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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How was this representation obtained from this submodule from Dummit and Foote?

On page 862 of D&F, we define the action of $S_{3}$ on $\mathbb{R}^{3}$ to be the permutation of the indices of the elementary basis vectors of $\mathbb{R}^{3}$. So we have a representation of $S_{...
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2 votes
1 answer
41 views

Decomposition of $GL_n$ representation

Let $V=\mathbb{C}^n$ be a vector space of dimension $n$, viewed as the standard representation of $GL_n$. I know how to decompose $S^2 V \otimes V=S^3 V\oplus S^{2,1}V$, where $S^\mu$ is the Schur ...
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Fusion rules for irreps evaluated at different group elements?

It is well-known that tensor products of irreducible representations of a finite group decompose into direct sums of irreducible representations according to fusion rules $$\Gamma_i \otimes \Gamma_j=\...
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Problem 5.17, Isaac's Character Theory Of Finite Groups

I couldn't find how I should go to the result in the following problem. ( Problem 5.17, Isaac's Character Theory Book ) Let $H \leq G$ and let $\chi = (1_H)^{G}$. Fix a positive integer $n$. For $g \...
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Irreps of $A_4$ [duplicate]

I try to find irreps of $A_4$ over $\mathbb{C}$, I know there are 4 irreps of dimensions $1,1,1, 3$. But I have no clue how to find them (except from the trivial one)
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1 answer
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A problem in group theory and representation theory.

Suppose that $H = \{1, h, h^2, \ldots, h^{n-1}\}$ is a normal subgroup of a finite non-abelian group $G$ having order $n$. It is known that $H$ is cyclic with generator $h$. Let $c_G$ be the number ...
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Irreducible representations of $S_3$ over $\mathbb{R}$

I want to calculate irreps of $S_3$ over $\mathbb{R}$. I tried to find first a formula to the dimensions of the irreps and I get $\sum dim_{\mathbb R}\mathbb{C}\cdot dim\pi_i^2=|S_3|$, which says that ...
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What are the properties of these "anti-split" numbers?

What are the algebraic properties of the following number system (which I would call "anti-split numbers")? Numbers are represented by pairs $(a,b)$, with $(1,0)$ being multiplicative unity ...
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Can a non-distributive algebraic system be represented as matrices?

For instance, can the following 4-dimensional "number system" (which I would call "anti-split numbers") be represented as matrices? It is commutative and associative but not ...
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Regular functions on torsors

Let $X \rightarrow Y$ be a torsor for a linear algebraic group $G$ (i.e. $X$ is a principal $G$-bundle over $Y$). Assume also that both $X$, and $Y$ are affine. What can be said about $\mathbb{C}[X]$ ...
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weight lattice vector which is not a weight of any representation

What is an example of a simply connected Lie group $ G $ and an integral vector $ \lambda $ (i.e. a vector in the weight lattice) such that $ \lambda $ is not a weight of any finite dimensional ...
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Tits' construction gives semisimple Lie algebras

Back to 1966, Jacques Tits gave a unified construction of the $5$ exceptional semisimple Lie algebras, and his work leads to the famous Freudenthal-Tits magic square. See, for example, https://arxiv....
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Multiplicity of highest weights and representations

I am wondering whether there is a nice and clean way of detecting the dimension of an irreducible representation of a complex simple Lie algebra simply from the linear expansion of the corresponding ...
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About Kernel of Induced Character

I'm stuck somewhere in the following claim, I would appreciate if you could help. ( Recall : Let $G$ be a group and $H \leq G$. If $\alpha$ is a character of this subgroup $H$, we define $\alpha^{G}(g)...
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Norm of the adjoint representation and invariant Riemannian distance

Let $G:=\text{SL}(d,\mathbb R)$ and let $Z$ denote the center of $G$ which is a finite cyclic subgroup. Consider a right $G$-invariant Riemannian distance $d$ on the homogeneous space $G/Z$. Let $\...
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Question concerning positive Weyl chamber

I would like to ask for a hint for exercise 22.5 in Bump's book "Lie groups". The setting is as follows: Let $G$ be a (semisimple, connected, simply connected) compact Lie group, choose a ...
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A relation for adjoint representation of $U(N)$ acting on product of matrices and $SU(2)$ generators.

Is the following relation true, and if so, what is the property that makes it so? \begin{align} \sum_{i=1}^3\mathrm{tr}\left([U^{-1}L_iU]\phi[U^{-1}L_iU]\phi\right) \stackrel{!}{=} \sum_{i=1}^3\...
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3 votes
1 answer
35 views

Are infinite-dimensional representations of semisimple Lie algebras semisimple?

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field of characteristic zero. It is well-known that $\mathfrak{g}$ is semisimple if and only if the category of finite-dimensional ...
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Is there a useful classification of the homogeneous spaces for real Lie groups?

Let $G$ be a compact semisimple real Lie group. For the complex case there is a very deep theory connecting $G$-homogeneous spaces with irreducible representations of $G$. My question is: Is there an ...
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Do $(\mathfrak{g},K)$-modules only deal with real Lie groups?

