Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [group-theory]

Questions on the group-theoretic functionality of Mathematica.

2
votes
1answer
30 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
37 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
33 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
38 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
27 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
108 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
82 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 ...
4
votes
0answers
40 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
95 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
105 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
37 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
73 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
0answers
20 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
36 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
152 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
42 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
30 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)...
2
votes
1answer
113 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{...
1
vote
0answers
98 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
62 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
65 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 <...
7
votes
2answers
330 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
36 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
71 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/...
2
votes
0answers
75 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
23 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: ...
2
votes
0answers
70 views

Listing all subgroups

Here's how to show all subgroups of $S_4$ in GAP: ...
0
votes
1answer
56 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
67 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
36 views

Combinatorica and MultiplicationTable in Mathematica 11

I try to use <<Combinatorica` or call IntervalSlider; Needs["Combinatorica`"] or call ...
2
votes
0answers
35 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
32 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 &...
0
votes
0answers
46 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
133 views

Proving uniqueness of group identity element

Start, as in the Mathematica 11.3 documentation, with: ...
1
vote
1answer
168 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),(...
0
votes
0answers
186 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
62 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?
4
votes
0answers
58 views

Wreath products of groups in Mathematica?

Looking online, I found this external package that can compute e.g. the CycleIndexPolynomial of wreath products of symmetry groups. Considering that this package is ...
5
votes
3answers
146 views

Generate all dihedral transformations of a matrix?

Given a matrix, e.g. matrix = Table[a[i, j], {i, 1, 3}, {j, 1, 3}]; I would like to have a function that takes matrix as ...
2
votes
1answer
53 views

Unified class of an object type “Group”?

Does Mathematica support an unified class for "group-type" objects? Or, less general, for groups with a fixed defined representation in Mathematica? For example: ...
10
votes
6answers
764 views

Is there concise code for the list operation I want to perform?

Is there any concise syntax for the following partitioning of a list. Given {1, 2, 3, 4}, I want to get the output as shown below. I have tried various function ...
1
vote
0answers
157 views

Fox Calculus/Fox Derivative

Is there a way to compute the Fox derivative in Mathematica? I know there are some methods to do it in Sage, but I'd like to use Mathematica instead. Any suggestions?
3
votes
2answers
136 views

How can I output list of permutation products?

I have A = Permutations[{1, 2, 3, 4}]. And c = Cycles[{{1, 2, 3, 4}}]. And I need to output for all $a \in A: a \cdot с \cdot a^{...
5
votes
1answer
145 views

How find out whether a graph is vertex transitive?

How can I use Mathematica to find out whether a graph is vertex transitive?
1
vote
0answers
137 views

display the elements of a quotient ring

How can I display the elements (polynomials) of the ring $$\mathbb Z_3[x]/\langle x^3+2x^2+1\rangle$$ Is there a built-in function that displays them?
2
votes
1answer
141 views

How to get the `CycleIndexPolynomial` of direct product of two symmetric groups?

If I want to obtain the CycleIndexPolynomial of the symmetric group $S_n$, I can just do i.e. ...
5
votes
1answer
255 views

Group Elements in Mathematica v10 using (x,y,z) notation

I have a group generated by this condition G = {(x,y,z): x^3=y^3=z^3=e}} and I want to compute all the possible elements i.e ...
3
votes
1answer
613 views

Hadamard Lemma and commutators algebra

I would like to implement the following formula, which goes under the name of Hadamard Lemma: $ e^A \, B \, e^{-A} = \sum_{k=0}^{+\infty} \frac{1}{k!} [A,B]_k $ where $ [A,B]_0 = B , \...
2
votes
1answer
259 views

Automorphism group of a cycle graph in Mathematica

From this lecture note, it is known that the automorphism group of the $n$-cycle $C_n$ is the dihedral group $D_n$ with $2 n$ elements. I use the following code to create a cycle graph of $6$ nodes. ...
0
votes
1answer
521 views

Finding the group of residue classes modulo n under multiplication

Is the group $(\mathbb{Z}/n\mathbb{Z})^\times = U(n)$ = the group of residue classes mod n under multiplication built into Mathematica? As a first step, I want Mathematica to list the elements of ...