# Tagged Questions

Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.

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### Let $G=AB$ where $(|A|,|B|)=1$ and $V$ be an $\mathbb{F}[G]$ module.

Under these assumptions it is a well-known fact that if $V_A$ and $V_B$ are faithful ($V_A$ denotes $V$ as an $\mathbb{F}[A]$-module) then $V$ is also faithful. Clearly if $V_A$ and $V_B$ is ...
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### Orbits of the permutation action of a subgroup on its cosets

Consider a finite group $G$ and a subgroup $H \subseteq G$. There is a transitive group action of $G$ on the set of left cosets $gH$ by left multiplication, and the stabilizer of $gH$ is $gHg^{-1}$. ...
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### what does $\ltimes$ in the context of representation theory mean?

I am considering the following sentence wich is part of a theorem: '' Let $V$ be a finite dimensional unitary representation of $H=\mathbb{Z}^{2} \rtimes$ SL$_2(\mathbb{Z})$." I have no background ...
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### What are the units of $U(\mathfrak{sl}_2)$?

Let $U(\mathfrak{sl}_2)$ be the Universal Enveloping Algebra of $\mathfrak{sl_2}$ over a field $K$, i.e. the (non-commutative) algebra generated by three generators $E,F,H$ subject to the commutator ...
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### Irreducible representations of Heisenberg group

Lately, I've been struggling with the following problem. Let $H$ be the 3 dimensional Heisenberg group and let $\rho:H\to\text{GL}(n,\mathbb{C})$ be a irreducible representation. Show that $n=1$. I ...
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### Expressing $\mathbb{C}^3$ as a direct sum

$S_3$ acts on $\lbrace 1,2,3 \rbrace$, so this affords a homomorphism $S_3\to GL_3(\mathbb{C})$ (acting on $\mathbb{C}^3$). I showed the only vector fixed by the action of $S_3$ is zero. Find two ...
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### Looking for examples of groups with complex representations realizable over $\mathbb{Q}$

I'm looking for examples of finite groups $G$ such that all the complex irreducible representations of $G$ are realizable over i.e the representing matrices can be chosen to have rational entries. One ...
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### Why is this paragraph so short?

$G$ is a connected, reductive linear algebraic group.The reference is Springer, Linear Algebraic Groups. I am having trouble making sense out of anything in this paragraph. Proposition 7.31(ii) ...
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### Finite dimensional irreducible representations of Sp(2).

I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$. I can ...
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### Schur test and it's relation to representation theory

I was told by analyst who doesn't know about such things that Schur's test relating to boundedness of integral operators is somehow a version of Schur's lemma on irreducible representations, the group ...
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### Involutions and Representation of Lie Algebras

In what follows I'm going to use $V_{\theta_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta_s$. Is easy to see that ...
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### Faithful monomial representation induced from faithful character

Let $\rho: G \rightarrow GL_n(\mathbb{C})$ be a faithful irreducible representation such that $\rho = Ind_N^G \phi$ for some 1-dimensional representation $\phi$ and normal subgroup $N$. Does $\phi$ ...
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### Correlation among a given representation of a finite group and a representation given by the composition with an automorphism

Assume that $G$ is a finite group, and $p : G \rightarrow GL_n(\mathbb{F})$ is a linear represenation. Furthermore assume that the image of $G$ lies inside a subgroup of $GL_n(\mathbb{F})$, say for ...
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### Tensor invariants constructed from identity tensor

It is evident that tensors constructed from copies of the identity tensor (and scalars) eg $t^{ij}_{kl} = 2 \delta^i_k \delta^j_l - \delta^i_l \delta^j_k$ are invariant under any matrix group, and ...
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### Showing a rep of $sl(2,\mathbb{K})$ is irreducible

Let $V$ be a $m+1$-dim $K$-vector space with char$K=0$. Let $(v_0,v_1,\dots,v_m)$ be a basis of $V(m)$. Now suppose I construct a representation of $sl(2,K)$ on this representation. How do I show ...
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### Restricted linear representations of abelian groups

If $G$ is a group (say finite for simplicity though the question applies to infinite groups as well), what can one say about the subgroup $G^*_n = \text{Hom}(G, \mu_n)$ of the group of all linear ...
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### Odd order subgroups of $PSL(2.q)$

Let q be an odd prime power. Is it true that every odd order subgroup of $PSL(2,q)$ is abelian ? If yes, how can it be proven ?
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### Centralizer of a Sylow $2-$subgroup of $PSL(2,q)$

Let q be an odd prime power. By a classic result, a Sylow 2−subgroup $P$ of $SL(2,q)$ is generalized quaternion. It is an irreducible subgroup of $GL(2,q)$ (since otherwise its natural ...
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### Can we do representation theory for algebras with >2 operations?

Suppose I define an algebra with 3 or more operations, perhaps using universal algebra. Would it be meaningful to talk about the representation theory of this algebra? In particular, I am interested ...
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### The complex numbers $\mathbb{C}$ are a Frobenius algebra over $\mathbb{R}$

I try to show that there is a functional lineal $f:\mathbb{C}\longrightarrow\mathbb{R}$ whose Kernel contains no nonzero left ideals. Define $f:\mathbb{C}\longrightarrow\mathbb{R}$ by $f(a+bi)=a$ ...
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### A Sylow $2-$subgroup of SL(2,q) is irreducible

Let $q$ an odd prime power. By a classic result, a Sylow $2-$subgroup of $SL(2,q)$ is generalized quaternion. How can I show that it is an (absolutely) irreducible subgroup of $GL(2,q)$ ? I have ...
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### Definition of induced representation

Definition. Suppose that $H$ is a subgroup of the finite group $G$ and $\sigma \colon H \to \operatorname{GL}(W)$ is a representation of $H$. Then the induced representation from $H$ up to $G$ denoted ...
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### All but a finite number of finite simple groups are groups of matrices over $\mathbb{F}_q$

In the introduction to this honors thesis, http://people.math.gatech.edu/~jrabinoff6/papers/building.pdf I found this statement: Matrix groups defined over the finite fields $\mathbb{F}_q$ ...
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### Stable equivalences preserving injective dimension

Let $A,B$ be two finite dimensional connected algebras and $F$ be a stable equivalence between their stable module categories (module category modulo projectives). Are there some natural conditions ...
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### Properties of non-abelian characters

I'm looking for some (short of) non-abelian generalization of the following result: Let $G$ be a finite abelian group and let $f$ be a function on $G$ with values in some field of characteristic ...
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### Representation of of $SO(3)$ in the vector space $V = \mathbb C^{2S+1}$

Certain part in my textbook implies that a representation of $SO(3)$ in the vector space $V = \mathbb C^{2S+1}$, where $S \in \mathbb Z$, is possible. I am trying to find a path that leads to this ...
It is well-known that the group algebra $F[G]$ is a direct sum of irreducible $G$-modules. The proof in my text book is as follows Write $F[G] = \oplus_{i=1}^n V_i$, where $V_i$ is a set of ...
Let $A$ be a finite dimensional $k$-algebra, and let $V$ be a finite dimensional $A$-module. How do we show that $V$ is semisimple (i.e. is the direct sum of simple submodules) if every maximal ...