Questions tagged [irreducible-representation]
An irreducible representation of a group is a group representation that has no nontrivial invariant subspaces.
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Changing the field of an irreducible representation leaves it irreducible.
I am currently learning representation theory of finite groups over arbitrary fields.
I've encountered an interesting result multiple times but never found a proof of it anywhere.
Let $E/F$ be a field ...
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Does an irr. rep. of finite $G$ have a basis of the form $\{hv : h \in H\}$ for $H$ subgroup of $G$?
Let $G$ be a finite group and $V$ be a (complex vector space) representation of $G$. Consider the following three facts:
Whenever a $V$ is an irreducible representation of $G$, the dimension of $V$ ...
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Generating a conjugate representation of an irreducible self-conjugate representation of $S_n$
Suppose we have a complex matrix representation $\Gamma_{ij}^\sigma \in \mathbb{C}^{d \times d}$ of dimension $d$ for the permutations $\sigma$ of the group $S_n$ of permutations of $n$ objects.
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Indecomposable complex finite dimensional representations of a compact Lie group are Irreducibles?
I am studying the book "Representations of Compact Lie Groups" by Theodor Brocker and Tammo tom Dieck. At page 68 they prove the following proposition:
Let $G$ be a compact group. If $V$ is ...
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Irreducible complex representations of some abelian Lie groups
I wanted to classify all irreducible complex representations of the following basic abelian Lie groups:
$\mathbb{S}^1$ the circle in the complex plane, $\mathbb{R}_{>0}$ the positive real numbers, $...
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Why is the set of linear combinations of irreps closed under matrix products?
I'm studying Barry Simon's book on representation theory. There's this one result he proves stating that the set of linear combinations of $D_{ij}^{(\alpha)}$ is closed under taking products. The $D_{...
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Further decomposition of isotypic components in a representation
Let $(V,\rho)$ be an orthogonal (resp. unitary) representation of finite group $G$ whose irreducible representations over the same field as $V$ are $W_i$ with character $\chi_i$.
We have $V \cong \...
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Scalar extension of endomorphism ring of representation
I am currently trying to learn representation theory of finite group over an arbitrary field and i stumbled on a statement that seemed very intuitive to me but i could not find a proof of it anywhere.
...
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Given a representation of a group, how does one determine the multiplicity of the irreps?
Let G be a semi-simple Lie group and $(\pi, V)$ be a representation of $G$. $V$ is decomposed by $G$'s irreps as
$$
V = \oplus_{\mu,\lambda} V_{\mu,\lambda},
$$
where $\mu$ labels different irreps and ...
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A question about inducing of group representations
Let $G$ be a compact Lie group, $K$ its closed subgroup, $\rho$ a finite-dimensional real irreducible representation of $K$, $c\rho$ its complexification, $\mathrm{Ind}^G_K(\rho)$
the real ...
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Extension of the base field of a simple module over a finite dimensional algebra stays completely reducible/ semisimple
As I was thinking about a previous question of mine, i asked myself if there was a way to prove the following statement:
Let $L\setminus K$ be a field extension and $A$ be a finite dimensional $K$-...
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Matrices invariant under rotations are always proportional to the identity?
Is this proof true?
Suppose we have a $3\times 3$ matrix $M^{ab}$ satisfying
$$M^{ab}=R^a\,_cR^b\,_dM^{cd},$$
i.e.
$$M=RMR^T,$$
for all rotations $R\in \mathrm{O}(3)$. Now, if denote representations ...
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Extension of the base field of an irreducible representation of a finite group stays completely reducible
I was reading chapter $9$ of "Character theory of finite groups" by Isaacs in which he explores the theory of representation of finite groups over arbitrary fields.
In his theorem $(9.2)$, ...
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A brief explanation on Representation Theory
I'm trying to read this beautiful paper Regularity and cohomology of determinantal thickenings by Claudiu Raicu but I'm getting in trouble with Representation Theory, since I have no knowledge of it.
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Projection of real representations onto the isotypic components
Let $(V, \rho)$ be a representation of a finite group $G$ over field $\mathbb{F}$ and $W_i$ be irreducible representations (irreps) of $G$ over $\mathbb{F}$ with dimension $d_i$ and character $\chi_i$....
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