Basic¶

numqi implements a minimum set of group theory operations required in other numqi modules. It is based on the sympy library.

Cayley table (multiplication table): a table of all possible products of group elements.

Left regular form: a permutation representation of a group.

One featured function is to give all inequivalent, irreducible representations of a finite group.

TODO

1. analytical results for Dihedral group and cyclic group (@book-ZhongqiMA)
2. su(n)
3. space group

In [1]:

import numpy as np

try:
    import numqi
except ImportError:
    %pip install numqi
    import numqi

import numpy as np try: import numqi except ImportError: %pip install numqi import numqi

Irreducible representations of $S_3$¶

In [2]:

N0 = 3
cayley_table = numqi.group.get_symmetric_group_cayley_table(N0)

print(f'Cayley Table:', cayley_table, sep='\n')
print(f'shape: {cayley_table.shape}, dtype: {cayley_table.dtype}')
print('#group elements:', cayley_table.shape[0])

N0 = 3 cayley_table = numqi.group.get_symmetric_group_cayley_table(N0) print(f'Cayley Table:', cayley_table, sep='\n') print(f'shape: {cayley_table.shape}, dtype: {cayley_table.dtype}') print('#group elements:', cayley_table.shape[0])

Cayley Table:
[[0 1 2 3 4 5]
 [1 0 4 5 2 3]
 [2 3 0 1 5 4]
 [3 2 5 4 0 1]
 [4 5 1 0 3 2]
 [5 4 3 2 1 0]]
shape: (6, 6), dtype: int64
#group elements: 6

Above shows the Cayley table of Symmetric group $S_3$. The group has 6 elements $g_0,g_1,g_2,g_3,g_4,g_5$, and the Cayley table is a 6x6 matrix. Each row of Cayley table represents one element of the group. For example, the third element of the group $g_2$ is represented as [2,3,0,1,5,4], which means $g_2g_0=g_2$, $g_2g_1=g_3$, $g_2g_2=g_0$, $g_2g_3=g_1$, $g_2g_4=g_5$, and $g_2g_5=g_4$.

Below, we first convert the Cayley table to the left regular form, which is a permutation representation of the group. Then we find all inequivalent, irreducible representations (irrep) of the group using the algorithm from Sheaves-blog

In [3]:

left_regular_form = numqi.group.cayley_table_to_left_regular_form(cayley_table)
irrep_list = numqi.group.reduce_group_representation(left_regular_form)
# irrep_list
dim_list = [x.shape[1] for x in irrep_list]
for ind0,irrep in enumerate(irrep_list):
    print(f'irrep[{ind0}]:', irrep.shape)

left_regular_form = numqi.group.cayley_table_to_left_regular_form(cayley_table) irrep_list = numqi.group.reduce_group_representation(left_regular_form) # irrep_list dim_list = [x.shape[1] for x in irrep_list] for ind0,irrep in enumerate(irrep_list): print(f'irrep[{ind0}]:', irrep.shape)

irrep[0]: (6, 1, 1)
irrep[1]: (6, 1, 1)
irrep[2]: (6, 2, 2)

Above, irrep_list is a list of 3-dimensional array, each array is of shape (#element, dim(irrep), dim(irrep)). From Cayley table, we know that $g_2g_4=g_5$ and we can also verify this from irrep.

In [4]:

for ind0, irrep in enumerate(irrep_list):
    print(f'irrep[{ind0}]')
    g2 = irrep[2]
    g4 = irrep[4]
    g5 = irrep[5]
    print(f'g2:', g2)
    print(f'g4:', g4)
    print(f'g5:', g5)
    print(f'g2*g4:', g2 @ g4, '\n')

for ind0, irrep in enumerate(irrep_list): print(f'irrep[{ind0}]') g2 = irrep[2] g4 = irrep[4] g5 = irrep[5] print(f'g2:', g2) print(f'g4:', g4) print(f'g5:', g5) print(f'g2*g4:', g2 @ g4, '\n')

irrep[0]
g2: [[-1.]]
g4: [[1.]]
g5: [[-1.]]
g2*g4: [[-1.]] 

irrep[1]
g2: [[1.]]
g4: [[1.]]
g5: [[1.]]
g2*g4: [[1.]] 

irrep[2]
g2: [[-0.5       -0.8660254]
 [-0.8660254  0.5      ]]
g4: [[-0.5       -0.8660254]
 [ 0.8660254 -0.5      ]]
g5: [[-0.5        0.8660254]
 [ 0.8660254  0.5      ]]
g2*g4: [[-0.5        0.8660254]
 [ 0.8660254  0.5      ]] 

From irrep_list, we can obtain the character table of the group, which is a 2-dimensional array of shape (#irrep, #class). According to ((TODO-theorem)) stackexchange-link, number of inequivalent irreducible representations #irrep equals to number of conjugacy classes #class.

