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Questions tagged [representation-theory]

Representation theory studies (among other things) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.

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Point groups where the tensor square of two-dim. irreps (over $\mathcal{O}(3)$) does not contain a two-dim. irrep in its decomposition

Which are the point groups where the tensor square of a two dimensional irreducible representation does not decompose into a sum that contains a two-dimensional irreducible representation? For ...
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What is the meaning of $\rho \nu_{\rho}$?

I am reading the paper. On page 14, section 3.1, I am trying to understand the notation $\rho \nu_{\rho}$, where $\rho$ is a supercuspidal representation $GL_n(F)$, $F$ is a non-archimedean local ...
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Equivariant projective modules and skew group algebras

This is a question related to the two dimensional McKay correspondence. Let $R = \mathbb{C}[x,y]$, and $G$ a finite group acting on $R$. Recall that a $G$-equivariant $R$-module is an $R$-module with ...
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When are indecomposable projective modules finitely generated?

What conditions to you need to put on your ring to guarantee that the indecomposable projective modules are all finitely generated? Edit: I was hoping there was some general result for this. If your ...
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Embeddings of Coxeter Groups of Rank $3$ into $\text{SO}(3)$

Let $W = \langle x_1, x_2, x_3 \;|\; (x_ix_j)^{m_{ij}} \rangle$ be an irreducible Coxeter group, i.e., the graph with vertices $v_1, v_2, v_3$ and edges between all pairs $(v_i, v_j)$ with $m_{ij} \ne ...
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Satake correspondence for groups over finite field

In Langlands' program, Satake correspondence gives a correspondence between unramified representation of a reductive group $G$ over a local field and conjugacy classes in the Langlands dual group ${}^{...
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Extending scalars from $\mathbb{Z}[G]$ to $\mathbb{Q}[G]$

Let $G$ be a finite group. Let $M$ and $N$ be finitely generated $\mathbb{Z}[G]$-modules such that $M$ is free as a $\mathbb{Z}$-module. Suppose that $\mathbb{Q}\otimes_\mathbb{Z}M$ and $\mathbb{Q}\...
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Every Borel contains a Cartan, and conjugacy theorems: A simple proof?

Conjugacy of Borel subalgebras $\newcommand{\ad}{\mathrm{ad}\,}$ Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$. A Borel subalgebra $\mathfrak{b} \subseteq \mathfrak{g}$ is a ...
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25 views

Group representation preserving finitely many generators

Let $\rho : G \rightarrow GL_n(\mathbb{C})$ be a representation of $G$. If $G$ is finitely generated as a group, does that mean that $im(\rho) \leq GL_n(\mathbb{C}) $ is finitely generated? Because, ...
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$\mathbb R$-points of semisimple real algebraic groups, connectivity, and Cartan involutions: some questions

I am reading about Cartan involutions on semisimple real Lie groups and have a point of confusion I am trying to reconcile with linear algebraic groups. Let $\mathbf G$ be a linear algebraic group ...
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Deciding whether a representation is orthogonal or symplectic

I'm trying to understand the proof of Proposition 7 part (iii) from this paper of Dadok https://www.ams.org/journals/tran/1985-288-01/S0002-9947-1985-0773051-1/S0002-9947-1985-0773051-1.pdf (Statement ...
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Unitary Representations and the Peter-Weyl Theorem

Consider part II of the Peter-Weyl Theorem (see this wikipedia page for more information): Let $\rho$ be a unitary representation of a compact $G$ on a complex Hilbert space $H$. Then $H$ splits ...
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Is this algebra semisimple?

Is this algebra, A, semisimple? And what are its simple modules? $$\begin{bmatrix}a&b&0\\c&d&0\\0&0&e\end{bmatrix} \subset M_3(k),$$ where $k$ is a field and $a,b,c,d,e$ are ...
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51 views

Direct sum of isomorphic simple modules

Let $R$ be a ring and $M$ an $R$-module. Suppose that $\{M_i\}_{i\in I}$ is a (possibly infinite) collection of simple submodules of $M$, which are pairwise isomorphic. Suppose that $M$ is the direct ...
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Certain Isomorphic Representations of the dihedral group $D_{3}$

Using the following presentation of the dihedral group $D_{3}$ \begin{equation} D_{3} = \left\langle r,s \mid r^{2} = s^{2} = (rs)^{3} = e \right\rangle \end{equation} There is one (...
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1answer
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Complex group algebra of $S_3$ isomorphic to $\mathbb{C}[x]/(p)$?

