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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 $ij$ ...
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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 ...
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1 vote

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=\... • 312k 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} & \...
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