Permutation and Permutation Group Symmetry

Documentation

Functions

bool TiledArray::operator== (const Permutation &p1, const Permutation &p2)
 Permutation equality operator. More...
 
bool TiledArray::operator!= (const Permutation &p1, const Permutation &p2)
 Permutation inequality operator. More...
 
bool TiledArray::operator< (const Permutation &p1, const Permutation &p2)
 Permutation less-than operator. More...
 
std::ostream & TiledArray::operator<< (std::ostream &output, const Permutation &p)
 Add permutation to an output stream. More...
 
Permutation TiledArray::operator- (const Permutation &perm)
 Inverse permutation operator. More...
 
Permutation TiledArray::operator* (const Permutation &p1, const Permutation &p2)
 Permutation multiplication operator. More...
 
PermutationTiledArray::operator*= (Permutation &p1, const Permutation &p2)
 return *this ^ other More...
 
Permutation TiledArray::operator^ (const Permutation &perm, int n)
 Raise perm to the n-th power. More...
 
bool TiledArray::symmetry::operator== (const PermutationGroup &p1, const PermutationGroup &p2)
 PermutationGroup equality operator. More...
 
bool TiledArray::symmetry::operator!= (const PermutationGroup &p1, const PermutationGroup &p2)
 PermutationGroup inequality operator. More...
 
bool TiledArray::symmetry::operator< (const PermutationGroup &p1, const PermutationGroup &p2)
 PermutationGroup less-than operator. More...
 
std::ostream & TiledArray::symmetry::operator<< (std::ostream &output, const PermutationGroup &p)
 Add permutation group to an output stream. More...
 
template<typename MultiIndex >
bool TiledArray::symmetry::is_lexicographically_smallest (const MultiIndex &idx, const PermutationGroup &pg)
 
PermutationGroup TiledArray::symmetry::conjugate (const PermutationGroup &G, const PermutationGroup::Permutation &h)
 Computes conjugate permutation group obtained by the action of a permutation. More...
 
PermutationGroup TiledArray::symmetry::intersect (const PermutationGroup &G1, const PermutationGroup &G2)
 
template<typename Set >
PermutationGroup TiledArray::symmetry::stabilizer (const PermutationGroup &G, const Set &f)
 

Function Documentation

◆ conjugate()

PermutationGroup TiledArray::symmetry::conjugate ( const PermutationGroup G,
const PermutationGroup::Permutation h 
)
inline

Computes conjugate permutation group obtained by the action of a permutation.

Conjugate of group $ G $ under the action of element $ h $ is a group $ \{ a: a = h g h^{-1}, \forall g \in G \} $ .

Note
Since all permutation groups are subgroups of the symmetric group on the "infinite" set $ \{ 0, 1, \dots \} $, this can be used to "shift" the domain of G by action of h that permutes elements in the domain of G with those outside its domain, e.g. if $ G = S_2 $ on domain $ \{0,1\} $, conjugation with $ h = {1->2, 2->1} $ will produce $ S_2 $ on domain $ \{0,2\} $.
Parameters
Ginput PermutationGroup
hinput Permutation
Returns
conjugate PermutationGroup

Definition at line 376 of file permutation_group.h.

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◆ intersect()

PermutationGroup TiledArray::symmetry::intersect ( const PermutationGroup G1,
const PermutationGroup G2 
)
inline

Computes intersect of 2 PermutationGroups

Note
The intersect is guaranteed to be a subgroup for both groups, hence this is the largest common subgroup.
Parameters
G1PermutationGroup
G2PermutationGroup
Returns
PermutationGroup

Definition at line 390 of file permutation_group.h.

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◆ is_lexicographically_smallest()

template<typename MultiIndex >
bool TiledArray::symmetry::is_lexicographically_smallest ( const MultiIndex &  idx,
const PermutationGroup pg 
)

determines whether a given MultiIndex is lexicographically smallest among all indices generated by the action of pg.

Template Parameters
MultiIndexa sequence type that is directly addressable, i.e. has a fast operator[]
Parameters
idxan Index object
pgthe PermutationGroup
Returns
false if action of a permutation in pg can produce an Index that is lexicographically smaller than idx (i.e. there exists i such that pg[i]*idx is lexicographically less than idx), true otherwise

Definition at line 351 of file permutation_group.h.

◆ operator!=() [1/2]

bool TiledArray::operator!= ( const Permutation p1,
const Permutation p2 
)
inline

Permutation inequality operator.

