Questions tagged [group-theory]
Questions on the group-theoretic functionality of Mathematica.
141
questions
0
votes
0
answers
35
views
Make GroupOrbits recognise same orbit
When going through a list of elements, GroupOrbits usually recognises if the orbit of an element has been calculated already and will skip it in that case to avoid ...
- 143
2
votes
1
answer
77
views
How to determine if a group H is a normal subgroup of group G?
A subgroup $H$ of the group $G$ is normal group in $G$ if and only if $\displaystyle ghg^{-1}\in H$ for all $\displaystyle g\in G$ and $\displaystyle h\in H$. How to use MMA to know the group $H$ is a ...
- 23.1k
1
vote
1
answer
51
views
Finding double cosets of a group
I am new to Mathematica, and I would love to know if there is a function that that returns the double cosets of a group $G$ w.r.t a subgroup $H$, where a double coset is defined as $HxH=\{h_1xh_2,\...
- 11
1
vote
1
answer
50
views
How to convert a PermutationGroup to a named group
We can convert any built-in named group into PermutationGroup by this code(such as AlternatingGroup[5]):
...
- 23.1k
3
votes
2
answers
106
views
How to know the Galois Group of a polynomial is a solvable group?
ResourceFunction["StauduharGaloisGroup"][2 x^5+3 x^4+10 x^3+15 x^2+8 x+12,x]["GaloisGroup"]
...
- 23.1k
2
votes
1
answer
101
views
How to draw a cycle graph of a group?
MMA can plot a Cayley graph by CayleyGraph directly, which can help us to visualize the group:
...
- 23.1k
3
votes
3
answers
191
views
How to get the action diagram when the group act on a set?
I previously thought the orbital map would achieve this, but as the current answer or discuss in the comment. I realized that was wrong. And the Close behavior advice tells me "Needs details or ...
- 23.1k
8
votes
2
answers
255
views
How to find all left cosets about a subgroup?
I have a group:
group = PermutationGroup[{Cycles[{{1, 3, 6}, {2, 4}}], Cycles[{{6, 7}}]}];
And I have s subgroup
...
- 23.1k
6
votes
1
answer
70
views
2
votes
0
answers
56
views
Are there irreducible representations of the crystallographic groups
This yields the list of the irreducible representations of the point crystallographic group Oh:
...
- 34.8k
3
votes
1
answer
63
views
Issues in FiniteGroupData Character Tables
Using $Version == "12.3.0 for Mac OS X x86 (64-bit) (May 10, 2021)" and executing
...
- 5,314
3
votes
2
answers
103
views
Replacing multiple variables according to the index
I'm currently working on the Polya's Enumeration Theorem implementation in Mathematica. As an example, of what I want to do, here's a formula I'm working with:
$$P_{C_{12}}(x_1,\ldots,x_{12})=\frac{1}{...
2
votes
0
answers
56
views
Reducing to Irreducible Representations
Group Theory Background / Utilities
Suppose I give you a list G of matrices which represent some group, in that the matrices are closed under multiplication. In ...
- 5,314
4
votes
1
answer
154
views
CharacterTable for symmetric groups $S_n$ with large $n$
I am looking for a package to generate character tables for symmetric groups $S_n$. At this moment I am using
...
- 361
0
votes
1
answer
50
views
Solving abstract factorization problem with Mathematica
I'm a Mathematica newbie.
Given something like:
$$(r_0 + r_1 a + r_2 a^2 + r_3 a^5)(r'_0 + r'_1b + r'_2 b^2 + r'_3 b^5) = 1 - a^5 b^5$$
I'd like to determine if there exist $r_i,r'_i$ that solve the ...
- 103
3
votes
0
answers
68
views
Packages for study presentation of groups
This question asks if there are tools/packages in Mathematica for the study of presentation of groups, but it is almost 7 years old and an answer suggests to use Combinatorica package which is now ...
- 6,104
7
votes
2
answers
212
views
How Can I Compute The Automorphism Group of a Matroid?
I want to compute the automorphism group of a matroid. This reduces to the following (more general) problem:
Suppose I have a list of sets $\{\{b_{11},\dots,b_{1k}\},\dots,\{b_{k1},\dots,b_{kk}\}\}$ ...
- 695
4
votes
1
answer
51
views
How to decompose a 5-cycles into a permutationproduct of two 3-cycles?
