# Questions tagged [representation-theory]

For questions about representations or any of the tools used to classify and analyze them. A representation linearizes a group, ring, or other object by mapping it to some set of linear transformations. A common goal of representation theory is classifying all representations of some type. Representation theory is a broad field, so questions not including the word "representation" may be appropriate.

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### Modular group with a finite order $T$

Let $G$ be the modular group. We know this can be described by the relations (in terms of the $S$ and $T$ transformations) given by $S^4 = I, (ST)^3 = S^2$. In my work matrix representations of $G$ ...
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### Difficulty parsing the meaning of polynomial coefficients of a group in a vector space (representation theory) - misunderstanding $\Bbb C[G]$ notation

I will be making reference to this answer by David E. Speyer. Representation theory, as taught by Artin's introductory text, has interested me greatly. Unfortunately he does not prove the result that ...
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### Decomposing a representation of $S_{3}$ over the vector space $\mathbb{F}_{2}[\{1,2,3\}]$.

Given the usual action of $S_{3}$ on $X:=\{1,2,3\}$, consider the linearization to obtain a representation of $S_{3}$ on $\mathbb{F}_{2}[X]$ (where $\mathbb{F}_{2}$ is the field with 1 element). ...
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### On Alperin's paper "The Green Correspondence and Brauer's Characterization of Characters" (aka what is a central factor?)

I was studying the paper "The Green Correspondence and Brauer's Characterization of Characters" by J. Alperin and I couldn't understand two of the passages. Hypotheses and notations $G$ is a ...
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### What is the importance of Cartan decomposition of a semi-simple Lie algebra?

I just started learning about Cartan decomposition of semi-simple Lie algebras, and I'm curious to know what are their applications in studying semi-simple Lie algebras. My guess was that it might be ...
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### If F < E are fields, how is it possible for a representation X, to be irreducible as an F-representation, but reducible as an E-representation?

Studding Character theory, and been bouncing back and forth between reading Dummit and Foote, and Character theory of finite groups by Martin Issacs. In section 18.1 of Dummit and Foote, we are given ...
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### Showing that a non-degenerate invariant bilinear form on a irreducible representation is either symmetric or anti-symmetric

Let $H \subset GL(n,\mathbb{C})$ be a group, acting irreducible on $V=\mathbb{C}^n$ and let $F$ be a non-degenerate bilinear form on $V=\mathbb{C}^n$, which is invariant under $H$. I now want to show, ...
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### For $\mathbb{Z}_+$ module over a based ring, indecomposability implies irreducibility.

This is exercise 3.4.3 in Etingof's Tensor Categories. Let A be a based ring with basis $\{b_i\}_{i\in I}$ and anti-involution $x\mapsto x^*$. Suppose $M$ is a indecomposable $\mathbb{Z}_+$ module ...
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### Modular representations of GL_n

I am interested in the irreducible representations of $G=GL_n(k)$ for a finite field $k$, over $\overline k$. For complex Lie group representations of $GL_n(\mathbb C)$, the irreducible ...
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### Dihedral and generalized quaternion groups have same character table

It is well known that $D_8$ and $Q_8$ are non-isomorphic groups with the same characters. I was wondering if this is true for $D_n$ and $Q_n$ in general were $n = 2^k$. Someone claimed that this is ...
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### Unique irreducible complex representation of Clifford algebra implies isomorphism with matrix algebra

Consider the Clifford algebra $\mathrm{Cl}(n)$ over Euclidean space $\mathbb{R}^n$ (with the standard inner product). Now, in the case that $n$ is even, it is known (cf. [1]), then $\mathrm{Cl}(n)$ ...
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### Primitive subgroup of $SU_n$ contained in maximal finite subgroup?

A subgroup $G$ of $GL_n(\mathbb{C})$ is reducible if we can write $\mathbb{C}^n=V_1 \oplus \dots \oplus V_k$ as a direct sum of smaller subspaces such that every $g \in G$ fixes the subspaces. ...
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### Number of simple modules of group algebras

I am attempting the following question: For each of the following algebras A, determine the number of non-isomorphic A-modules, and describe each simple A-module by giving a vector space basis and by ...
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### How does the weyl group act on weights\roots

Let the Weyl group be: $$W=N(T)/T$$ where $T$ is the maximal torus of some lie group $G$ and $N(T)$ is the normalizer of $T$. I saw that in this question that the Weyl group acts on weights by: (w.\...
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### Question about finite dimensional representations of a semi-simple Lie group

I've encountered the following paragraph while reading page 3 of this paper https://link.springer.com/article/10.1007/BF01232026?noAccess=true Let $G$ be a semi-simple Lie group with a maximal ...
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