Parameters
p1The left-hand permutation to be compared
p2The right-hand permutation to be compared
Returns
true if any element of p1 is not equal to that of p2, otherwise false.

Definition at line 419 of file permutation.h.

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◆ operator!=() [2/2]

bool TiledArray::symmetry::operator!= ( const PermutationGroup p1,
const PermutationGroup p2 
)
inline

PermutationGroup inequality operator.

Parameters
p1The left-hand permutation group to be compared
p2The right-hand permutation group to be compared
Returns
true if any element of p1 is not equal to that of p2, otherwise false.

Definition at line 228 of file permutation_group.h.

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◆ operator*()

Permutation TiledArray::operator* ( const Permutation p1,
const Permutation p2 
)
inline

Permutation multiplication operator.

Parameters
p1The left-hand permutation
p2The right-hand permutation
Returns
The product of p1 and p2 (which is the permutation of p2 by p1).

Definition at line 460 of file permutation.h.

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◆ operator*=()

Permutation& TiledArray::operator*= ( Permutation p1,
const Permutation p2 
)
inline

return *this ^ other

Definition at line 465 of file permutation.h.

◆ operator-()

Permutation TiledArray::operator- ( const Permutation perm)
inline

Inverse permutation operator.

Parameters
permThe permutation to be inverted
Returns
perm.inverse()

Definition at line 452 of file permutation.h.

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◆ operator<() [1/2]

bool TiledArray::operator< ( const Permutation p1,
const Permutation p2 
)
inline

Permutation less-than operator.

Parameters
p1The left-hand permutation to be compared
p2The right-hand permutation to be compared
Returns
true if the elements of p1 are lexicographically less than that of p2, otherwise false.

Definition at line 429 of file permutation.h.

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◆ operator<() [2/2]

bool TiledArray::symmetry::operator< ( const PermutationGroup p1,
const PermutationGroup p2 
)
inline

PermutationGroup less-than operator.

Parameters
p1The left-hand permutation group to be compared
p2The right-hand permutation group to be compared
Returns
true if the elements of p1 are lexicographically less than that of p2, otherwise false.

Definition at line 238 of file permutation_group.h.

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◆ operator<<() [1/2]

std::ostream & TiledArray::operator<< ( std::ostream &  output,
const Permutation p 
)
inline

Add permutation to an output stream.

Parameters
[out]outputThe output stream
[in]pThe permutation to be added to the output stream
Returns
The output stream

Definition at line 439 of file permutation.h.

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◆ operator<<() [2/2]

std::ostream& TiledArray::symmetry::operator<< ( std::ostream &  output,
const PermutationGroup p 
)
inline

Add permutation group to an output stream.

Parameters
[out]outputThe output stream
[in]pThe permutation group to be added to the output stream
Returns
The output stream

Definition at line 248 of file permutation_group.h.

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◆ operator==() [1/2]

bool TiledArray::operator== ( const Permutation p1,
const Permutation p2 
)
inline

Permutation equality operator.

Parameters
p1The left-hand permutation to be compared
p2The right-hand permutation to be compared
Returns
true if all elements of p1 and p2 are equal and in the same order, otherwise false.

Definition at line 408 of file permutation.h.

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◆ operator==() [2/2]

bool TiledArray::symmetry::operator== ( const PermutationGroup p1,
const PermutationGroup p2 
)
inline

PermutationGroup equality operator.

Parameters
p1The left-hand permutation group to be compared
p2The right-hand permutation group to be compared
Returns
true if all elements of p1 and p2 are equal, otherwise false.

Definition at line 218 of file permutation_group.h.

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◆ operator^()

Permutation TiledArray::operator^ ( const Permutation perm,
int  n 
)
inline

Raise perm to the n-th power.

Constructs the permutation $ P^n $, where $ P $ is the permutation perm.

Parameters
permThe base permutation
nExponent value
Returns
This permutation raised to the n-th power

Definition at line 476 of file permutation.h.

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◆ stabilizer()

template<typename Set >
PermutationGroup TiledArray::symmetry::stabilizer ( const PermutationGroup G,
const Set &  f 
)
inline

Computes the largest subgroup of a permutation group that leaves the given set of indices fixed.

Template Parameters
Seta set of indices
Parameters
Ginput PermutationGroup
finput Set
Returns
the fixed set subgroup of G

Definition at line 406 of file permutation_group.h.

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