b : List Matrix gf4 := sp4.2 fricas -- Both are absolutely irreducible... Type: Booleanfricas isAbsolutelyIrreducible? dA6d8b Type: Booleanfricas -- and they are not equivalent... fricas -- So we have found five absolutely irreducible representations of A6 -- in characteristic 2. Type: List(Matrix(PrimeField?(2)))fricas dA6d4a Type: List(Matrix(PrimeField?(2)))fricas dA6d4b Type: List(Matrix(PrimeField?(2)))fricas dA6d8a fricas dA6d8b fricas -- And here again is the irreducible, Type: List(Matrix(PrimeField?(2))) |