In [5]:

character,class_list,character_table = numqi.group.get_character_and_class(irrep_list)
print(np.array2string(character_table, precision=3))

character,class_list,character_table = numqi.group.get_character_and_class(irrep_list) print(np.array2string(character_table, precision=3))

[[ 1.00e+00  1.00e+00 -1.00e+00]
 [ 1.00e+00  1.00e+00  1.00e+00]
 [ 2.00e+00 -1.00e+00 -1.11e-16]]

The following function can print the character table in a more readable format (rounded to nearest integer).

In [6]:

numqi.group.pretty_print_character_table(character_table, class_list)

numqi.group.pretty_print_character_table(character_table, class_list)

| $\chi$ | 1 | 2 | 3 |
| :-: | :-: | :-: | :-: |
| $A_{0}$ | 1 | 1 | -1 |
| $A_{1}$ | 1 | 1 | 1 |
| $A_{2}$ | 2 | -1 | 0 |

NumQI have several more built-in finite groups, and user can also input their own Cayley table to find the irreducible representations.

In [7]:

# cayley_table = numqi.group.get_multiplicative_group_cayley_table(N0)
# cayley_table = numqi.group.get_symmetric_group_cayley_table(N0, alternating=True)
# cayley_table = numqi.group.get_dihedral_group_cayley_table(N0)
# cayley_table = numqi.group.get_cyclic_group_cayley_table(N0)
# cayley_table = numqi.group.get_klein_four_group_cayley_table()
# cayley_table = numqi.group.get_quaternion_cayley_table()

# cayley_table = numqi.group.get_multiplicative_group_cayley_table(N0) # cayley_table = numqi.group.get_symmetric_group_cayley_table(N0, alternating=True) # cayley_table = numqi.group.get_dihedral_group_cayley_table(N0) # cayley_table = numqi.group.get_cyclic_group_cayley_table(N0) # cayley_table = numqi.group.get_klein_four_group_cayley_table() # cayley_table = numqi.group.get_quaternion_cayley_table()

Character table¶

Here, we show the character tables solved by NumQI.

Klein four-group¶

wiki-link

$\chi$	1	1	1	1
$A_0$	1	-1	-1	1
$A_1$	1	-1	1	-1
$A_2$	1	1	-1	-1
$A_3$	1	1	1	1

Symmetric group¶

wiki-link

$S_n$	order	dim(irrep)
$S_2$	2	1,1
$S_3$	6	1,1,2
$S_4$	24	1,1,2,3,3
$S_5$	120	1,1,4,4,5,5,6
$S_6$	720	1,1,5,5,5,5,9,9,10,10,16

$S_2$

$\chi$	1	1
$A_0$	1	-1
$A_1$	1	1

$S_3$

$\chi$	1	2	3
$A_0$	1	1	-1
$A_1$	1	1	1
$A_2$	2	-1	0

$S_4$

$\chi$	1	3	6	6	8
$A_0$	1	1	-1	-1	1
$A_1$	1	1	1	1	1
$A_2$	2	2	0	0	-1
$A_3$	3	-1	-1	1	0
$A_4$	3	-1	1	-1	0

$S_5$

$\chi$	1	10	15	20	20	24	30
$A_0$	1	-1	1	1	-1	1	-1
$A_1$	1	1	1	1	1	1	1
$A_2$	4	-2	0	1	1	-1	0
$A_3$	4	2	0	1	-1	-1	0
$A_4$	5	-1	1	-1	-1	0	1
$A_5$	5	1	1	-1	1	0	-1
$A_6$	6	0	-2	0	0	1	0

$S_6$

$\chi$	1	15	15	40	40	45	90	90	120	120	144
$A_0$	1	-1	-1	1	1	1	-1	1	-1	-1	1
$A_1$	1	1	1	1	1	1	1	1	1	1	1
$A_2$	5	3	-1	2	-1	1	1	-1	0	-1	0
$A_3$	5	-1	3	-1	2	1	1	-1	-1	0	0
$A_4$	5	-3	1	2	-1	1	-1	-1	0	1	0
$A_5$	5	1	-3	-1	2	1	-1	-1	1	0	0
$A_6$	9	-3	-3	0	0	1	1	1	0	0	-1
$A_7$	9	3	3	0	0	1	-1	1	0	0	-1
$A_8$	10	2	-2	1	1	-2	0	0	-1	1	0
$A_9$	10	-2	2	1	1	-2	0	0	1	-1	0
$A_{10}$	16	0	0	-2	-2	0	0	0	0	0	1