In Bump's Automorphic Forms and Representations, p. 200, he gives the definition of a $(\mathfrak{g},K)$-module for $\mathfrak{g}=\mathfrak{gl}_n\mathbb{R}$ and $K=O(n)$ being the maximal compact ...
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When is an image of a Hopf algebra a Hopf algebra?

Suppose $R\subseteq S$ is a flat extension of rings and $A/R$, $B/S$ are flat Hopf algebras. Let $\varphi:A\otimes_R S\to B$ be a surjective $S$-Hopf algebra homomorphism. When is it the case that $\...
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Invariance of trace of a squared commutator under U(N) conjugation; SU(2) representations

Let the matrices $L_i$, with $i=1,2,3$, be the generators of $SU(2)$ in the $N$-dimensional irreducible representation. Then suppose I have an $N\times N$ Hermitian matrix $\phi$, I am trying to show ...
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1 answer
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Correspondence between G-covers and grupoid homomorphisms

I have a question concerning this article (https://math.berkeley.edu/~qchu/TQFT.pdf) about TQFTs and representation theory of finite groups. In the beginning of the third section, it is stated that, ...
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Adjoint representation in second tensor power of faithful representation

Let $ G $ be a simple Lie group and $ (\pi,V) $ a faithful finite dimensional representation of $ G $. Consider the action of $ G $ on $ V \otimes V $ by $$ g \cdot (v_1 \otimes v_2)= gv_1 \otimes ...
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1 vote
1 answer
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Tensor product of modules over Kronecker algebra

Let $\mathbf{k}$ be a field and $A=\begin{bmatrix}\mathbf{k}&0\\\mathbf{k}^2&\mathbf{k}\end{bmatrix}$ be the Kronecker algebra. Let $M$ and $N$ be the left and right $A$-modules (respectively),...
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The regular representation of $\mathbb{Z}_{3}$ over $\mathbb{Z}_{3}$ is indecomposable but not irreducible.

Let, $F = G = \mathbb{Z}_{3}$. We then let $V$ be an $FG$ module where $G$ is a basis for $V$, and the action is: $$g' \cdot g = g'g, ~ ~ g \in G, g'\in V.$$ We then obtain the representation, $\phi:G ...
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4 votes
1 answer
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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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2 votes
1 answer
42 views

What is the Coxeter plane?

I'm having trouble in understanding what the Coxeter plane is and how to get graph like this one. Suppose I have a Coxeter group with presentation Suppose I have The Coxeter group $$H_{4}=\left\...
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1 vote
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Schur's orthogonality relations on the symmetric Group $S_3$

I have the following proposition which I have to show through an example of the symmetric group $S_3$: Proposition. Let $G$ be a finite group. Let $\varphi^{(1)}, \ldots, \varphi^{(s)}$ be a complete ...
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3 votes
1 answer
54 views

Why are highest weight modules of integral highest weight $B$-equivariant?

Suppose $G$ is a connected semisimple algebraic group over $\mathbb{C}$, $B \subset G$ is a Borel subgroup, and $T \subset B$ is a maximal torus. Write $\mathfrak g$, $\mathfrak b$ and $\mathfrak h$ ...
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Decomposition of the representation generated by words with m copies of n letters

Let $V$ be the (complex) vector space generated by words of length $nm$ where each letter from $1$ to $m$ appears exactly $n$ times. For example, if $m=2$ and $n=3$, then $$ V = \mathbb{C}\{111222, ...
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3 votes
1 answer
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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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6 votes
3 answers
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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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1 vote
1 answer
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Unitary dualization functor continuous?

Let $G$ be a topological group and denote its unitary dual by $\hat{G}:=\{\pi:G\to\text{U}(\mathcal{H})\text{ irreducible unitary representation}\}/_\cong$. If $H$ is another topological group and $\...
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3 votes
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Pontryagin dual and unitary dual of abelian group homeomorphic?

Let $G$ be an abelian topological group, $G^\vee:=\{f\in\text{Hom}(G,\mathbb{T})\text{ continuous}\}$ its Pontryagin dual and $\widehat{G}:=\{\pi:G\to\text{U}(\mathcal{H})\text{ irreducible unitary ...
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  • 105
3 votes
1 answer
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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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3 votes
1 answer
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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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3 votes
2 answers
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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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0 votes
0 answers
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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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1 answer
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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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2 votes
1 answer
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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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1 answer
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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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2 votes
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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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1 answer
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Orthogonal Complement of a subrepresentation is a subrepresentation

Let $\rho: G \to GL(V)$ be a linear representation of $G$ and assume $(x\mid y)$ is a scalar product satisfying $(x\mid y) = (\rho_gx\mid \rho_gy)$ for all $g \in G$ If $W \subset V$ is stable under $...
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2 votes
1 answer
51 views

Sum on product of two charcteres, which runs on symmetric generating set

Let $G$ be a finite (not necessarily abelian) group and let $S$ be a symmetric generating set of $G$, i.e. if $s\in S$ then $s^{-1} \in S$. Let $\chi$ be an irreducible character of $G$. I have ...
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3 votes
1 answer
79 views

Why do we restrict ourselves to continuous representations of galois groups?

When studying Galois representations, we always assume that our representations are continuous. I'm new to studying these objects and am a bit struck by this assumption. What is the reasoning behind ...
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