# 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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### About decompositions of induced characters

Suppose $G$ is a finite group, $H\leqslant G$ is a subgroup. $\chi_1,...,\chi_s$ are all the irreducible characters of $G$ and $\psi$ is an irreducible character of $H$. Prove that if ...
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### Every irreducible representation of $G_2$ appears in some tensor power of the standard representation

In the Book "Representation Theory" by Fulton and Harris, this fact ist stated on page 353 after looking at the weight diagrams of the complex Lie-Algebra $G_2$. The authors deduce that with ...
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### Modules generated by primitive idempotent elements

Assume that A is a finite dimensional k-algebra, and $e \in A$ is a primitive idempotent element. Is it true that the submodule of $A$ namely $<e>$ is simple $A$-module? If it is, how do we ...
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### references of modular representations for finite group

What is modular representation for finite groups? I tried to find a book to understanding that but I could not find a good one. Are there any useful references?
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### Finite dimensional representations of the Weyl algebra in characteristic $p>0$

I'm working through representation theory course notes of P. Etingof. In problem 1.26 it is asked to find all finite dimensional irreducible representations of the algebra ...
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### Every irreducible representation is either even or odd. [closed]

Let $V$ be any $n$-dimensional complex vector space and $SL(2,\Bbb{Z})$ is special linear group. Let $\rho:SL(2,\Bbb{Z}) \rightarrow GL(V)$ be a representation. It is even if $\rho(-I)=\Bbb{id}_V$ and ...
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### Dual of a faithful representation

A representation $\sigma$ of a finite group G is said to be faithful if Ker$\sigma={1}$. Then is it true that dual of a faithful representation is also faithful?
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### Composition Series of the regular A-module

Assume A is a finite-dimensional algebra over field K. How can we prove that any simple A-module occurs, as a composition factor (up to isomorphism) of an arbitrary composition series of A, as module ...
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### Mackey's criterion and double cosets of $A_3$ and $S_3$

State Mackey's criterion $Ind_{H}^{G}$ is irreducible $\iff p$ is irreducible $p^s$ and $p$ are disjoint representations of $H \cap sHs^{-1}$ for any $s \in T$\ $\{1\}$ Find the double ...
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### Show $Res_H^G(sgn_G)=sgn_H$ where $G=S_4$ and $H=S_3$

Let $G=S_3$ and $H=S_2$. Show that $Res^G_H(sgn_G)=sgn_H$ The symmetric group $G=S_3$ has three irreducible representations $1_G, sgn_G$ and $V$ where $1_G$ denotes the trivial ...
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### Comultiplication in a tensor algebra.

Let $V$ be a vector space. Then we have the tensor algebra $TV = \oplus_{i=0}^{\infty} T^i V$. In the webpage, it is said that the comultiplication $\Delta: TV \to TV \otimes TV$ is given by the ...
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### Frobenius reciprocity and induced representations

In representation theory, we consider the restriction functor for any group $G$ and subgroup $H$. This is: $Res_H^G : Rep(G) \rightarrow Rep(H)$ This gives a representation of $H$ The Induced case ...
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### Generators in adjoint representation are structure constants

Given that $g T_a g^{-1} = D^b_a T_b$ one can show that the generators in the adjoint representation of a group $G$ are the structure constants of the lie algebra satisfied by the $T_a$. Write $g$ ...
### Compute the associated induced Lie algebra action $\text{d}\pi$
Let $G=\mathrm{SL}_2(\mathbb{C})$ and consider the action of $G$ on the space of smooth functions on column vectors $\mathbb{C^2}$ given by $\big(\pi(g)\phi\big)(v)=\phi\left({g^\top}\,v\right)$ for ...
It seems to me like the coordinate statement of Schur's Orthogonality relations  \sum_{R \in G}^{|G|} \Gamma^{(\lambda)}(R)_{nm}^* \Gamma^{(\mu)}(R)_{n'm'} = \delta_{\lambda \mu} \delta_{n n'} ...