Alternating group¶

wiki-link

$A_2=\lbrace e\rbrace$

$A_3=C_3$

$A_n$	order	dim(irrep)
$A_2$	1	1
$A_3$	3	1,1,1
$A_4$	12	1,1,1,3
$A_5$	60	1,3,3,4,5
$A_6$	360	1,5,5,8,8,9,10

$A_3$: $a=e^{i2\pi/3}$

$\chi$	1	1	1
$A_{0}$	1	$a$	$a^{-1}$
$A_{1}$	1	$a^{-1}$	$a$
$A_{2}$	1	1	1

$A_4$: $a=e^{i2\pi/3}$

$\chi$	1	3	4	4
$A_{0}$	1	1	$a$	$a^{-1}$
$A_{1}$	1	1	$a^{-1}$	$a$
$A_{2}$	1	1	1	1
$A_{3}$	3	-1	0	0

$A_5$: $a=\frac{\sqrt{5}-1}{2}\simeq 0.618$

$\chi$	1	12	12	15	20
$A_{0}$	1	1	1	1	1
$A_{1}$	3	$-a$	$a^{-1}$	-1	0
$A_{2}$	3	$a^{-1}$	$-a$	-1	0
$A_{3}$	4	-1	-1	0	1
$A_{4}$	5	0	0	1	-1

$A_6$: $a=\frac{\sqrt{5}-1}{2}\simeq 0.618$

$\chi$	1	40	40	45	72	72	90
$A_{0}$	1	1	1	1	1	1	1
$A_{1}$	5	-1	2	1	0	0	-1
$A_{2}$	5	2	-1	1	0	0	-1
$A_{3}$	8	-1	-1	0	$-a$	$a^{-1}$	0
$A_{4}$	8	-1	-1	0	$a^{-1}$	$-a$	0
$A_{5}$	9	0	0	1	-1	-1	1
$A_{6}$	10	1	1	-2	0	0	0

Dihedral group¶

wiki-link

$D_n$	order	dim(irrep)
$D_3$	6	1,1,2
$D_4$	8	1,1,1,1,2
$D_5$	10	1,1,2,2
$D_6$	12	1,1,1,1,2,2
$D_7$	14	1,1,2,2,2
$D_8$	16	1,1,1,1,2,2,2

$D_3$ ($S_3$)

$\chi$	1	2	3
$A_0$	1	1	-1
$A_1$	1	1	1
$A_2$	2	-1	0

$D_4$

$\chi$	1	1	2	2	2
$A_0$	1	1	-1	-1	1
$A_1$	1	1	-1	1	-1
$A_2$	1	1	1	-1	-1
$A_3$	1	1	1	1	1
$A_4$	2	-2	0	0	0

$D_5$

$a=\frac{\sqrt{5}-1}{2}\simeq 0.618$

$\chi$	1	2	2	5
$A_0$	1	1	1	-1
$A_1$	1	1	1	1
$A_2$	2	$-a$	$a^{-1}$	0
$A_3$	2	$a^{-1}$	$-a$	0

$D_6$

$\chi$	1	1	2	2	3	3
$A_0$	1	-1	-1	1	-1	1
$A_1$	1	-1	-1	1	1	-1
$A_2$	1	1	1	1	-1	-1
$A_3$	1	1	1	1	1	1
$A_4$	2	2	-1	-1	0	0
$A_5$	2	-2	1	-1	0	0

$D_7$

$a=2\cos(\frac{2\pi}{7})\simeq 1.247$

$b=\frac{\sin(5\pi/7)}{\sin(\pi/7)}\simeq 1.802$

$c=\frac{1}{a+1}\simeq 0.445$

$\chi$	1	2	2	2	7
$A_0$	1	1	1	1	-1
$A_1$	1	1	1	1	1
$A_2$	2	$-b$	$a$	$-c$	0
$A_3$	2	$-c$	$-b$	$a$	0
$A_4$	2	$a$	$-c$	$-b$	0

$D_8$

$\chi$	1	1	2	2	2	4	4
$A_0$	1	1	-1	1	-1	-1	1
$A_1$	1	1	-1	1	-1	1	-1
$A_2$	1	1	1	1	1	-1	-1
$A_3$	1	1	1	1	1	1	1
$A_4$	2	-2	$-\sqrt{2}$	0	$\sqrt{2}$	0	0
$A_5$	2	2	0	-2	0	0	0
$A_6$	2	-2	$\sqrt{2}$	0	$-\sqrt{2}$	0	0