We know that every 3-cycles can be expressed as the product of two commutations.
Cycles[{{1, 2, 3}}] ==
PermutationProduct[Cycles[{{1, 3}}], Cycles[{{3, 2}}]]
In ...
1
vote
0
answers
26
views
3
votes
3
answers
169
views
How to delete duplicate graphics of the same kind?
A054247: Number of n X n binary matrices under action of dihedral group of the square D_4.
...
4
votes
2
answers
304
views
How to correctly enumerate all the schemes of this cube coloring problem?
This problem is the fifth question of 1996 Chinese High School Mathematics League or Chinese Mathematical Olympiad in Senior:
Choose several colors from the given six different colors to dye six faces ...
0
votes
1
answer
73
views
How to create a set (of matrices) which will be used as an finite group to minimize a function?
I'm new here so I'm a little lost.
I need to minimize a function considering that the minimizing parameter belongs to a preestablished set. It all involves matrices.It's something like this:
Where ...
3
votes
2
answers
91
views
Orbits of a set $X$ under the action of cyclic permutation $T$
Let $X$ be a set defined as
$$X = \{\{\sigma_1, \dots, \sigma_L\} \;|\; \sigma_i = 0,\dots ,n-1\}.$$ Furthermore, let $T:X\longrightarrow X$ be a cyclic permutation
$$ T\cdot\{\sigma_1, \dots, \...
- 233
5
votes
1
answer
187
views
Finding invariant matrix given group elements
I have a question about speeding up / optimising the following calculation. I have a feeling there is a way to rewrite it but I can't quite see what to do.
In $d=3$ dimensions, given a set of ...
- 109
2
votes
0
answers
45
views
Does Mathematica limit the size of a set of permutation group generators?
I tried to generate general and special linear groups PGL2 and PSL2 over finite fields using straightforward algorithms over finite fields: The generators are permutations defined as the additive ...
- 1,730
6
votes
1
answer
82
views
Finding sequence of group generators that yields group element
Here is my permutation group that acts on lists of length 4, defined in terms of four generators:
...
- 19k
5
votes
2
answers
200
views
Symmetry unique atom coordinates
One thing I love about Mathematica is how easily I can go from the name of a molecule to estimated coordinates of its atoms, with a command like
...
- 277
5
votes
0
answers
54
views
How to compute with Mathematica the space representation of a group for a given point set?
A set of points $P = \{p_1,p_2,\dots\} \subset \mathbb{R}^3$ in 3D is given. A symmetry transformation $S \in \mathbb{R}^{3 \times 3}$ is considered here as a matrix which maps the elements of $P$ to ...
- 5,283
0
votes
1
answer
139
views
How to solve problem 599 of Project-Euler
Problem 599 of the Project-Euler is described as follows:
The well-known Rubik's Cube puzzle has many fascinating mathematical properties. The 2×2×2 variant has 8 cubelets with a total of 24 visible ...
4
votes
1
answer
124
views
Calculating Rubiks $ 2 \times 2 \times 2 $ Permutation using Cycles
The help page of PermutationGroup shows a neat example on calculating the permutations of a $3\times 3\times 3$:
...
- 219
6
votes
4
answers
271
views
Illustration of Motzkin numbers: how to delete duplicates
The $n$-th Motzkin number is the number of different ways of drawing non-intersecting chords between $n$ points on a circle (not necessarily touching every point by a chord -- see https://en.wikipedia....
- 3,644
4
votes
2
answers
365
views
How to use Mathematica to prove that isotropic materials have only two independent parameters
Posts on related issues can be found from here or here.
Index symmetries:
A stiffness tensor $C$ is a fourth-order tensor with components $c_{ijkl}$ which maps symmetric second-order tensors into ...
2
votes
1
answer
89
views
Delete duplicates when cycle both position and element
Cycle of position means: {a, b, a, a} and {a, a, b, a} is the same.
Cycle of element means: ...
- 3,644
1
vote
0
answers
60
views
How to evaluate an expression [closed]
I have a 3-dimensional array of numbers say A. I want to evaluate the following expression.
$\displaystyle\sum_{\sigma,\tau \in S_6}(\Pi_{i=1}^6A(i,\sigma(i),\tau(i)))$. $S_6$ is the symmetric group ...
