As of May 31, 2023, we have updated our Code of Conduct.

# Tag Info

Accepted

### Issue with a "proof" for Maschke's Theorem

You seem to be assuming that there is a unique projection onto a subspace of a vector space. That is not true. For instance, if $W$ is two-dimensional space with basis $\{e_1,e_2\}$, then for any ...
• 316k
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### Isaacs Character Theory - exercise 4.11

For Question 1, to show that $t = |G|/q$, it is sufficient to show that all involutions in $G$ are conjugate. There is a hint on how to do that in the book. If not, choose two non-conjugate ...
• 84.8k
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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?

I think this is based in a misunderstanding of how we can convert between $E$ and $F$ vector spaces. For concreteness, let's work with $\mathbb{C}$ and $\mathbb{R}$, but you'll see that the same idea ...
• 32.6k

### There are no irreducible representations of dimension less than $7$ for the Lie algebra $\mathfrak g_2$.

One possibility is to use the fact that the formal character of an irreducible representation is invariant under the action of the Weyl group. In the case of $\mathfrak{g}_2$ the Weyl group has order ...
• 128k
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### Regular Representation of infinite groups

For an abstract infinite group $G$ the definition is the same as in the finite case; over a field $k$ you can always consider the action of the group algebra $k[G]$ on itself by left multiplication (...
• 399k
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### Representations of abelian groups

It is not true in the infinite-dimensional case. An irreducible representation of an abelian group $A$ over a field $k$ is the same thing as a simple module over the commutative $k$-algebra $k[A]$. ...
• 399k
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### Perfect groups whose character degrees square divide its order

If you extend to perfect groups, this is pretty easy. Let $G$ be your favourite simple group, and let $A$ be a very large abelian group. A semidirect product $X=A\rtimes G$ has character degrees ...
• 10.7k

### Motivation for representation theory

Have you tried googling to find motivatations for the subject? Some are here, here, and here. Historical background for the subject is here. The question that first led Frobenius to develop ...
• 39.6k

• 17k
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### Two-dimensional representation of $Q_8$ is not "real"

By the sounds of it you want an explanation rather than a proof, because the proof is just the fact that such a representation has to be irreducible, and there is only one by counting the squares of ...
• 10.7k

### Possible degrees of group characters

Yes, for instance the symmetric group $S_{n+1}$ has an irreducible representation of degree $n$. Precisely, take the canonical permutation representation $V$ of $S_{n+1}$; it has degree $n+1$, and we ...
• 24.2k
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### Subadditivity of the multiplicity in a composition series?

The following certainly works for modules, and should therefore also work in $\mathsf{C}$ by the Freyd–Mitchell embedding theorem: \begin{align*} \newcommand{\im}{\mathrm{im}} [\im(ϕ + ψ) : S] &...
• 15.7k

• 45.7k

### There are no irreducible representations of dimension less than $7$ for the Lie algebra $\mathfrak g_2$.

If you are willing to use that $\mathfrak g_2$ is simple, then you can use the fact that any non-zero representation is injective and that $\mathfrak g_2$ has dimension $14$. So for a representation ...
• 19.2k
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### When can the basis of a Lie algebra always be made Hermitian?

To write things in mathematical language and notation here, we are considering an $n$-dimensional matrix Lie algebra $\mathfrak{g} \subseteq \mathfrak{gl}_d(\mathbb{C})$ (to a mathematician a "...
• 399k

In a commutative ring, nilpotent elements annihilate every simple module, because any nilpotent element belongs to the Jacobson radical. This is generally false for noncommutative rings. For instance, ...
• 235k
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### Determine the group structure from its character table: a group of order $24$ as an example

As you know that $P$ is normal, all that is left to prove that $Q_1 P = G$ and $Q_1 \cap P =1$, using the characterization of internal semidirect products. Both of these follow from order ...
• 17k
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### Endomorphisms of a Lie algebra representation

Things are not quite as easy. If you have non-zero eigenvectors for different weights, then the sum of these spaces is not an eigenspace. Moreover, any homomorphism of $\mathfrak g$-modules preserves ...
• 19.2k

### representation about Lie algebra and Lie group

Well for a start we cannot write every element of $G$ as a matrix nor as an exponential. Lie algebras can always be written as matrices by Ado's theorem but that is not true for Lie groups. Of course ...
• 3,287
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### The character of the alternating sum of the exterior powers of the standard representation

If $V$ is a representation of a group $G$ of degree $d$, $g\in G$ and $p\geq0$, then the character of $\Lambda^pV$ evaluated at $g$ is $$\chi_{\Lambda^pV}(g)=\sigma_p(\lambda_1,\dots,\lambda_m),$$ ...
Accepted

### Tensor product of Representations of Lie algebras

Your second definition just doesn't define a Lie algebra representation at all; a Lie algebra representation must in particular be a linear map, and what you've written down isn't linear. Conceptually ...
• 399k

### Representation of $\mathrm{Diff}(M)$ onto $C^\infty(M)$

The corresponding Lie algebra representation "ought to be" the action of vector fields by the Lie derivative. The sign is supposed to be there. The issue, as far as I can tell, is that there ...
• 399k
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### Is the alternating group $A_8$ a subgroup of the exceptional lie group $G_2$?

$G_2$ is famously the group of symmetries of an antisymmetric trilinear form in its 7 dimensional representation. The 7 dimensional representation $V$ of $A_8$ does not preserve any antisymmetric ...
• 10.7k