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What is the meaning of an "irreducible representation"?
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An irreducible representation of $SO(3)$ will be a representation of $SO(3)$ that is irreducible. $SO(3)$ acts naturally on the vector space $\mathbb{R}^3$: it consists of all automorphisms of $\mathbb{R}^3$ that respect the inner product, so this is itself a representation of $SO(3)$ (which is irreducible, because no proper subspace of ...Explore further
linear algebra - Irreducible vs. indecomposable representation ...
WebFeb 1, 2019 · While an indecomposable representation is a representation that is not isomorphic to any direct sum of other representations. Wikipedia and other sources …
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Decomposability and irreducibility for representations
WebA representation $\rho:G \to GL(V)$ is irreducible if it has no invariant subspace apart from the trivial ones. It is indecomposable if we cannot write the representation as a …
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What are the irreducible representations of the cyclic group
WebConclusion: The irreducible real representations of $C_n$ are either: 1-dimensional, with matrix a real $n$ th root of unity (i.e., either $1$ or $-1$ ); 2-dimensional, with matrix …
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Irreducibility of the standard representation of $S_n$.
WebLet $\mathbb{R}^n$ be generated by $\{e_i\}_{1\leq i\leq n}$. Basically, when $n=2$ it's clear that $V$ is irreducible, so let $n>2$ and use induction. Note that the subgroup that fixes …
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Wiki claim - Irreducible representations with highest weight
WebMay 31, 2023 · How can I even be sure this $S_n$ is a representation of $\mathfrak{sl}_3$? Does something about the behavior of the Cartan subalgebra on …
Irreducible representation : question of the definition
WebMay 9, 2017 · On the french page of wikipedia it is said that an irreducible representation is a representation that has only $V$ and ${0}$ as stable spaces. But as I understood (it …
Restriction of Irreducible Representation to Subgroups
WebJan 6, 2024 · 1. (1, 2, 3) ↦ (−1 −1 1 0), (1, 2) ↦ (0 1 1 0). ( 1, 2, 3) ↦ ( − 1 1 − 1 0), ( 1, 2) ↦ ( 0 1 1 0). The restriction to H:= (1, 2, 3) H := ( 1, 2, 3) is equivalent to the sum of two 1 1 …
A question about dimension and irreducibility of a representation
WebSince dim(v) = | H | ≥ | {gv: g ∈ H} |, {gv: g ∈ H} is a basis for V (As a remark, this is isomorphic to the regular representation). But then V has a non-trivial proper …
Is there a general way to prove a representation is irreducible?
WebSep 2, 2019 · I'm working on simply proving the D(e) = 1, D(a) = e(2iπ/3), D(b) =e(4iπ/3) D ( e) = 1, D ( a) = e ( 2 i π / 3), D ( b) = e ( 4 i π / 3) an irreducible representation of Z3 Z …
Irreducible representations (over - Mathematics Stack Exchange
WebAnalysis of irreducible representations of $D_n$ One dimensional irreducibles. The commutators of $D_n$ look like $[r^i, sr^j]$ or the inverse of such, and $$[r^i, sr^j] = r^{ …
Low-dimensional Irreducible Representations of $S_n$
WebOct 2, 2011 · For any subset $X\subset\{1,\dots,n\}$ with $|X|=n-1,$ the restriction to $S_X$ defines a representation $S_{n-1}\simeq S_X\to U_m.$ This representation is faithful …
abstract algebra - Representations of the symmetric group $S_3 ...
WebOct 17, 2015 · STANDARD REPRESENTATION: $V=\{(x_1,...,x_n)\in k^n\mid\sum_ix_i=0\}\subset k^n$, $G\rightarrow GL_n(k),\sigma\mapsto P_\sigma$ …
Definition of "direct sum of irreducible representations"
WebOct 25, 2017 · I am struggling to understand the precise definition of "direct sum of irreducible representations", when speaking of representations of Lie algebras. Given …
representation theory - What is the intuition behind simple Lie …
WebMay 8, 2022 · By Weyl's Theorem, finite-dimensional linear representations of semisimple Lie algebras are completely reducible, i.e., a direct sum of irreducible representations. …
What is the difference between a reducible representation and a ...
WebApr 29, 2017 · As @Derek Holt mentioned in the comments a counterexample to your statement that every reducible representation can be written as a direct sum of …
Every Irreducible Representation of a Compact Group is Finite ...
WebNov 25, 2017 · I know it's true that every irreducible representation of a compact Lie group is finite-dimensional, and I've seen it mentioned that this result is a corollary of the Peter …
orthogonal matrices - Irreducible representations of $SO(N ...
WebSep 21, 2020 · In general you can check that if a real Lie algebra $\mathfrak{g}$ acts on a real vector space $V$ such that the representation of $\mathfrak{g} \otimes \mathbb{C}$ …
Irreducible characters of SU (3) - Mathematics Stack Exchange
WebJul 15, 2018 · Yes, in fact there is! The formula for the character of the irreducible representation of SU(3) with highest weight (p, q) is. χp, q(θ, ϕ) = eiθ ( p + 2q) p + q ∑ k …
Two representations with a common irreducible component
WebFeb 28, 2024 · Let M˜ M ~ be the subgroup of GL(n,Fq) G L ( n, F q) consisting of block diagonal matrices such that the main-diagonal blocks are invertible matrices. Let M = M˜ …
Infinite dimensional irreducible complex representations of
WebYou can prove stronger statement: every complex irreducible representation of a compact group is finite dimensional. Since finite group is compact, the statement for finite group …
Irreducible representation of dimension $5$ of $S_5$
WebThe irreducible $\mathbb{C}$-representations of $S_n$ are very interesting. The family $\{ S_n\}_{n\geq 1}$ of Symmetric groups is an infinite family of groups whose all …
Irreducible Representation of Lie algebra - Mathematics Stack …
WebAug 31, 2020 · 1. Let $g$ be a lie algebra, and $V$ be a vector space over $F.$ Assume that $\operatorname {char} (F) =0,$ and $\dim (V)=n>1.$ I want to show that $ (\rho, V)$ …
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