Questions tagged [group-theory]

Questions on the group-theoretic functionality of Mathematica.

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6
votes
1answer
53 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: ...
4
votes
2answers
138 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 ...
5
votes
0answers
30 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 ...
0
votes
1answer
105 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 ...
2
votes
1answer
89 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 $3\times 3\times 3$: ...
0
votes
0answers
14 views

Why does FiniteGroupData[{“SymmetricGroup”, 5}, “ConjugacyClasses”] return Missing[NotAvailable]?

This command works as I expected: FiniteGroupData[{"SymmetricGroup", 5}, "ConjugacyClassSizes"] {1, 10, 15, 20, 20, 30, 24} However, this one does not: ...
2
votes
1answer
69 views

Motzkin circles with rotational symmetry

Motzkin Circle means circles that any strings in a circle do not intersect. If we consider rotation symmetry, the rotation coincidence is the same, how to delete duplicate graphics? Following is the ...
4
votes
2answers
264 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
1answer
74 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: ...
1
vote
0answers
49 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 ...
2
votes
1answer
44 views

What is the cycle index polynomial for the trivial group

CycleIndexPolynomial[Cycles[{}], {Subscript[x, 1]}] returns 1. I was expecting ...
2
votes
2answers
168 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^{-\...
0
votes
0answers
69 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: ...
2
votes
1answer
36 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 ...
1
vote
1answer
46 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 a group: ...
1
vote
1answer
38 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],...
3
votes
1answer
47 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)\...
2
votes
0answers
35 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 ...
3
votes
1answer
343 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^{-...
4
votes
2answers
105 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 ...
6
votes
2answers
186 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 ...
2
votes
1answer
124 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
1answer
138 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 ...
1
vote
1answer
67 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 &...
4
votes
1answer
79 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 ...
3
votes
1answer
48 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
vote
0answers
59 views

How to form sets of non-adjacent subgraphs from all possible subgraphs and the adjacency matrix?

I have connected graph with $72$ nodes. The binary adjacency matrix of the connected graph is $A$. The size of $A$ is $72\times 72$ and its symmetric. I have generated all possible subgraphs ($17000$...
1
vote
1answer
179 views

How to form subgraphs from a connected graph

I have a connected graph with 72 nodes and its binary adjacency matrix. I need to find all possible connected subgraphs with exactly 6 nodes, i.e., I need to find all connected subgraphs and each ...
1
vote
0answers
53 views

Function ellipticAdd to add points belonging to an elliptic curve $y^2 = x^3 + a\,x^2 + b\,x$ [duplicate]

In the EllipticLog WRI web page, there is a function called"ellipticMultiply defined as: ...
2
votes
1answer
50 views

How to specify discrete group through permutation cycles

I would like to define a group of order 12 in Mathematica by giving a list of 12 elements in cycle notation, $$ S = \{(e), (123), (132), (12)(45), (13)(45), (23)(45), (13), (23), (12), (45), (123)(45)...
3
votes
1answer
181 views

LieArt — 8 dimensional Irreducible representation of $\mathrm{SO}(8)$ and their decompositions - No.2

This is the followed up question of LieArt --- 3 different 8 dimensional Irreducible representation of SO(8) and their decompositions, Since $$ \mathrm{SO}(8) \supset \mathrm{SU}(2) \times \mathrm{...
4
votes
1answer
177 views

LieArt — 3 different 8 dimensional irreducible representation of $\mathrm{SO}(8)$ and their decompositions

I am using the LieArt which you can download freely online https://arxiv.org/pdf/1206.6379.pdf There are three different 8 dimensional $\mathrm{SO}(8)$ irreducible representations, formally it is ...
1
vote
1answer
96 views

How to generate addition table for $GF(2)[x] \mod x^3 + 1 = 0$ [closed]

I have been playing with Mathematica for a while. I tried generating addition table for a simple ring $R$ such that $R = \mathbb{Z}_{15}$ as asked in my last question. However, I am completely ...
2
votes
1answer
86 views

How to generate addition table for ring $\mathbb Z_{15}$?

