Maple package PSL

 
 

The main functions of PSL,  
The main functions of PSL for dealing with trace representation ideals,  
Important functions of PSL for dealing with trace representation ideals,  
Functions of PSL for dealing with representation ideals ,  
The most important low level functions of PSL ,  

 

Main commands of PSL

 

Command Description
L2Quotients Find all L2-quotients of the given finitely presented group group
AllCorrAssPrimes All minimal associated prime ideals corresponding to all trace representation ideals (for all corresponding sign systems)
SplitOfRel Compute the decomposition of relations with the corresponding sign systems
MinimalAssociatedPrimesZZ Compute minimal associated prime ideals of an ideal in a polynomial ring over integers
SubgroupOfIndex2 Compute generators of subgroup of index 2 for the given group
GaloisDescent Check whether a matrix representation over a subfield is possible: if possible compute matrices that fulfill the same relations over a smaller field

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The main functions of PSL for dealing with trace representation ideals

 

Command Description
CondOnTrace Compute the set of relations for traces 2x2 matrices fulfilling the given relations
IsA4S4A5 Test, if a given trace representation ideal yields a representation isomorphic to A_4,S_4,A_5
OnlyPSLorPGL From a given list of ideals remove those yielding irreducible representations or representations isomorphic to dihedral groups, A_4,S_4,A_5 (i.e. leave only ideals corresponding to representations isomorphic to groups of L2-type)
IsPSLp Test, if in a given characteristic p the trace representation ideal yields a group isomorphic to PSL(2, K) or PGL(2, K)
JTRepres Compute a matrix ansatz using the given trace representation ideal
Irred For a given ideal I compute all ideals containing I that yield irreducible representations
IsIrred Test, if the given trace representation ideal corresponds to an irreducible representation
GrowthParameters2 Compute the growth parameters (cf. PlF (2009)), implemented for the case of groups given on two generators

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Important functions of PSL for dealing with trace representation ideals

 

Command Description
RedTraceCond Condition on the trace representation ideals to yield a reducible representation (for group presentations on two generators)
DihedralTraceCond Conditions on the trace representation ideals to yield a representation isomorphic to a dihedral group (for group presentations on two generators), 3 possibilities
A4TraceCond Conditions on the trace representation ideals to yield a representation isomorphic to A_4 (for group presentations on two generators), 16 possibilities
S4TraceCond Conditions on the trace representation ideals to yield a representation isomorphic to S_4 (for group presentations on two generators), 18 possibilities
A4TraceCond Conditions on the trace representation ideals to yield a representation isomorphic to A_5 (for group presentations on two generators), 76 possibilities
A4TraceCond3 Conditions on the trace representation ideals to yield a representation isomorphic to A_4 (for group presentations on three generators), 520 possibilities
DihedralTraceCond3 Conditions on the trace representation ideals to yield a representation isomorphic to a dihedral group (for group presentations on three generators)
AlgebraicDep3 The condition on algebraic dependence between traces of three 2x2 matrices of determinant 1

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Functions of PSL for dealing with representation ideals

 

Command Description
RelationsE Write the group relations for the given sign system
ListOfRelations Compute the list of relations for the given sign system and a given matrix ansatz
IsAbelian Test, if the group generated by the given matrices is abelian
IsDnOrCn Test, if the group generated by the given matrices is a dihedral or a cyclic group
IsSemiDirectProd Test, if the group generated by the given matrices is a semidirect product of an elementary abelian p-group and a cyclic group
IsA4S4A5 Test, if the group generated by the given matrices is isomorphic to A_4, S_4, A_5

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The most important low level functions of PSL

Command Description
Ansatz1, Ansatz2 Matrix ansatz
Ansatz1T, Ansatz2T Matrix ansatz using conditions on traces
Comm, comm Commutator defined as a^{-1}b^{-1}ab (in the Inv /in the standard notation)
CommA, commA Commutator defined as aba^{-1}b^{-1} (in the Inv /in the standard notation)
Inv Rewrite the given relation into the Inv-notation
PSI The minimal polynomial of x+x^{n-1} in Q[x]/Phi_n(x), where Phi_n(x) denotes the n-th cyclotomic polynomial, (cf. PlF (2009))
PafnutyPoly Compute a generalized Chebyshev polynomial for the given relation

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