ASKSAGE: Sage Q&A Forum - Latest question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 29 May 2019 13:09:55 -0500Orbits on group actions acting on setshttps://ask.sagemath.org/question/9652/orbits-on-group-actions-acting-on-sets/Hello!
I am wondering how to solve the following problem efficiently.
I have a Permuation Group $G$ acting on $A = \{1,\ldots,n\}$ and I wish to compute the orbits of $G$ but not the ones of $G$ acting on $A$ but rather for $G$ acting on some $S \subseteq A \times A$ in the natural way. That is if $g \in G$ and $ x = \{a,b\} \in S$ then $x^g = {a^g,b^g\} \in S$
Other software for permuation groups (magma, gap) allows to do this by specifing an additional option "on sets/on tuples" to compute the specifed orbits.
I am wondering how could I do the same in sage, given a permuation group $G$ and an $S$ as described above.
Thanks!SGQWed, 26 Dec 2012 03:56:11 -0600https://ask.sagemath.org/question/9652/Constructing group representationshttps://ask.sagemath.org/question/46701/constructing-group-representations/I've got a vector space $V= \mathbb{F}_p^n$ and I want to construct rings such as the symmetric algebra $\mathrm{Sym}^*(V)$, the divided power algebra generated by $V$, and so forth. I can do this, but the thing I'm not sure how to do is to build these objects along with the induced action of $\mathrm{GL}_n(\mathbb{F}_p)$ by algebra homomorphisms.
This brings two more general questions:
1. (Likely simple) If I've got a finite group $G$, how do I build a vector space $V$ with an action of $G$? In particular, it would be nice if I could build $V$ with named generators $v_1, v_2, \ldots$.
2. If I've got a vector space $V$ with an action of a group $G$, and I want to define an algebra $\mathcal{F}(V)$ which is functorial in $V$, then how do I port the action of $G$ over?
Alternatively, it might be computationally a lot more efficient to simply construct the divided power algebra over $\mathbb{F}_p$ on $n$ generators from scratch, but if I do this then how do I tell sage how $\mathrm{GL}_n(\mathbb{F}_p)$ acts on it?
(The divided power algebra on one generator $y$ is a ring with polynomial generators $y_1, y_2, y_3, \ldots$ subject to the relation $y_iy_j=\binom{i+j}{i}y_{i+j}$. So intuitively, one thinks of $y_n=\frac{y_1^n}{n!}$. Over a field of characteristic $p$, this amounts to being an algebra on $y_1, y_p, y_{p^2}, \ldots$ where the $p$-th power of each generator equals zero.)ksankarWed, 29 May 2019 13:09:55 -0500https://ask.sagemath.org/question/46701/group 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:
$$e_0 = \hat{G}$$
$$e_1=\hat{b}−\widehat{(\text{ideal generated by }a)∗ (\text{ideal generated by }b)}$$
$$e_2=\widehat{a−G}$$
$$e_3=\widehat{(\text{ideal generated by }a_p)∗ (\text{ideal generated by }b)}−\hat{G}$$bandanaTue, 16 Apr 2019 04:04:47 -0500https://ask.sagemath.org/question/46218/How to treat a vector space as a group?https://ask.sagemath.org/question/33990/how-to-treat-a-vector-space-as-a-group/
I need to use a module as a group, so that I can define a group algebra over this module.
Essentially, I want to take the group of 2-dimensional complex vector space and define a group algebra over this. I cannot find appropriate direction on the internet and sage gives me the ridiculous "False" as below.
sage: V=FreeModule(CC,2)
sage: V in Groups()
False
Nihar GargavaFri, 01 Jul 2016 12:27:43 -0500https://ask.sagemath.org/question/33990/