I need some help to understand some proofs which can be found in the book Introduction to Representation Theory by Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina.

  • Definition 2.6, I don't understand how dual representation works, I was thinking that this it's such a way similar to dual maps in linear algebra, but It doesn't look like, this definition it's the central ideal in the proof of theorem 2.6.
  • Lemma 2.8, I don't know how I can conclude that $V_{i}/V_{i-1}$ are irreducible $\forall i$. My attempt: $V_{i}/V_{i-1}=\pi^{-1}(U_{i-1})/\pi^{-1}(U_{i-2})\cong \pi^{-1}(U_{i-1}/U_{i-2})$ but I'm not able to show that isomorphism, and It doesn't have sense since $U_{i-1}/U_{i-2}\not\subseteq U$, in the same way, I was trying to use a sequence to deduce this, but I can't.
  • Proposition 2.16, I don't know how I can conclude that $\text{End}_{A}(A)\cong\bigoplus_{i}\mathbb{M}_{n_i}(k)$ in the implication $(5)\Rightarrow (3)$ having in mind that $\text{End}_{A}(V_i)\cong k$ and no copy of $V_i$ in $A$ can be mapped to a distinct $V_j$.
  • Theorem 2.19 (Krull-Schmidt theorem), I don't know how can I show existence of such descomposition, I tried by induction over $\dim V$ and stuck even in base step, because if $\dim V=1$ then $V$ it's irreducible, but it doesn't mean that $V$ it's indecomposable.

I'm grateful for any hint in any item.

  • 2
    $\begingroup$ By the way, the usual expectation is for each question to be a separate question. Since there is a vague relationship between the questions, and because I was familiar with that text, I answered, but in the future please make sure your question is a question. $\endgroup$ – Kyle Miller Jun 11 '18 at 4:58

For Definition 2.7, they use without saying that $V^*$ as a vector space is the usual dual vector space $\hom_k(V,k)$. The definition says how this is a right $A$-module: $f\cdot a:=(v\mapsto f(a\cdot v))$. In other words, if $\langle-,-\rangle$ is the pairing on $V^*$ and $V$, then $\langle f\cdot a,v\rangle=\langle f,a\cdot v\rangle$ gives the definition of the dual representation. Basically, we want $\langle-,-\rangle:V^*\otimes_k V\to k$ to also be a map $V^*\otimes_A V\to k$. (If $A$ is a group algebra, then we can use inverses for the opposite algebra, as in $\langle g^{-1}\cdot f,v\rangle=\langle f,g\cdot v\rangle$ with $g$ a group element. By substituting $h=g^{-1}\cdot f$, we get $\langle h,v\rangle=\langle g\cdot h,g\cdot v\rangle$, meaning that the dual is such that $\langle-,-\rangle:V^*\otimes V\to k$ is an intertwiner for group algebras, with $k$ being the trivial representation.)

For Lemma 2.8, I see this as a consequence of the correspondence theorem. Since there are no modules between $U_{i-1}$ and $U_{i}$, there are no modules between $V_{i-1}$ and $V_i$. Thus, $V_i/V_{i-1}$ is simple (no possible kernels).

For Proposition 2.16, $\operatorname{End}_A(A)\cong ((\bigoplus_i n_iV_i)^*\otimes (\bigoplus_j n_jV_j))^A\cong \bigoplus_i\bigoplus_j (n_iV_i^*\otimes n_jV_j)^A$, where $-^A$ means $A$-invariant elements. By Schur's lemma, $(V_i^*\otimes V_j)^A\cong \delta_{ij}k$, so by composing with the standard inclusions $V_i\to n_iV_i$ and projections $n_jV_j\to V_j$, we can get that when $i\neq j$ then $(n_iV_i^*\otimes n_jV_j)^A\cong 0$. Hence, $\operatorname{End}_A(A)\cong\bigoplus_i (n_iV_i^*\otimes n_i V_i)^A\cong \bigoplus_i\operatorname{End}_A(n_iV_i)$. Then, $\operatorname{End}_A(n_iV_i)$ is a matrix algebra over $\operatorname{End}_A(V_i)\cong k$ for various reasons. A hands-on approach is to use the standard inclusions and projections to extract matrix entries from an endomorphism.

For Theorem 2.19, strong induct on dimension like so: if $V=W\oplus W'$ is a nontrivial decomposition, then since $W$ and $W'$ both have dimension less than the dimension of $V$, they decompose into indecomposables, and thus $V$ does as well. If $V$ is indecomposable, then $V=V$ is the decomposition (every such indecomposable gives a base case).

As a point of fact, irreducible representations are indecomposable. The $W$ above would be a non-trivial subrepresentation of $V$.


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