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Questions tagged [group-theory]

Questions on the group-theoretic functionality of Mathematica.

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The central product and the permutation representation of the Pauli group of order 16

I am interested in obtaining a permutation representation of the Pauli Group $G_1 = \langle X, Y, Z \rangle$. I think this would be easy enough as a "Regular representation" but then I learn ...
Phillip Dukes's user avatar
1 vote
0 answers
48 views

Why is SymmetrizedArray taking so much memory?

The context is: I take the tensor product of 2 totally symmetric tensors of rank 8, in 2 dimensions(in this case there are only 9 independent components): ...
Felipe's user avatar
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1 vote
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Efficient chain rule implementation

(there is a fair amount of context here) I am implementing a generalized chain rule to do some work, say for order 3 in derivatives ($\partial_i = \partial/\partial x_i$): \begin{align} \partial_i \...
Felipe's user avatar
  • 499
5 votes
3 answers
175 views

Calculating the basis set of quotient spaces

Having a polynomial $f(x,y)$, I would like to compute the following quantity \begin{equation*} {\mathbb C}[X,Y,Z]/\langle f_{x}, f_{y}, f_{z} \rangle, \end{equation*} where $f_{x},f_{y},f_{z}$ are, ...
Shasa's user avatar
  • 1,043
2 votes
1 answer
47 views

By what criteria does Mathematica generate the list of group elements in `GroupElements[group]`?

I ask because aside from always giving the identity element first, I have found a "pattern" associated with the list given by GroupElements[group]. This &...
Phillip Dukes's user avatar
3 votes
2 answers
70 views

Bug in Point Group Conjugacy Class Data

I noticed that in $Version == "13.2.0 for Mac OS X x86 (64-bit) (November 18, 2022)" Mathematica has conflicting ...
evanb's user avatar
  • 6,141
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0 answers
43 views

Defining function through a formula

I would like to define a cocycle on a group that maps into the unit circle. That is, I want to be able to define a function $f:G\times G\to\mathbb{T}$ such that $f(e_G,a)=f(a,e_G)=1$ for all $a\in G$ ...
aqwer's user avatar
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1 vote
1 answer
93 views

Elements of a group that send one element to another

If I have a permutation group, say $S_{10}$, how do I get all the permutations that send the set {1, 2, 3} to {5, 6, 7}? I know ...
JRV's user avatar
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4 votes
3 answers
188 views

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.
lapcal's user avatar
  • 531
4 votes
2 answers
78 views

How to find a set of generators with the smallest number of elements in a permutation group?

Given a permutation group with many generators ...
lapcal's user avatar
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4 votes
2 answers
359 views

How to generate all permutation matrices for 4 qubits?

In quantum mechanics, a qubit can be understood as a 2 by 1 vector denoted by Dirac notation as $$|0\rangle \equiv \left( \begin{array}{c} 1\\ 0\\ \end{array} \right) ,|1\rangle \equiv \left( \...
narip's user avatar
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3 votes
1 answer
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Why am I having issues working with large group multiplication tables?

My mathematica running on Wolfram Cloud seems to break down if I use a SymmetricGroup greater than 6. As a basic example, when I run, ...
nj869's user avatar
  • 31
3 votes
2 answers
211 views

Cayley for SL group

In this paper, they are using an expander graph. It seems like it's just a Cayley graph for $SL(2,Z_p)$, where $P$ is a prime number. How do I go about making a Cayley graph as shown in the first of ...
sra's user avatar
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4 votes
1 answer
249 views

How to convert a polynomial into monic form of a polynomial

The function ResourceFunction["StauduharGaloisGroup"] can get a Galois Group about a monic irreducible integer polynomial. But I want to know the Galois ...
yode's user avatar
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5 votes
1 answer
152 views

How to find all the vertices of graph that have the same status as a vertex $v$?

We sometimes say that in a graph $G$ two vertices $a$ and $b$ look the same. In layman's terms, this means that $a$ and $b$ are of same status. Precisely, there exists a auto-isomorphic mapping of $G$...
licheng's user avatar
  • 2,039
4 votes
1 answer
102 views

Obtain a group with desired properties using RandomEntity

This is my current try: ...
yode's user avatar
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4 votes
1 answer
142 views

How can I remove the redundant generators in PermutationGroup?

Consider: ...
yode's user avatar
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1 vote
0 answers
51 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 ...
Thrash's user avatar
  • 405
2 votes
1 answer
182 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 ...
yode's user avatar
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4 votes
1 answer
132 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,\...
fp1's user avatar
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2 votes
1 answer
134 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]): ...
yode's user avatar
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5 votes
3 answers
306 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"] ...
yode's user avatar
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3 votes
1 answer
223 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: ...
yode's user avatar
  • 26.8k
5 votes
5 answers
263 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 ...
yode's user avatar
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12 votes
3 answers
607 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 ...
yode's user avatar
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6 votes
1 answer
83 views

How to hightlight CayleyGraph?

...
yode's user avatar
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2 votes
2 answers
455 views

Source files for the book Exploring Abstract Algebra with Mathematica

I have been looking for the latest version from 2020. The way back machine did not save the file. The files that I’m looking for are : AbstractAlgebraDownloadsV9x.zip And Master.m Here is the link ...
goznevi's user avatar
  • 21
2 votes
0 answers
108 views

Are there irreducible representations of the crystallographic groups

This yields the list of the irreducible representations of the point crystallographic group Oh: ...
Alexei Boulbitch's user avatar
3 votes
1 answer
117 views

Issues in FiniteGroupData Character Tables

Using $Version == "12.3.0 for Mac OS X x86 (64-bit) (May 10, 2021)" and executing ...
evanb's user avatar
  • 6,141
3 votes
2 answers
225 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}{...
mikemykhaylov's user avatar
3 votes
0 answers
188 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 ...
evanb's user avatar
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4 votes
1 answer
453 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 ...
mikis's user avatar
  • 391
0 votes
1 answer
60 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 ...
Ivan Pong's user avatar
  • 103
3 votes
0 answers
116 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 ...
mattiav27's user avatar
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7 votes
2 answers
295 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 $\{\{b_{11},\dots,b_{1k}\},\dots,\{b_{k1},\dots,b_{kk}\}\}$ ...
Madeline Brandt's user avatar
4 votes
1 answer
97 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 ...
A little mouse on the pampas's user avatar
1 vote
0 answers
29 views

Mathematica gives invalid generators of a group? [closed]

...
David Collins's user avatar
3 votes
3 answers
199 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. ...
A little mouse on the pampas's user avatar
4 votes
2 answers
406 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 ...
A little mouse on the pampas's user avatar
0 votes
1 answer
111 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 ...
Carollice's user avatar
3 votes
2 answers
138 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, \...
Heidar's user avatar
  • 263
5 votes
1 answer
318 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 ...
ala10's user avatar
  • 109
2 votes
0 answers
50 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 ...
Slepecky Mamut's user avatar
6 votes
1 answer
163 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: ...
QuantumDot's user avatar
  • 19.7k
5 votes
2 answers
326 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 ...
user54038's user avatar
  • 277
5 votes
0 answers
96 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 ...
Mauricio Fernández's user avatar
0 votes
1 answer
188 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 ...
A little mouse on the pampas's user avatar
4 votes
1 answer
168 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 $3\times 3\times 3$: ...
Tino 's user avatar
  • 219
6 votes
4 answers
299 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....
Aster's user avatar
  • 3,846
4 votes
2 answers
447 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 ...
A little mouse on the pampas's user avatar