2 votes
Accepted

Help understanding link between module and representation.

Equation (4) is wrong: the $ij$ entry of the product matrix $\rho(g)\rho(h)$ is not $\rho_{ij}(g)\rho_{ij}(h)$ (you don't multiply matrices by multiplying them entry-by-entry). Instead, the $ij$ ...
2 votes
Accepted

Two examples for projective resolutions for finite dimensional algebras

There is no example satisfying (a). Suppose $(P_\bullet,d_\bullet)$ is a minimal projective resolution of $M$, so that $\Omega^i(M)=\operatorname{im}d_i=\ker d_{i-1}=N\oplus N'$. Then the composition ...
1 vote

Primitive idempotents of a semisimple algebra PAP

This is false. For instance, if $A=M_2(\mathbb{C})$, then you could have $e=\begin{pmatrix} 1&0 \\ 0&0 \end{pmatrix}$ and $P=\begin{pmatrix} 0&1 \\ 0&1\end{pmatrix}$, and then $eP=\...
1 vote
Accepted

Help showing associativity when multiplying group element by vector.

Replace $$x \boldsymbol{v}_i=\sum_j \rho_{i j}(x) \boldsymbol{v}_j$$ by $$x \boldsymbol{v}_j=\sum_i\rho_{i j}(x) \boldsymbol{v}_i.$$
  • 18.3k
1 vote
Accepted

Computationally representing a Fuchsian group

[…] But Mathematica didn't express the answer in terms of $\phi$. There seems to be some theory that I'm missing. Personally I would take the distance for the translation not from the matrix equation ...
  • 40.5k
1 vote

Can GAP determine whether a local algebra is Frobenius?

The MeatAxe works for finite dimensional modules of associative algebras, given by matrices for algebra generators, describing the action on the module. Build matrices for the regular module (by ...
  • 16.8k
1 vote
Accepted

Irreducible Characters and the Dual Space

Here's the situation for complex representations of finite groups: $$ \begin{array}{l|l|l} \textrm{space} & \textrm{basis} & \textrm{dual basis} \\ \hline \textrm{all functions} & \...
  • 19.4k

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