Articles of representation theory

Non-Examples for Krull-Schmidt-Azumaya

I am looking for some rings $A$ and finitely generated $A$-modules $M$, where the conclusion of the Krull-Schmidt-Azumaya Theorem does not hold for $M$, i.e. where $M$ can be written as direct sums of indecomposable modules in multiple ways (the summands not being unique up to order and isomorphism). To be specific, I am looking […]

The simple modules of upper triangular matrix algebras.

Let k be a field. $A= \begin{pmatrix} k& k& k\\ 0&k&k\\ 0&0&k \end{pmatrix}$ is the upper triangular matrix algebra. Then the projective module $P(2)=Ae_2=\begin{pmatrix} k\\k\\0\end{pmatrix}$ has a submodule $P(1)= \begin{pmatrix} k\\0\\0\end{pmatrix}$. Also we know that the cokernel of the inclusion map $i: P(1) \rightarrow P(2)$ is $S(2)$(the simple module correspond to $e_2$), I thought $S(2)$ […]

Do we have $(g \wedge g)^g = 0$?

Let $g$ be a simple Lie algebra. Let $(g \wedge g)^g = \{a \wedge b \in g \wedge g: x.(a \wedge b) = [x,a] \wedge b + a \wedge [x,b] = 0\}$ be the set of $g$ invariants under the adjoint action. Do we have $(g \wedge g)^g = 0$? If $g$ is a semisimple […]

How to differentiate a homomorphism between two Lie groups

Let $G$ and $H$ be two Lie groups and $\rho: G \to H$ be a homomorphism. How to differentiate $\rho$ to obtain a Lie algebra homomorphism $d\rho_e: T_eG \to T_eH$?

The comultiplication on $\mathbb{C} S_3$ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=S_3$, the algebra structure of $\mathcal{A}$ is $\mathbb{C} \oplus \mathbb{C} \oplus M_2(\mathbb{C})$. Let $\{ e_1,e_2,a_{11}, a_{12}, a_{21}, a_{22} \}$ be a matrix […]

Structure constants for and the adjoint representation and meaning in $sl(2,F)$

First, what I know is that given the basis: $$e = \left(\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right),f = \left(\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}\right),h = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)$$ I want to find the ‘structure constants’, but furthermore that the adjoint representation of $sl(2,F)$, […]

counterexample: degree of representation $\leq$ index of normal subgroup

if I have a finite group $G$ with an abelian normal subgroup $N$ and an irreducible representation $\pi$ of $G$ over $K$. Then I know, that $deg(\pi) \leq [G:N]$, if $K$ has positive characteristic and is a splitting field for $G$. My professor claimed (without a proof), that this is also true if I take […]

Does the induced representation always contain a non-trivial representation

Let $H$ be a proper subgroup of a finite group $G$ – not normal. Does $Ind_H^G 1$ contain a non-trivial representation? The Frobenius character formula was my original approach, but I can’t rule out that the trace of $Ind_H^G 1$: $$g \mapsto \# \{ x \in H\backslash G: x^{-1}gx \in H \}$$ is not the […]

Corollary of Schur's Lemma – why abelian

Corollary (of Schur’s Lemma): Every irreducible complex representation of a finite abelian group G is one-dimensional. My question is now, why has the group to be abelian? As far as I know, we want the representation $\rho(g)$ to be a $Hom_G(V,V)$, where $V$ is the representation space. Isn’t this always the case (i.e. even if […]

Clarifications on the faithful irreducible representations of the dihedral groups over finite fields.

I would like to clarify a few things in the answer to this question: Faithful irreducible representations of cyclic and dihedral groups over finite fields 1) When a representation extends, what does the remaining generator map to under the representation? 2) Why does a representation extend if and only if $z^{-1} = z^{p^{d}}$ for some […]