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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Faithful irreducible character and Sylow subgroup

I am trying to solve the (very nice) exercise 5.25 from Isaacs, character theory. Assume that every Sylow subgroup of $G$ has a faithful irreducible character. Show that $G$ has one also. The ...
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Equivalence between Representations

Asseume that $k$ is an algebraically closed field of a strictly positive characteristic $p$, G is a finite group of order $p$ and that $p:G \rightarrow GL(V)$ is a representation of $G$. Then $p(g)$ ...
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Representation of $Q_8$ over $\mathbb{R}$

I'm trying to solve the following problem, Give an example of a finite group $G$ and its irreducible representation $L$ over $\mathbb{R}$ such that the division algebra $Hom_G(L, L)$ is isomorphic ...
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some confusions about the concepts of algebra

Recently I tried to learn Algebra(Revised third edition) with the book written by Serge Lang. Since I have not covered all topics in the elegant book but now just view it as a reference for some ...
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On the converse of Schur's Lemma

Let $G$ be a finite group and $F$ a field with $\mathrm{char}(F)=0$ or coprime to $|G|$. Let $V$ be a $FG$-module in a way that every $FG$-homomorphism $f : V \to V$ is given by $f(x)= \lambda x$. ...
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Characters of permutation representations for $S_4$

I am going through the lecture note How to get character tables of symmetric groups. On page 2, it computes the character table of $S_4$. The procedure starts with building the table of the ...
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How to write $R_{ij}$ as a matrix?

Suppose that $V$ is a vector space of dimension $n$ and $R: V \otimes V \to V \otimes V$ a linear map. Then we can write $R$ as a $n^2 \times n^2$ matrix. Let $R_{ij}: V^{\otimes m} \to V^{\otimes m}$ ...
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Faithful representation of a $p$-group

Suppose $G$ is a nontrivial $p-group$. Let $H$ be the intersection of the center of $G$ and the set of elements in $G$ of exponent $p$. Let $\rho: G\rightarrow GL(V)$ be a representation. Show that if ...
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Considering $Res^G_{H_\rho}$ instead of $G$ in quantum Fourier sampling

I am going through the proof of theorem 4 in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. Here, they are trying to calculate the probability of measuring the ...
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Classify subrepresentations in finite dimensional semisimple representations

Quoted from "forgetfulfunctor": I'm following the notes by Prof. Etingof, linked here, and am stuck on a detail from Prop. 2.2, on page 23. To briefly recap what is in the notes, we have a ...
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What does the set of dominant integral elements in a Cartan sub algebra look like?

I'm reading about the theorem of the highest weight: Irreducible finite dimensional representations of a complex semisimple Lie algebra (with a fixed Cartan sub algebra, $\frak{h}$ and choice of ...
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A question of the paper“Crystallizing the q -Analogue of Universal Enveloping Algebras”?

I'm reading the paper "Crystallizing the q -Analogue of Universal Enveloping Algebras" written by Masaki Kashiwara. But there is something I don't know. Can anyone tell me how to use the ...
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Representation decomposition over $GL_2(\mathbb{C})$

I have found that $Sym^2(V) \otimes Sym^2(V)$ can be decomposed over the special linear group as follows: $Sym^2(V) \otimes Sym^2(V) \simeq Sym^4(V) \oplus Sym^2(V) \oplus 1$ This is done using the ...
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elments of a linear algebraic group agreeing on a vector

Let $G \subset \mathrm{GL}_n(k)$ be a connected affine algebraic group over a field $k$ with the following property: for any two distinct elements $g,h \in G$ there exists a vector $x \in k^n, x\neq 0$...
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What is the space $\operatorname{Sym}^2(V)$ and how does it act on the vector space $V$?

If $V$ is a vector space over $\mathbb{C}$ with basis vectors $e_i$, what is the space $\operatorname{Sym}^2(V)$? I am hoping someone can give me some insight into this space; perhaps by describing ...
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Permutation and Linear Representation of Finite Group

By a permutation representation of a finite group $G$, we mean a homomorphism from $G$ to $S_n$, the (full) permutation group on $n$ letters. By a linear representation of a finite group $G$, we mean ...
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Find an irreducible extension of an irreducible representation

Let $G$ be a finite group and $C$ the center of $G$. Let$μ:C→F^×$ be a character of $C$. Prove that there is an irreducible representation $ρ : G → GL(V )$ such that $ρ(c)(v) = μ(c)v$ for all $c ∈ C$ ...
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Representation of Sylow which does not extend

Let $H$ be a subgroup of a finite group $G$ and $\rho$ a representation of $G$ such that the restriction of $\rho$ to $H$ is invariant under conjugation in $G$, in the sense that its character is ...
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Holomorph of Cyclic $p$-groups

Assume that $G$ is a cyclic $p$-group (or up to isomorphism more explicitly $\mathbb{Z}_{p^r}$). Do you know if there does exist any explicit way to describe the holomorph of the above group? Moreover,...