Projective objects in BGG category O\mathcal{O} are projective U(g)U(\mathfrak{g})-modules?

Let g be a finite dimensional semi-simple complex Lie algebra. Then, BGG category O is defined to be the full subcategory of finitely generated U(g)-modules of those modules which are weight modules and locally U(n)-finite. It is known that O is not extension-closed in the category of (finitely generated) U(g)-modules, see e.g. this math.stackexchange question. … Read more

Auslander–Reiten quiver

Consider the quiver 1⇇. I am trying to find the form of the Auslander–Reiten quiver. So I got : P(1)= \begin{matrix} 1 \end{matrix}, P(2) =\begin{matrix} 2 &\\ 1 & 1 \end{matrix}, I(1) =\begin{matrix} 2 & 2\\ 1 & \end{matrix}, I(2)= \begin{matrix} 2 \end{matrix}, S(1)=1, S(2)=2. I am just curious about how the Auslander–Reiten quiver looks … Read more

Representation theory of Lie algebras

A representation of the Lie algebra g on the vector space V≠{0} is a Lie algebra homomorphism ρ:g→gl(V),x↦ρ(x) So is the representation actually ρ? Where since this is mapping into the general linerar Lie group ρ is a matrix? I am having trouble understanding what a representation actually is. So we have a map from … Read more

Exterior power of irreducible representation

I am new to representation theory. Suppose that $G$ is a finite group with an irreducible representation over a (real or complex) vector space $V$. In my application, $G$ is a symmetric group and the representation is faithful. What can be said about the representation of $G$ over the $k$-th exterior power $\Lambda^k V$ of … Read more

Character of an $\mathbb{R}G$-module constructed from a $\mathbb{CG}$-module

I have been reading Representations and Characters of Groups by Gordon James and Martin Liebeck. I encountered the following construction of an $\mathbb{R}G$-module from a $\mathbb{C}G$-module. Next, there’s a proposition related to this on the next page. The proof of Proposition 23.6(1) is clear from the first image. I don’t understand the proof of Proposition … Read more

what does $\ltimes$ in the context of representation theory mean?

I am considering the following sentence wich is part of a theorem: ” Let $V$ be a finite dimensional unitary representation of $H=\mathbb{Z}^{2} \rtimes $ SL$_2(\mathbb{Z})$.” I have no background in represantion theory but i know what a unitary representation of a group is. What does the symbol $\rtimes$ mean? Answer This is the notation … Read more

Reference request: Representation theory over fields of characteristic zero

Many representation theory textbooks and online resources work with the field of complex numbers or more generally algebraically closed fields of characteristic zero. I am looking for a good textbook on representation theory which works with fields of characteristic zero directly from the beginning. (In case there are many such textbooks, I would prefer “modern” … Read more

Computing an expression involving characters

Suppose that $G$ is a finite group, and let $\rho: G\to \text{GL}(V)$ be an irreducible representation with $\dim V = n$ (let’s work over complex numbers, so $V$ is a $\mathbb{C}$-vector space). I will denote $\chi$ to be the character of $\rho$. (Recall that $\chi: G\to \mathbb{C}$ is defined by $\chi(s) = \operatorname{Tr}(\rho(s))$ for each … Read more

Weight space corresponding to $\lambda$ what is $-\lambda$?

Let $G$ be an algebraic group and let $T\subset G$ be a torus. Let $\lambda:T\to \Bbb C^\times$ be a character. Let $V$ be a $G$-module, and $V’$ be a nonzero $\lambda$-weight space of $G$ (pick the $\lambda$ at the start so it is a weight). What does it mean to say “consider the $-\lambda$ weight … Read more

All linear representations of a finite group are quiver representations?

Definitions A linear representation of a finite group G is a group homomorphism: ρ:G→GL(V), where we shall assume that V is a finite dimensional vector space. We call V the representation space of ρ. Let Q be a finite quiver with (finite) vertex set Q0 and (finite) edge set Q1. Then a representation M, of … Read more