For the symmetric group on two letters, $S_2$, there exists an isomorphism from the complex group algebra $\mathbb{C}[S_2]$ to the complex polynomial algebra $\mathbb{C}[x]/(x^2 - 2x)$ by taking $e + (...
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Crystal operators

Define the operator $s_i$ on tableaux: Consider letters $i$ and $i + 1$ in row reading word of the tableau. Successively “bracket” pairs of the form (i + 1, I ). Left with word of the form $i^r (i +...
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Representations of $\mathbb{G}_m$

I know that the multiplicative affine group scheme $\mathbb{G}_m$ is diagonalizable, since the algebra that represents it is $k[X,X^{-1}]$, which is isomorphic to the group algebra $k[\mathbb{Z}]$. ...
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For which graphs does this “+1 game” terminate?

Consider this game on simple graphs described by Allen Knutson: Begin by assigning a $1$ to a single node and a $0$ to each other node in the graph. Then, while such a node exists, choose a node with ...
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Characters of $\mathbb{G}_m$

Fix a field $k$. Let $\mathbb{G}_m$ be the multiplicative affine group scheme over $k$. A $k-$character $\chi$ of $\mathbb{G}_m$ is an endomorphism of affine group schemes $\mathbb{G}_m \to \mathbb{G}...
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Simple formula for the dimension of weight spaces of Verma module?

Let $\mathfrak{g}$ be a simple Lie algebra (e.g. $\mathfrak{sl}_n$), and let $M_\lambda$ be the Verma module with highest weight $\lambda$. Is there a simple formula for $\dim (M_\lambda)_\mu$, where ...
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1answer
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Invariant subspaces of the representation of $T(\sigma)[x_1,…,x_n] = [x_{\sigma^{-1}(1)},…,x_{\sigma^{-1}(n)}]$

Where $\sigma \in S_n$ and the representation is over the vector space $\mathbb{C}^n$ I'm trying to find as many invariant subspaces of this representation as possible. I don't know how to find these ...
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structure of subrepresentations of (infinite) sums of irr. representations

Let $G$ be a (locally compact) group and $ ( \pi_1 , V_{\pi_1} ) , ( \pi_2 , V_{\pi_2} ) , \ldots$ irreducible, unitary representations. I think, I can show, that for the (finite) direct sum ...
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Show that $\dim V^{G} = 1/|G| \sum_{g \in G} \chi_{T} (g). $

Let $T$ be a complex linear representation of the group $G$ in a space $V$ Show that $\dim V^{G} = 1/|G| \sum_{g \in G} \chi_{T} (g).$ If I know the definition of $V^G$ from this corollary: Which ...
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1answer
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Representation of sigma complete Boolean algebras

In Terrence Tao's article 245B notes 4: The Stone and Loomis-Sikorski representation theorems he gives a proof that not each sigma-complete Boolean algebra can be realized as a $\sigma$-complete ...
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Group ring $R[G]$ semisimple if and only if $J = J^2$?

Let $G$ be an abelian group and let $R$ be a commutative ring and consider the group ring $R[G]$ of finite formal linear combinations of elements of $G$ with coefficients in $R$. Let $J = (1 - g ~|~ g ...
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How is Schur's lemma being used here?

I have a question about a small point in Vigneres' introductory notes on the trace formula. Here $G$ is a finite group, $L$ is an algebraically closed field (if necessary assume $\operatorname{char}...
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Calculate the characters of the left and right regular representationsof an arbitrary finite group.

Calculate the characters of the left and right regular representations of an arbitrary finite group. The answer of the question is given below: But I do not know why the character of the left and ...
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Computing the inner product $(\chi_{R},\chi_{R})$.

Let $\chi_{R}$ denote the character of the right regular representation $R$. Compute the inner product $(\chi_{R},\chi_{R})$ directly and also by decomposing $R$ into a sum of irreducible ...
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Decomposition of Induced and Restricted Modules

I am taking a course in representation theory, and as I missed many classes I am having a lot of trouble in dealing with the decomposition of induced and restricted representations. I have no idea how ...
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1answer
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How to find the matrix generators of higher dimensional irreducible representations of $\operatorname{SU}(2)$?

Using the most general form of a $2\times 2$ unitary matrix $U$ of determinant $+1$ and using the formula $$T^a=-i\frac{\partial U}{\partial\theta^a}|_{\{\theta_a\}=0} ~{\rm with}~ a=1,2,3. $$ In this ...
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Characters of $(\mathbb{Z}/2\mathbb{Z})^m$

I just started to read the book 'Representation theory of finite groups' by B. Steinberg and I'm trying to solve Exercise 4.5., which discusses the characters of $(\mathbb{Z}/2\mathbb{Z})^m$. For a ...
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Schur Index for Quaternion Algebra

I learned form this question and this answer that Schur index in GAP can be found using LoadPackage("wedderga") the functions "SchurIndex". But I am working on the field $K=\mathbb Q (\sqrt{-39})$ ...
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Why is $\Omega (M)$ a superfluous submodule?

