Quantum groups#

This module provides tools for computations of integrals of polynomials on the quantum groups.

Quantum groups Python interface

References

[1] Banica, T., & Collins, B. (2007). Integration over compact quantum groups. Publications of the Research Institute for Mathematical Sciences, 43(2), 277-302.

haarpy.quantum.haar_integral_free_symmetric(sequences, group_dimension)[source]#

Returns the integral of the free symmetric group under the Haar measure

Parameters:

sequences (tuple[tuple[int, ...], ...]) – Sequences of matrix elements
group_dimension (Symbol) – The dimension of the free symmetric group

Returns:

The integral under the Haar measure

Return type:

Expr

Examples

>>> from sympy import Symbol
>>> from haarpy import haar_integral_free_symmetric
>>> d = Symbol("d")
>>> sequences = ((0, 1, 2), (2, 1, 0))
>>> haar_integral_free_symmetric(sequences, d)
1/(d*(d - 2)*(d - 1))
>>> haar_integral_free_symmetric(sequences, 4)
1/24

See also

haarpy.partition.gram_matrix(): Generates the Gram matrix of a given input set of partitions
haarpy.partition.non_crossing_partitions(): Yields non crossing partitions of the set \([n] = \{1,2,...,n\}\)

haarpy.quantum.haar_integral_free_orthogonal(sequences, group_dimension)[source]#

Returns the integral of the free orthogonal group under the Haar measure

Parameters:

sequences (tuple[tuple[int, ...], ...]) – Sequences of matrix elements
group_dimension (Symbol) – The dimension of the free orthogonal group

Returns:

The integral under the Haar measure

Return type:

Expr

Examples

>>> from sympy import Symbol
>>> from haarpy import haar_integral_free_symmetric
>>> d = Symbol("d")
>>> sequences = ((0, 1, 1, 0), (0, 0, 1, 1))
>>> haar_integral_free_symmetric(sequences, d)
-1/(d*(d - 1)*(d + 1))
>>> haar_integral_free_symmetric(sequences, 4)
-1/60