All Questions
Tagged with finite-groups symmetric-groups
19 questions with no upvoted or accepted answers
14
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0
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262
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Which irreducible representations of the symmetric group are eigenspaces of class sums?
In the setting of complex representations of finite groups, a class sum $1_C=\sum_{g\in C} g$ acts on an irreducible representation $V$ as $\lambda(C,V)\operatorname{Id}$, where $\lambda(C,V)=|C|\...
- 2,306
9
votes
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114
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Smith normal form of conjugacy class actions
This question was inspired by Smith Normal Form of a Cayley Graph of the Symmetric Group.
Let $\mathbb{Q}S_n$ denote the group algebra over $\mathbb{Q}$ of the
symmetric group $S_n$. Identify a ...
- 50.8k
8
votes
0
answers
236
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Alternate proof in Fulton–Harris of representation theoretic version of Littlewood–Richardson rule
$\DeclareMathOperator\Ind{Ind}$Let $d = d_1 + d_2$ with $d_1$, $d_2$ positive integers. Let $\lambda$ be a partition of $d_1$ and $\mu$ a partition of $d_2$, so that the Young symmetrizer construction ...
- 241
8
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188
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Non-zero group determinant for symmetric group
Let $G$ be a finite group. Given complex numbers $x=\{x_g: g\in G\}$, one can define a $|G|\times |G|$ matrix $X$, with entries $X_{g,h} = x_{gh^{-1}}$.
Let's consider $G$ being the symmetric group $...
- 1,839
5
votes
0
answers
200
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Subgroups of the symmetric group and binary relations
Motivation
The following came up in my work recently. (NB this is the motivation, not the question I'm asking. You can skip to the actual question below, which is self-contained, but not self-...
- 756
5
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0
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418
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What are the relation between Rep(G) and Rep(S_n)?
Let G be a finite group. We know it can be written as a subgroup of S_n. On the other hand, people sometimes say Rep(G) --- the category of all finite dimensional representations, are more interesting ...
- 2,040
4
votes
0
answers
266
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Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)
In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...
- 24.9k
3
votes
0
answers
157
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Faithful representation into $\operatorname{GL}(9,3)$
Take $T=\big(\left< (123) \right> \times \left< (456) \right> \times \left< (789) \right>\big) \rtimes \left< (147)(258)(369) \right> \leq S_9$.
Does there exist an injective ...
- 381
3
votes
0
answers
400
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Character table of the symmetric group $S_n$ according to James
In James, "The representation theory of the symmetric groups" an algorithm is described to produce the character table of a symmetric group. The proof involves the equation (pp. 22,23)
$$\...
- 2,050
3
votes
0
answers
133
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Is there some sort of formula for $t(S_n)$?
Let $G$ be a finite group. Define $t(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > t(G)$ and $\langle X \rangle = G$, then $XXX = G$.
Is there some sort of formula for $t(S_n)...
- 2,618
3
votes
0
answers
56
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Groups that can occur as graph automorphisms of a fixed size graph
From theorem $4$ and corollary $1$ in this book we have that graph isomorphism has to do with automorphism group of a graph. We also know every group is the automorphism group of a graph by Frucht's ...
- 13.9k
3
votes
0
answers
164
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Generating sets of the symmetric group that yield isomorphic Cayley graphs
Let $S$ and $S'$ be subsets of size $k$ of $\mathfrak{S}_n$.
Are there any necessary or sufficient conditions to determine whether or not $S$ and $S'$ yield isomorphic Cayley graphs?
Assuming we ...
- 1,164
3
votes
0
answers
226
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$S_n$ action on the sequences of transpositions
It is well-known, that any element $\rho$ of the symmetric group $S_n$ with $n-p$ cycles admits a unique presentation as a product of a sequence of transpositions $\{(a_i\,b_i)\}_{i = 1}^p$ with $a_i &...
- 401
2
votes
0
answers
220
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Characters of alternating groups
I am studying certain properties of characters of alternating groups and I have found precisely three characters not satisfying it up to $A_{15}$. These are:
A character of dimension $3.696$ of $A_{...
- 370
2
votes
0
answers
85
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Permutation factorizations according to number of generated orbits
Let $\pi$ be a permutation in $S_n$ with cycle type $\lambda$.
How many factorizations into two factors $\pi=\sigma_1\sigma_2$ are there, such that the subgroup $\langle \sigma_1,\sigma_2\rangle$ ...
- 2,552
2
votes
0
answers
156
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Special sets of involutions generating ${\rm S}_n$
For which positive integers $k$ and $r$ are there involutions $g_{n,i} \in {\rm S}_n$
$(n \in \mathbb{N}, \ i = 1, \dots, k)$ such that the following hold?:
for any $n$, the $g_{n,i}$ $(i = 1, \dots, ...
- 19.6k
1
vote
0
answers
72
views
Scalars by which symmetrizations of cyclic permutations act on Specht modules
Let $S_n$ be the symmetric group. Pick $a \in 2,\ldots,n$ and denote by $c_a \in \mathbb{C}[S_n]$ the symmetrization of the element $(12\ldots a)$ i.e. $c_a$ is the sum of cycles of type $a$.
Let $\...
- 163
1
vote
0
answers
213
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Is there any research on the action of a subgroup on the whole finite group by conjugation?
I want to know whether there are any research on the orbits of the action of a subgroup by conjugation on the whole group, when the group is finite. (Especially whole symmetric group.)
I'm especially ...
- 1,013
1
vote
0
answers
285
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Classification of transitive subgroups of finite symmetric groups generated by double transpositions
I want to classify (up to isomorphism) all transitive subgroups of symmetric group $S_n$ which are generated by double transpositions (product of two transpositions). Is there a characterization for ...
- 11