I'm struggling with a result that seems intuitive and that authors don't even bother to prove, so I think there's something stupid that I'm not seeing. Let $M$ be a finitely generated module over $...
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Isomorphism of modules implies isomorphism of tensor products

Let $R$ be a Noetherian ring, $M$ a finitely generated left $R$-module and $I,J$ two sided ideals in $R$ such that $I \subset J$. Suppose that $IM=JM$. What are some conditions that will allow us to ...
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1answer
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A difficulty in understanding a step in finding all irreducible representations of $S_{3}.$

Here is a part of the proof: But I do not understand why to find sub-representations it is enough subspaces of V invariant under the action of both $\sigma $ and $\tau$? ...... could anyone explain ...
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How can I show that a representation of $S_n$ is reducible

Considering the representation of $S_n$ where the objects being permuted are the basis vectors of an $n$ dimensional vector space $$ |1 \rangle, |2 \rangle, \:... \:, |n \rangle$$ If the ...
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“If $g$ is semisimple, It is not too hard to see that $H^2(g,a)=0$. With a little supplementary argument…”

This is a statement made in Knapp, Lie groups, Lie algebras, Cohomology Chpt 4 last paragraph of Sec 2. $H^i(g,a)$ is the $i-$th cohomology group of complex $Hom(\wedge^i g,a)$ with $a$ abelian lie ...
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Conjugacy of Borel Subalgebras: Proof in Humphreys' Introduction to Lie Algebras and Representation Theory

In the title referenced above a proof of the conjugacy of Borel subalgebras is given on page 84: We assume $L$ semisimple and let $B$ be a standard Borel subalgebra and $B'$ any other Borel ...
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Representation of Lie algebra of germs of smooth/holomorphic functions

$\def\O{\mathcal{O}} \def\g{\mathfrak{g}}$ Suppose $G$ is a real or complex Lie group, with Lie algebra $\g$. Write $\O_{G,1}$ (resp. $\O_{\g,0}$) for the ring of germs of smooth/holomorphic function ...
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Why does representing groups as linear operators provide insight about the groups themselves?

An important aspect of group theory is how groups can be represented as linear operators acting on vector spaces. While I understand how this works and how the (at least basic) tools are defined, what ...
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Proof or counterexample for isomorphism of group representations

Let $(\pi , V_\pi)$ be an irreducible unitary representation of the (locally compact) group $G$. Let $V_\pi^n = V_\pi \oplus \ldots \oplus V_\pi$ be the $n$-fold direct sum of $V_\pi$ on which we have ...
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Prove that every finite group of order larger than 2 has more than two irreducible complex representations. [closed]

prove that every finite group of order larger than 2 has more than two irreducible complex representations. could anyone give me a hint on how to solve this please?
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Unitary representation of the loop group LSU(2)

Let $\pi$ is a standard representation of G = SU(2) on $\mathbb{C}$$^2$. For every z $\in$ $\mathbb{S}$$^1$, define an irreducible unitary representation $\pi$$_z$ of the loop group LG on $\...
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direct sum of representation of product groups

Given two finite groups $G_1$ and $G_2$, and some representations $\rho_1: G_1 \to V_1$ and $\rho_2: G_2 \to V_2$, it seems the standard way to create a representation for $G_1 \times G_2$ is to use ...
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What is the alternative of an algebraic direct sum?

I'm working through some Lie theoretic representation theory, and the definition of a $(\mathfrak{g},K)$-module states that the representation $V$ decomposes into an algebraic direct sum of finite-...
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Difference between 2 questions on the alternating group $A_{4}$.

I have to answer 2 questions on the alternating group $A_{4}$: One of them asks me to find all irreducible complex representations of the group $A_{4}$ (and calculate their characters) while the ...
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A difficulty in understanding the definition of “Spaces of Matrix Elements.”

The definition is given below: If the definition of the Spaces of Matrix Elements is as given below: But I do not understand why: 1- Any linear combination of matrix elements can be expressed ...
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A consequence of Schanuel's lemma

In Carlson's Cohomology and representation theory, the author states Schanuel's lemma, and then derives a consequence that I cannot understand. They define, for a $kG$ module $M$, $\Omega (M)$ to be ...
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Construct a basis of the matrix elements in the space $\mathbb{C}[S_{3}]$

Construct a basis of the matrix elements in the space $\mathbb{C}[S_{3}]$. If the definition of the Spaces of Matrix Elements is as given below: And the answer of the question at the back of ...