All Questions
24 questions
8
votes
0
answers
236
views
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
2
votes
0
answers
220
views
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
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
10
votes
7
answers
2k
views
Representations of products of symmetric groups
I'm writing a paper and want to cite some references to efficiently prove that over any field $k$ of characteristic zero, every irreducible representation of a product of symmetric groups, say
$$ S_{...
- 22.3k
10
votes
2
answers
547
views
Arbitrarily large finite irreducible matrix groups in odd dimension?
I consider a finite irreducible matrix group $\Gamma\subseteq\mathrm{GL}(\Bbb R^d)$. I am interested in the maximal size of $\Gamma$ depending on $d$. But this question makes only sense if there is an ...
- 13.6k
1
vote
0
answers
213
views
Is there any research on the action of a subgroup on the whole finite group by conjugation?
I want to know whether there are any research on the orbits of the action of a subgroup by conjugation on the whole group, when the group is finite. (Especially whole symmetric group.)
I'm especially ...
- 1,013
3
votes
0
answers
157
views
Faithful representation into $\operatorname{GL}(9,3)$
Take $T=\big(\left< (123) \right> \times \left< (456) \right> \times \left< (789) \right>\big) \rtimes \left< (147)(258)(369) \right> \leq S_9$.
Does there exist an injective ...
- 381
3
votes
1
answer
143
views
Permutation representation of a finite $p$-group
In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book," Theory of Groups Of Finite Order". The group ($\mathbb{Z_{p^{2}}}\rtimes \mathbb{Z_{p^{}}}) ...
- 381
3
votes
0
answers
400
views
Character table of the symmetric group $S_n$ according to James
In James, "The representation theory of the symmetric groups" an algorithm is described to produce the character table of a symmetric group. The proof involves the equation (pp. 22,23)
$$\...
- 2,050
8
votes
0
answers
188
views
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
14
votes
0
answers
262
views
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
4
votes
1
answer
745
views
Decomposition into irreducible of a representation of the wreath product $S_d\wr S_n$
Let $S_d, S_n$ be the permutation groups of $d,n$ elements.
An intuitive representation of the wreath product $S_d\wr S_n$ is $V_1\otimes...\otimes V_n$, where each $V_i$ is of dimension $d$. Writing ...
- 583
10
votes
3
answers
734
views
Low-dimensional irreducible 2-modular representations of the symmetric group
I apologize if this question is a little too basic for MathOverflow, but it's somewhat outside of my background and I'm frustrated that the answer doesn't seem to be explicit in the literature even ...
- 1,298
16
votes
1
answer
484
views
Irreducible representations occuring in $\mathrm{Ind}_G^{S_{|G|}}1$ for $G$ finite group
Let $G$ be a finite group with $|G|=n$, let $S_G=S_n$ be the group of $n!$ permutations of the set $G$. Then $G$ is a subgroup of $S_G$ via left-translation (i.e. $g\in G$ corresponds to the ...
- 691
6
votes
1
answer
341
views
Sum of Young symmetrisers of a given shape
Preliminaries and notation:
Let $n\in \mathbb{Z}_{>0}$ and $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_s)\vdash n$ be a partition. Given a Young diagram of shape $\lambda$, we can associate it ...
- 87
15
votes
2
answers
838
views
factorization of the regular representation of the symmetric group
Let $\mathbb{C}[S_n]$ be the regular representation of the symmetric group $S_n$, and let $\mathbb{C}^n$ be the vector representation.
Question: Does there exist a representation $V$ (of dimension $(...
- 2,537
21
votes
1
answer
1k
views
Okounkov-Vershik approach to representation theory of $S_n$
This is a rather soft question. I was wondering if someone could explain on a fundamental and intuitive level, what the Okounkov-Vershik approach to representation theory of $S_n$ is all about. It's ...
- 291
8
votes
2
answers
464
views
Faithful projective representations of symmetric groups
This is a reference request.
Do you know where I can find the dimensions of the faithful projective representations of $S_n$ and $A_n$ for $n\ge 5$?
Thank you in advance.
- 845
8
votes
2
answers
1k
views
In general, are 'Young symmetrisers' given by Littlewood-Richardson 'Orthogonal projection Operators'?
Consider $V^{\otimes n}$ where $V$ is vector space and the representation of GL(V) acting in the usual way. Now if I consider tensor products or plethysms of irreducible spaces, this is not in general ...
- 349
1
vote
1
answer
226
views
Homomorphisms from irreducible spaces to reducible spaces
Let $P_{\lambda}$ be a Young symmetriser associated to the following tableau $(a_1 a_2 a_3 b_3 ; b_1 b_2)$ where the entries seperated by the ; belong to first and second COLUMNS of the tableau. Take $...
- 349
9
votes
2
answers
860
views
What is natural about the well-known bijection between conjugacy classes and irreps of a symmetric group?
Symmetric groups possess a well-known bijection between conjugacy classes and irreducible representations. More precisely, both sets are indexed by Young diagrams.
Question: To what extent is this ...
- 24.9k
11
votes
5
answers
2k
views
Structure of the adjoint representation of a (finite) group (Hopf algebra) ?
Every group acts on itself by conjugation $h \mapsto g h g^{-1}$. Respectively considering functions on a group we obtain a linear representation.
Question 1: what is known about this representation ...
- 24.9k
8
votes
1
answer
400
views
Decomposition of $\mathrm{End}(V)$ as $S_n\times S_n$-module
Let $V$ be a finite-dimensional, complex vector space and set $\newcommand{\Gl}{\mathrm{Gl}}G:=\Gl(V)\times\Gl(V)$. Let $E:=\mathrm{End}(V)$ and consider its coordinate ring $\mathbb C[E]$, the space ...
- 4,902
5
votes
0
answers
418
views
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