Symmetric group

numqi.group._symmetric

numqi.group.permutation_to_cycle_notation(index_tuple)

Permutation to cycle notation

Parameters:

Name	Type	Description	Default
index_tuple	tuple[int]	tuple of int, e.g. (2,3,0,1)	required

Returns:

Name	Type	Description
ret	tuple[tuple[int]]	tuple of tuple of int, e.g. ((0,2),(1,3))

numqi.group.get_symmetric_group_cayley_table(n, alternating=False)

Cayley table of symmetric group

Parameters:

Name	Type	Description	Default
n	int	the order of symmetric group	required
alternating	bool	whether to get the alternating group	False

Returns:

Name	Type	Description
ret	ndarray	the Cayley table of symmetric group, dtype=np.int64, shape=(n!,n!)

numqi.group.get_sym_group_num_irrep(N, return_full=False)

Number of irreducible representations of symmetric group

Parameters:

Name	Type	Description	Default
N	int	the order of symmetric group	required
return_full	bool	whether to return the full table	False

Returns:

Name	Type	Description
ret0	int	the number of irreducible representations
ret1	ndarray	the full table of the number of irreducible representations, dtype=np.int64, shape=(N+1,N+1), only if return_full is True

numqi.group.get_sym_group_young_diagram(N)

Young diagram of symmetric group, almost to N=50 (10 seconds)

Parameters:

Name	Type	Description	Default
N	int	the order of symmetric group	required

Returns:

Name	Type	Description
ret	ndarray	the Young diagram of symmetric group, dtype=np.int64, shape=(#Young,N)

numqi.group.get_young_diagram_mask(young, check=True)

Young diagram mask

Parameters:

Name	Type	Description	Default
young	tuple[int]	the Young diagram, e.g. (3,2,1)	required
check	bool	whether to check the input	True

Returns:

Name	Type	Description
ret	ndarray	the Young diagram mask, dtype=np.int64, shape=(len(young),young[0])

numqi.group.check_young_diagram(np0)

Check the Young diagram

Parameters:

Name	Type	Description	Default
np0	tuple[int]	the Young diagram, e.g. (3,2,1)	required

numqi.group.get_hook_length(*int_tuple, check=True)

Hook length wiki-link

Parameters:

Name	Type	Description	Default
int_tuple	tuple[int]	the Young diagram, e.g. (3,2,1)	()
check	bool	whether to check the input	True

Returns:

Name	Type	Description
ret	int	the hook length

numqi.group.get_young_diagram_transpose(np0, check=True)

Transpose of Young diagram

Parameters:

Name	Type	Description	Default
np0	tuple[int]	the Young diagram, e.g. (3,2,1)	required
check	bool	whether to check the input	True

Returns:

Name	Type	Description
ret	ndarray	the transpose of Young diagram, dtype=np.int64, shape=(len(np0),)

numqi.group.get_all_young_tableaux(young, check=True)

All Young tableaux

Parameters:

Name	Type	Description	Default
young	tuple[int]	the Young diagram, e.g. (3,2,1)	required
check	bool	whether to check the input	True

Returns:

Name	Type	Description
ret	ndarray	all Young tableaux, dtype=np.int64, shape=(#tableaux,len(young),young[0])

numqi.group.young_tableau_to_young_symmetrizer(young, tableau)

Young tableau to Young symmetrizer

Parameters:

Name	Type	Description	Default
young	tuple[int]	the Young diagram, e.g. (3,2,1)	required
tableau	ndarray	the Young tableau, dtype=np.int64, shape=(len(young),young[0])	required

Returns:

Name	Type	Description
symmetrizer	ndarray	the Young symmetrizer, dtype=np.int64, shape=(#?,sum(young))
sign	ndarray	the sign of Young symmetrizer, dtype=np.int64, shape=(#?,)

numqi.group.print_all_young_tableaux(N0)

Print all Young tableaux

Parameters:

Name	Type	Description	Default
N0	int	the order of symmetric group	required