ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Thu, 25 Apr 2019 00:46:40 +0200group algebrahttps://ask.sagemath.org/question/46218/group-algebra/Can anyone help in writing code to find the list of idempotent and primitive elements of a group algebra?
The examples goes like this. Let $p$ be an odd prime such that $\bar2$ generates $U(Z_{p^2})$ and let $G =(\text{ideal generated by }a) ∗ (\text{ideal generated by }b)$ an abelian group, with $o(a)=p^2$ and $o(b)=p$.
Then $F_2G$ has four inequivalent minimal codes, namely, the ones generated by the idempotents:
$$e_0 = \hat{G}$$
$$e_1=\hat{b}−\widehat{(\text{ideal generated by }a)∗ (\text{ideal generated by }b)}$$
$$e_2=\widehat{a−G}$$
$$e_3=\widehat{(\text{ideal generated by }a_p)∗ (\text{ideal generated by }b)}−\hat{G}$$Tue, 16 Apr 2019 11:04:47 +0200https://ask.sagemath.org/question/46218/group-algebra/Comment by John Palmieri for <p>Can anyone help in writing code to find the list of idempotent and primitive elements of a group algebra?</p>
<p>The examples goes like this. Let $p$ be an odd prime such that $\bar2$ generates $U(Z_{p^2})$ and let $G =(\text{ideal generated by }a) ∗ (\text{ideal generated by }b)$ an abelian group, with $o(a)=p^2$ and $o(b)=p$.</p>
<p>Then $F_2G$ has four inequivalent minimal codes, namely, the ones generated by the idempotents:</p>
<p>$$e_0 = \hat{G}$$
$$e_1=\hat{b}−\widehat{(\text{ideal generated by }a)∗ (\text{ideal generated by }b)}$$
$$e_2=\widehat{a−G}$$
$$e_3=\widehat{(\text{ideal generated by }a_p)∗ (\text{ideal generated by }b)}−\hat{G}$$</p>
https://ask.sagemath.org/question/46218/group-algebra/?comment=46253#post-id-46253Is your group finite or infinite? If it's finite, you could hope to write down a linear map, the kernel of which is the space of primitives. If it's infinite, then you need a computational method to implement.Thu, 18 Apr 2019 06:58:45 +0200https://ask.sagemath.org/question/46218/group-algebra/?comment=46253#post-id-46253Comment by slelievre for <p>Can anyone help in writing code to find the list of idempotent and primitive elements of a group algebra?</p>
<p>The examples goes like this. Let $p$ be an odd prime such that $\bar2$ generates $U(Z_{p^2})$ and let $G =(\text{ideal generated by }a) ∗ (\text{ideal generated by }b)$ an abelian group, with $o(a)=p^2$ and $o(b)=p$.</p>
<p>Then $F_2G$ has four inequivalent minimal codes, namely, the ones generated by the idempotents:</p>
<p>$$e_0 = \hat{G}$$
$$e_1=\hat{b}−\widehat{(\text{ideal generated by }a)∗ (\text{ideal generated by }b)}$$
$$e_2=\widehat{a−G}$$
$$e_3=\widehat{(\text{ideal generated by }a_p)∗ (\text{ideal generated by }b)}−\hat{G}$$</p>
https://ask.sagemath.org/question/46218/group-algebra/?comment=46350#post-id-46350Note: also asked as
- [Math Stack Exchange question 3174517: Algebra using Sage](https://math.stackexchange.com/q/3174517)Thu, 25 Apr 2019 00:46:40 +0200https://ask.sagemath.org/question/46218/group-algebra/?comment=46350#post-id-46350