Questions tagged [group-theory]
Questions on the group-theoretic functionality of Mathematica.
108
questions
1
vote
0answers
44 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
42 views
What is the cycle index polynomial for the trivial group
CycleIndexPolynomial[Cycles[{}], {Subscript[x, 1]}]
returns 1.
I was expecting ...
2
votes
2answers
88 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
18 views
Which groups exactly include the class “ComputedFiniteGroup”?
Which groups exactly include the class "ComputedFiniteGroup" ?
Help says "A finite group derived by computation", but it's not very specific. Moreover, the examples given in the help mostly show ...
0
votes
0answers
65 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
35 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
41 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
36 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
42 views
The product of two symmetric groups acting on a function
Considering the rational function
$$
\small
\begin{align*}
f&(x_1,x_2,x_3;y_1,y_2,y_3)\\
&=\frac{\left(1-\frac{y_1}{x_1}\right)\left(1-\frac{y_2}{x_1}\right)\left(1-\frac{y_3}{x_1}\right)\...
2
votes
0answers
29 views
How to iterate over Orbit of PermutationGroup?
Another day, another permutation group question.
I'm given a g=PermutationGroup[...], and a list l={1, 3, 3, 2, ...}. I know I ...
3
votes
1answer
206 views
Definition of WignerD function?
On Wikipedia, elements of Wigner's D-matrix are defined as
$$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
97 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 ...
5
votes
0answers
86 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
111 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
117 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
47 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:
$$\begin{bmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 \\
0 & 0 & 1 &...
4
votes
1answer
75 views
What does this graph mean?
Mathematica 11.3.0 includes a new command FindEquationalProof having good prospects. Studying it, I consider a somewhat modified example from the help
...
3
votes
0answers
21 views
Not too obsolete package extending the built-in functionality of Permutation Groups
Is there a modern package geared to permutation groups? With functions, for instance, for group-isomorphism (either w.r.t. permutation representation & w.r.t. group structure), direct, semidirect ...
1
vote
0answers
48 views
How to form sets of non-adjacent subgraphs from all possible subgraphs and the adjacency matrix?
I have connected graph with $72$ nodes.
The binary adjacency matrix of the connected graph is $A$. The size of $A$ is $72\times 72$ and its symmetric.
I have generated all possible subgraphs ($17000$...
1
vote
1answer
165 views
How to form subgraphs from a connected graph
I have a connected graph with 72 nodes and its binary adjacency matrix.
I need to find all possible connected subgraphs with exactly 6 nodes, i.e., I need to find all connected subgraphs and each ...
1
vote
0answers
45 views
Function ellipticAdd to add points belonging to an elliptic curve $y^2 = x^3 + a\,x^2 + b\,x$ [duplicate]
In the EllipticLog WRI web page, there is a function called"ellipticMultiply defined as:
...
2
votes
1answer
36 views
How to specify discrete group through permutation cycles
I would like to define a group of order 12 in Mathematica by giving a list of 12 elements in cycle notation,
$$ S = \{(e), (123), (132), (12)(45), (13)(45), (23)(45), (13), (23), (12), (45), (123)(45)...
2
votes
1answer
132 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
125 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
75 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
76 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
346 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
37 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
94 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
98 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
25 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
97 views
0
votes
1answer
60 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
47 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.
$$e =\begin{pmatrix}
1 & 0 \\
0 & 1
\end{...
1
vote
0answers
38 views
Solving some constraints on a subgroup of a Lie group [closed]
Let $M$ be a rank-3 matrix, I am interested in searching all the group elements $g \in$ SU(3) Lie group, such that,
$$
g^T M g =M.
$$
Example 1. Let
$$
M[1]=
\left(
\begin{array}{ccc}
0 & 1 &...
0
votes
0answers
57 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
144 views
Proving uniqueness of group identity element
Start, as in the Mathematica 11.3 documentation, with:
...
1
vote
1answer
184 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
297 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
64 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
66 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
154 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
55 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
771 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
187 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
143 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
162 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
145 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?
