# 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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### Why is the Fourier transform of a non-Abelian finite group the weighted superposition over all irreps?

I am going through the lecture note of Andrew Childs on Nonabelian Fourier analysis. I would like to quote from the note: My question: Why does it have to be weighted superposition and not equal ...
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### Degree one irreducible representations

In section 2.5 of his Linear representations of finite groups (I have the french copy), Serre gives an example of determination of the character table of a group $G$. The group $G$ is taken to be $S_3$...
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### Action of $sl(2,\mathbb{C})$ on Dual of Polynomials does not Exponentiate

Let $V$ be the space of holomorphic polynomial functions in two complex variables $\xi,\eta$ and let $V^\ast$ be its dual space with subspace $W$ of linear functionals of the form $Df(1,0)$ where $D$ ...
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### What is the use of a right-module?

It seems that only the left-module provides a representation of a group. So what is the use of a right-module?
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### Exterior power of irreducible representation

I am new to representation theory. Suppose that $G$ is a finite group with an irreducible representation over a (real or complex) vector space $V$. In my application, $G$ is a symmetric group and the ...
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### Degrees of irreducible characters of an extension of $A_5$ by an elementary abelian 5- group

I'm reading a recent paper of G. Navarro, The set of character degrees of a finite group does not determine its solvability, in which he construct two finite groups $H$ and $G$ with $cd(G)=cd(H)$ such ...
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### What are some Group representation of the rubik's cube group?

The Rubik's cube corresponds to valid sequences of moves of the Rubik's cube. What are some group representations of this group (with respect to finite dimensional vector spaces on finite fields)? ...
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### Definition of the category of group representations

One usually considers the category of complex linear group representations for a fixed group $G$. It is defined as the category whose objects are group morphisms $G \rightarrow GL(V)$ where $V$ is a ...
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### Young tableaux of partition $3+1+1$ for the conjugacy classes of $S_5$

I just computed the Young tableaux of partition $3+1+1$ for the conjugacy classes of $S_5$. It would be nice if anyone could confirm it's correctness. Thanks.
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### Does the supposed to exist functor considered in Langlands program bear a peculiar name?

I'm trying to figure out what a very rough sketch of the Langlands program could be. From what I (think I) understand, objects called reductive algebraic groups together with related so-called ...
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### $\text{Hom}$ to a projective $D[G]$-module for a complete DVR $D$

Suppose you have a complete DVR $D$ and a finite group $G$ with $D[G]$-modules $A$ and $B$. Does $B$ being projective imply that $\text{Hom}_{D[G]}(A,B)$ is $D$-free? Or should it be $A$ that's ...
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### Representation of a group on a vector space induces a representation on another representation space?

Caveat: this is a very basic question. Suppose you have a representation of a group $G$ on a vector space $V$, let's say to be concrete $\mathbb{R}^n$. How is this representation related to the one ...
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### Primitive of the matrix elements of irreducible representations of Lie groups

I am interested in the matrix coefficients $U_{ij}(g)$ of unitary irreducible representations of a Lie group $G$. In my case, these coefficients arise from the Peter-Weyl theorem. I would like to ...
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### Tensor product of $Spin(2k)$ representations

I am trying to find the tensor product of spinor representations of $SO(2k)$. Labels are given as $$(n+I/2,I/2,\ldots,I/2,s)\otimes(I/2,\ldots,I/2).$$ Where $I$ and $n$ positive integers. How can ...
### $G$ and $G/H$ representations
It is known that if a group $G$ has an invariant subgroup $H$ and the factor group $G/H$ has a known representation then this representation is also a representation of group $G$. But, how can we ...