Intereting Posts

Characterization of the Subsets of Euclidean Space which are Homeomorphic to the Space Itself
Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$
Is $\sum_{n=1}^{\infty}\frac{(\log\log2)^n}{n!}>\frac{3}{5}$
Are all mathematical statements true or false?
Is $W_0^{1,p}(\Omega)$ complemented in $W^{1,p}(\Omega)$?
Geometric interpretation of hyperbolic functions
A closed form of $\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$
Interpolation, Extrapolation and Approximations rigorously
Finiteness of the dimension of a normed space and compactness
Why is this weaker then Uniform Integrability?
Example $g\circ f=id_A$ but $f\circ g\neq id_B$
Why SVD on $X$ is preferred to eigendecomposition of $XX^\top$ in PCA
show that $f(z)+f(z^2)+\cdots + f(z^n)+\cdots$ converges locally uniformly to an analytic function in the unit disk.
Let $f(x)=\int_0^1|t-x|t~dt$ for all real $x$. Sketch the graph of $f(x)$, what is the minimum value of $f(x)$
Are there infinitely many primes of the form $4n^{2}+3$?

$\DeclareMathOperator{\End}{End}

\DeclareMathOperator{\Ker}{Ker}

\DeclareMathOperator{\Hom}{Hom}

\DeclareMathOperator{\Mat}{Mat}

\DeclareMathOperator{\Irr}{Irr}$

**Definition.**

An $A$-module $V$ is called *semisimple* if $V$ can be decomposed as a direct sum of irreducible submodules.

**Definition.**

Let $\Irr(A)$ denote a complete set of representatives for the equivalence classes of irreducible representations of $A$.

**Proposition.**

*Let $V=V_{1}\oplus \cdots \oplus V_{n}$ be a finite dimensional semisimple $A$-module. For each $U\in \Irr(A)$ let
$$
n_U:=\dim_{k} (\Hom_A(U,V)) \in \Bbb{Z}_{\geq 0}.
$$
Show that $n_U$ is equal to the number of $V_{i}$ that are equivalent to $U$.*

- Choices of decompositions of a representation into irreducible components (Serre, Ex. 2.8)
- Are there any fields with a matrix representation other than $\mathbb{C}$?
- Minimal embedding of a group into the group $S_n$
- What does the group ring $\mathbb{Z}$ of a finite group know about $G$?
- Group action of linear algebraic group $G$ on itself induces a representaion of $G$ on $Lie(G)$
- Reps of $Lie(G)$ lift to universal cover of $G$. Reps of $G$ descend to highest weight reps of $Lie(G)$?

I’ve been staring to this for quite some time, but nothing that really comes to my mind. My book says that it has to do with Schur’s Lemma. My main problem is that I don’t really get what the dim actually means in this context I guess. I would say that $\dim_{k} (\Hom_k(U,V))$ is just $\dim_k(U)\dim_k(V)$.

Edit: After staring a little bit more to it, I think I can reduce this problem by Schur’s lemma to showing that:

$$\dim_{k}(\Hom_A(U,V))=\sum _{i}\dim(\Hom_A(U,V_{i})$$

I’m not sure how to show that, but it seems to do with this proposition:

**Proposition.**

*Let $U,V$ be finite dimensional $k$-vector spaces with direct sum decompositions $U=\bigoplus_{j=1}^nU_{j}$ and $V=\bigoplus_{i=1}^mV_{i}$ and with corresponding complete sets of idempotents $e_{j}\in \End_{k}(U)$ and $f_{j}\in \End_{k}(V)$.*

*We have a $k$-linear isomorphism $M$ between $\Hom_{k}(U,V)$ and the $k$-vector space consisting of $m\times n$ matrices
\begin{align*}
\begin{pmatrix}\phi _{1,1} &\cdots &\phi _{1,n} \\
\vdots & & \vdots \\
\phi _{m,1} &\cdots &\phi _{m.n} \end{pmatrix}
\end{align*}
with $\phi _{i,j}\in \Hom_{k}(U_{j},V_{i})$. This isomorphism $M$ is defined by $\phi \mapsto (\phi _{i,j})$ where $\phi _{i,j}=f_{i}\phi |_{U_{j}}$.*

- Choices of decompositions of a representation into irreducible components (Serre, Ex. 2.8)
- Reference request for algebraic Peter-Weyl theorem?
- Group of order 24 with no element of order 6 is isomorphic to $S_4$
- Categorical description of algebraic structures
- The comultiplication on $\mathbb{C} S_3$ for a matrix basis?
- Prove that the augmentation ideal in the group ring $\mathbb{Z}/p\mathbb{Z}G$ is a nilpotent ideal ($p$ is a prime, $G$ is a $p$-group)
- Complex Galois Representations are Finite
- What are spinors mathematically?
- Do we have $End(V \otimes V) = End(V) \otimes End(V)$?
- Prove that $Q_8 \not < \text{GL}_2(\mathbb{R})$

- Prove that $A/B$ is regular when $A$ is regular and $B$ is regular or not regular.
- Difficulty evaluating complex integral
- Is $x^{1-\frac{1}{n}}+ (1-x)^{1-\frac{1}{n}}$ always irrational if $x$ is rational?
- How find $a_{n}$ if the sequence $a_{n}=2a_{n-1}+(2n-1)^2a_{n-2},n\ge 1$
- Number of binary strings containing at least n consecutive 1
- Is every compact subset of $\Bbb{R}$ the support of some Borel measure?
- Evaluate the sum $\sum_{k=0}^{\infty}\frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)}$?
- Derivation of Lagrange-Charpit Equations
- Is the integral closure of local domain a local ring?
- upper limit of $\cos (x^2)-\cos (x+1)^2$ is $2$
- How can I visualize division of negative numbers
- What is mathematical basis for the percent symbol (%)?
- If A, B, C, D are non-invertible $n \times n$ matrices, is it true that their $2n \times 2n$ block matrix is non-invertible?
- Second Countability of Euclidean Spaces
- Question About Orthoganality of Hermite Polynomials