Questions tagged [group-theory]
Questions on the group-theoretic functionality of Mathematica.
33
questions
3
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2
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What is the command to find function invariant?
What is the command to find function invariant?
http://demonstrations.wolfram.com/AFunctionInvariantUnderAGroupOfTransformations/
what is algorithm it use to calculate this?
Edit
there is a book ...
7
votes
2
answers
8k
views
Defining a non-commutative operator algebra in Mathematica
I am trying to develop a function(s) to do some commutator algebra to compute the enveloping algebra and ideals of a Lie algebra. For example if I have $SU(2)$ algebra generated by $L_i$ ($i=1,2,3$), ...
14
votes
4
answers
4k
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Finding elementary cycles of (directed) graphs
I need to enumerate all the simple cycles (i.e. elementary cycles where no vertex is repeated other than the starting) of a graph which has both directed and undirected edges, where we can treat the ...
19
votes
4
answers
3k
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Symmetry-finding packages
Where can I find the most up-to-date or whatever you consider to be the most useful symmetry-finding package for differential equations? I do not intend to restrict to, but would like to include those,...
7
votes
6
answers
491
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Total by a criteria
I am developing a weighted KNN algorithm. In a step, I need to do the sum of weights of each class. For example:
...
10
votes
2
answers
3k
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How to generate a matrix group?
I have three $7\times 7$ matrices (with real entries, lots of zeros) and I'd like to check if they generate a finite group (or, more precisely, if the group they generate is of precise order). Would ...
10
votes
1
answer
277
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How to transform abstract finite group to permutation group?
It seems that Mathematica only has group functionality for permutation groups? Then there is a step to transform the abstract finite one to a permutation one.
As an example, consider the following ...
8
votes
2
answers
796
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Plotting All Possible Points Belonging to a Group Orbit
Given that $$X = \{(x,y,z) \in \mathbb{R}^3 |\, x^2 + y^2 + z^2 - 2(xy + xz + yz) = k\}\,,$$ where $k$ is a constant. Also given that a group $G$ is represented by $$\langle g_1,g_2,g_3|\, g_1^2 = g_2^...
6
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2
answers
3k
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Package for representation of groups (group theory)
I wanted to know if there is a package which allows to compute representations of a group like the definition representation, adjoint and so on (for example the Pauli matrix for $SU(2)$ if I specify ...
5
votes
3
answers
275
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How the solve the parameter of the conjugate permutations
As we know the definition of conjugate permutations is:
$$\exists p \quad p^{-1} \alpha p=\beta$$
When I have an alpha=Cycles[{{1,4},{2,5,6,3}}] and a ...
5
votes
2
answers
749
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Mathematica 11 Symmetry Packages
Are there any symmetry finding packages (for differential equations) that work with Mathematica 11.1? I've tried MathLie but from looking at the documentation it had instructions for Windows 95... I ...
4
votes
2
answers
441
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How to use Mathematica to prove that isotropic materials have only two independent parameters
Posts on related issues can be found from here or here.
Index symmetries:
A stiffness tensor $C$ is a fourth-order tensor with components $c_{ijkl}$ which maps symmetric second-order tensors into ...
14
votes
2
answers
776
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How can I compute the representation matrices of a point group under given basis functions?
Take the $C_{3v}$ point group for example:
...
9
votes
1
answer
1k
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Definition of WignerD function?
On Wikipedia, elements of Wigner's D-matrix are defined as
$$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^{-...
8
votes
1
answer
567
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Generating abstract group from direct product of two abstract groups
In group theory one can calculate some abstract groups through the direct product of two other abstract groups. An example for such a generation is the product $A_5\times Z_2$ with order 120, or $Z_4\...
7
votes
2
answers
626
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Listing all subgroups
Here's how to show all subgroups of $S_4$ in GAP:
...
6
votes
1
answer
1k
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Irreducible representations for the symmetry group $T_d$
I am trying to get the explicit two dimensional irreducible representation matrices for the symmetry group $T_d$. I need the matrix representation for each element in the group. Are there any ...
5
votes
3
answers
291
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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"]
...
4
votes
3
answers
165
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How to list all subgroups of symmetry group S_6?
I learned from https://oeis.org/A005432 that $S_6$ has $1455$ subgroups, how can I list them all in mathematica?
As the comment says, the direct approach cannot solve the problem.
4
votes
1
answer
168
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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 $3\times 3\times 3$:
...
4
votes
1
answer
349
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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 ...
4
votes
1
answer
138
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4
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2
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398
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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 ...
4
votes
1
answer
2k
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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 , \...
3
votes
1
answer
215
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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:
...
3
votes
3
answers
199
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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.
...
3
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0
answers
178
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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 ...
2
votes
1
answer
75
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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 & ...
2
votes
1
answer
181
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 ...
2
votes
1
answer
156
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 <...
1
vote
1
answer
239
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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:
<...
1
vote
1
answer
452
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Symmetric group action on polynomials
I am working with polynomials in several variables with the obvious action of $S_n$. That is, given a polynomial $f$ in the variables $x_1, \dots, x_n$, a permutation $\sigma \in S_n$ acts on $f$ by ...
0
votes
1
answer
67
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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 ...