- 111
2
votes
1
answer
48
views
What is the cycle index polynomial for the trivial group
CycleIndexPolynomial[Cycles[{}], {Subscript[x, 1]}]
returns 1.
I was expecting ...
- 573
3
votes
2
answers
281
views
How to remove global phase from matrices
I have a list of 3 by 3 complex matrices. I would like to remove the matrices that are similar up to a global phase factor.
For instance, I have $ M, \mathrm e^{\mathrm i2\pi/3}M $ and $ \mathrm e^{-\...
- 33
0
votes
0
answers
71
views
How to list all the possible functions of given structure?
I can run this command :
Tuples[{ Tuples[{x, y, z}, 2], {a, b, c}}]
to list all combinations that looks like:
...
- 101
2
votes
1
answer
38
views
$2+2+\dots+2$ cycle conjugation classes of the symmetric group $S_n$
Command GroupElements[SymmetricGroup[4]] gives me all cycles of all conjugation classes. But I'm only interested in the 2+2 class ($2+2$ being one of the integer ...
- 23
1
vote
1
answer
58
views
What is the mathematical operation corresponding to this definition of the Dirichlet Character group operation?
The Wolfram Language & System Documentation Center page for DirichletCharacter indicates Dirichlet characters modulo k form ...
- 238
1
vote
1
answer
42
views
Show result of matrix operation in terms of user defined matrices
I have defined some matrices K[0],K[1],...,K[9] and S[1],...
- 11
3
votes
1
answer
52
views
The product of two symmetric groups acting on a function
Considering the rational function
$$
\small
\begin{align*}
f&(x_1,x_2,x_3;y_1,y_2,y_3)\\
&=\frac{\left(1-\frac{y_1}{x_1}\right)\left(1-\frac{y_2}{x_1}\right)\left(1-\frac{y_3}{x_1}\right)\...
- 251
2
votes
0
answers
46
views
How to iterate over Orbit of PermutationGroup?
Another day, another permutation group question.
I'm given a g=PermutationGroup[...], and a list l={1, 3, 3, 2, ...}. I know I ...
- 141
8
votes
1
answer
605
views
Definition of WignerD function?
On Wikipedia, elements of Wigner's D-matrix are defined as
$$D_{m'm}^{j}(\alpha,\beta,\gamma)=\langle jm'|e^{-i\alpha J_z}e^{-i\beta J_y}e^{-i\gamma J_z}|jm\rangle=e^{-im'\alpha}d_{m'm}^j (\beta)e^{-...
- 233
4
votes
2
answers
120
views
The set of polynomials under the action by a symmetric group
Let
$$f(x_1,x_2,x_3)=\frac{x_1^r x_2^r \left(1-x_1 x_3\right) \left(1-x_2 x_3\right)}{\left(1-\frac{x_2}{x_1}\right) \left(1-\frac{x_3}{x_1}\right)\left(1-\frac{x_3}{x_2}\right)},$$
where $r$ is a ...
- 251
7
votes
1
answer
315
views
Irreducible representations and conjugacy classes for the octahedral group
We need a few things about the octahedral group (including reflection, perhaps even using double cover later on):
Conjugacy classes
Irreducible representations (irreps) in the form of matrices. Just ...
- 265
2
votes
1
answer
147
views
How to create a group via group relations?
I have a group described by group relations. For simplicity group has a presentation $<a | a^n = e>$. How can I create this group in Mathematica?
6
votes
1
answer
162
views
Generate list of tuples, modulo PermutationGroup
I have a permutation group, e.g.
g = PermutationGroup[{Cycles[{{1, 2}}]}]
but not necessarily limited to a single generating cycle.
What I want is to create a ...
- 141
1
vote
1
answer
84
views
Find a matrix $X$ that block-diagonalizes a particular group of matrices
Essentially, I want to find a single matrix $X$ such that conjugation by $X$ sends:
$$\begin{bmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 \\
0 & 0 & 1 &...
- 160
4
votes
1
answer
91
views
What does this graph mean?
Mathematica 11.3.0 includes a new command FindEquationalProof having good prospects. Studying it, I consider a somewhat modified example from the help
...
- 16.6k
3
votes
1
answer
56
views
Not too obsolete package extending the built-in functionality of Permutation Groups
Is there a modern package geared to permutation groups? With functions, for instance, for group-isomorphism (either w.r.t. permutation representation & w.r.t. group structure), direct, semidirect ...
- 1,730