All Questions
12 questions
2
votes
1
answer
167
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Bound on the size of group related to a matrix basis
Let $ G $ be a group of $ n \times n $ matrices. Suppose that some subset $ \{ g_j: 1 \leq j \leq n^2 \} $ of $ G $ is a basis for the space of all $ n \times n $ matrices. Furthermore suppose that ...
- 4,081
13
votes
2
answers
801
views
Irreducible representation of $S_n$: contained in tensor powers of the standard representation?
Let $S_n$ be the permutation group and $V = \operatorname{Fun}(X,\mathbb{k})$ functions from $X=\{1,\dotsc,n\}$ to some field $\mathbb{k}$. How can I prove that every irreducible representation of $...
- 179
4
votes
2
answers
433
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What is the current status of representation theory of $n$-ary groups in terms of hypermatrices?
An $n$-ary group is a generalization of the usual concept of a group where the binary operation (2-argument operation) is instead an $n$-ary ($n$-argument) operation. More info here on Wikipedia.
I ...
- 2,689
0
votes
1
answer
596
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Definition of an irreducible subgroup, and how to tell if any subgroup of $\mathrm{GL}_{n}(p)$ is irreducible [closed]
I'm not entirely sure what the proper definition of an "irreducible subgroup". I want an intuitive definition what an irreducible subgroup is in the simplest, most pedagogical terms as ...
- 113
5
votes
2
answers
387
views
Presentations of $\mathbf{PGL}_3(\mathbb{F}_2)$ by three involutions
I am searching for (two) presentations of the group $\mathbf{PGL}_3(\mathbb{F}_2)$ for which the generators are involutions $a, b, c$, and such that the following relations are present [among extra ...
- 4,547
5
votes
2
answers
424
views
Is the Perron-Frobenius dimension of a G-Set given by its cardinality?
Given a ring $R$ with finite additive basis $\{e_i\}_{i=1}^{n}$, such that $e_i e_j=\sum c_{ijk}e_k$ with $c_{ijk}\in \mathbb{N}$, we define the Perron-Frobenius dimension $FPDim(e_i)$ of a basis ...
- 1,949
6
votes
1
answer
934
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Finite groups with all irreducible representations one dimensional
Let $k$ be a field of characteristic $p$ and $G$ a finite group. This question might be a dulicate of this question:
Which finite groups have no irreducible representations other than characters?
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- 137
6
votes
1
answer
205
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Can a projective solvable group be transitive?
Let $p > 3$ be a prime number, and let $G \leq \mathrm{PGL}_2(\mathbb{F}_p)$ be a solvable subgroup.
Is it possible that the action of $G$ on $\mathbb{P}^1(\mathbb{F}_p)$ is transitive?
- 11.3k
5
votes
1
answer
507
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Matrix Elements of Real Representations
I asked this question over at Math.StackExchange and despite having had a bounty on it I did not receive an answer.
Suppose that $G$ is a finite group and we have a unitary irreducible representation ...
- 1,037
0
votes
2
answers
1k
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Similarity about unitary matrices
Suppose $G_1, \ldots, G_k$ are unitary, Hermitian, and anti-commuting matrices, and assume the same for $F_1, \ldots, F_k$. Suppose these matrices are similar, i.e. there exists $T \in GL_n(\mathbb{C})...
- 651
2
votes
2
answers
199
views
Permutation covering of a $G$-lattice
Let $G$ be a finite group.
By a $G$-lattice we mean a finitely generated free abelian group $L$ with an action of $G$.
We say that $L$ is a permutation $G$-lattice if $L$ has a ${{\mathbf{Z}}}$-basis ...
- 14.2k
1
vote
1
answer
1k
views
trace of a matrix of finite order
Let $A$ be an $n$ by $n$ real matrix of order $d$. i.e. $d$ is the smallest positive integer greater than $1$ that makes $A^{d}=I_{n}$.
The set of trace zero real matrices form $n^{2}-1$ dimensional ...
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