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Results tagged with modules
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user 4366
For questions on modules over rings.
8
votes
Free $k[x_1, \dots, x_n]^{S_n}$-module?
Let $\mathcal{P}_n$ be the polynomial ring $k\left[x_1, x_2, \ldots, x_n\right]$. The symmetric group $S_n$ acts on $\mathcal{P}_n$ from the left by the formula
$${}^\pi f = f\left(x_{\pi\left(1\right …
- 1,472
2
votes
Constructing a simple $A$-module
Now, using the isomorphism $C(2)\cong C(1)\otimes C(1)$ and Lemma 2.9, the 4-dimensional $C(2)$-supermodule $U(1)\otimes U(1)$ decomposes as two 2-dimensional isomorphic absolutely irreducible modules … By the same reasoning, if $n=2k+1$, there are two nonisomorphic irreducible modules of dimension $2^k$, namely, $U_+(n)=U(2)^{\otimes k}\otimes U_+$ and $U_-(n)=U(2)^{\otimes k}\otimes U_-$. …
- 1,472