How do I generate an addition table for ring R such that $R = \mathbb{Z}_{15}$ or generally speaking, how to generate an addition table for any polynomial ring <...
8
votes
2answers
361 views

How to represent a product of cycles in matrix form?

I have a permutation a in a product of disjoint cycles form as follows $a = {{(1,9,3,7)(2,11,6)(4,8,5,10)}}$ I want to represent it in a matrix form ...
5
votes
0answers
39 views

Efficient Parallelization of PermutationReplace

I am generating the orbit of a partition of a set of size $n$ under the action of the symmetric group $S_n$. PermutationReplace won't allow itself to be parallelized. Why is this ? Is it because the ...
3
votes
0answers
98 views

Package like combinat

I started using Mathematica and want to do some computation involving Characters of the symmetric group. In maple, I used to use the package combinat. The link is below https://www.maplesoft.com/...
3
votes
0answers
116 views

Check the data of representations of Lie groups

Is there a way in Mathematica to check the data of representation of Lie groups? For example, the number of generators. the rank (or the dimension) of the representation matrices. and a list of all ...
1
vote
0answers
27 views

Calculating degrees of representations of sporadic groups

I am trying to calculate a list of the degress of irreducible representations of the sporadic groups, i.e A001379 for groups other than the monster. The GAP code to achieve that would be: ...
3
votes
0answers
142 views

Listing all subgroups

Here's how to show all subgroups of $S_4$ in GAP: ...
0
votes
1answer
62 views

Relabeling/matching variables of big data (in MultiplicationTable)

Let A and B be two sets of tables (from multiplication tables of a group, 24 by 24 as rows by columns), how can we effectively (and possibly also efficiently) find a way to map between them, if two ...
2
votes
1answer
70 views

Relabeling and matching variables (in MultiplicationTable)

Suppose we have a set of data given here as the multiplication table of 8 elements labeled {1,2,3,4,9,10,11,12}: $$ \begin{array}{cccccccc} 1 & 2 & 3 & 4 & 9 & 10 & 11 & ...
0
votes
0answers
54 views

Combinatorica and MultiplicationTable in Mathematica 11

I try to use <<Combinatorica` or call IntervalSlider; Needs["Combinatorica`"] or call ...
2
votes
0answers
36 views

Search the full group $G$ based on the partial list of its matrix representations

Suppose I have a list of matrices that may represent the partial list of the full group $G$. And here are the given set of 7 matrix elements. $$e =\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{...
1
vote
0answers
48 views

Solving some constraints on a subgroup of a Lie group [closed]

Let $M$ be a rank-3 matrix, I am interested in searching all the group elements $g \in$ SU(3) Lie group, such that, $$ g^T M g =M. $$ Example 1. Let $$ M[1]= \left( \begin{array}{ccc} 0 & 1 &...
1
vote
0answers
82 views

Function that Generates all Normal Subgroups of a Group

I am trying to create a function which, given a group structure, generates a list of all the normal subgroups contained within it. But I am not sure how to proceed. For now, I have the following: <...
9
votes
1answer
163 views

Proving uniqueness of group identity element

Start, as in the Mathematica 11.3 documentation, with: ...
1
vote
1answer
216 views

Group table for permutation group

I have got a permutation group: $(\{(e,((1, 4),(2,6),(3,5)),((1, 6),(2,5),(3,4)),((1, 5),(2,4),(3,6)),((1, 3,2),(4,6,5)),((1,2,3),(4,5,6)),((1, 4),(2,5),(3,6)),((1, 3),(4,6)),((2, 3),(5,6)),((1, 2),(...
1
vote
1answer
525 views

Defining an Arbitrary Group

I learned today, while doing my homework, that Mathematica can understand group theory. Combing through the documentation though just gave the examples of permutation groups. I would like to know how ...
-3
votes
1answer
74 views

$GL(n,C)$ as a Lie group [closed]

I understand that $SO(3)$, $SU(3)$ are Lie groups since they are associated with rotations (The S excludes reflection). Why is $GL(n,C)$ also a Lie group?