ASKSAGE: Sage Q&A Forum - Latest question feedhttp://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 -0500Constructing group representationshttp://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 -0500http://ask.sagemath.org/question/46701/Branching to Levi Subgroups in Sagehttp://ask.sagemath.org/question/45691/branching-to-levi-subgroups-in-sage/In the Sage computer package, there useful exist tools for branching representations of a simple Lie group to a Levi subgroup. See for example the root system $branching Rules \subseteq $ combinatorics in the Sage manual
Explicitly, one is branching to subgroup corresponding to a Dynkin sub-diagram, obtained by removing a single node.
For example, we can branch from $\operatorname{SL}(n)$ to the subgroup $\operatorname{SL}(n-1)$.
However, $\operatorname{SL}(n-1)$ can be considered as "living" in the larger subgroup
$\operatorname{SL}(n-1) \times \operatorname{U}(1)$. This is true for every subgroup coming from a deleted node, i.e. one can always take the product of the subgroup with $\operatorname{U}(1)$, to obtain a larger subgroup.
How does one branch to this subgroup in Sage. For example, it is done in the LieArt program for mathematica: see A3 of the ArXiv version of Lie Art.
Is this also possible in Sage?nadiasusyWed, 06 Mar 2019 15:10:48 -0600http://ask.sagemath.org/question/45691/Using GAP package in Sagehttp://ask.sagemath.org/question/26163/using-gap-package-in-sage/I would like to use [this GAP package](http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap1.html#X7DB566D5785B7DBC) to do some computations of representations, in particular [these commands](http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap4.html). However, I want to use Sage and not GAP directly for various reasons (such as not having to switch back and forth between systems for where the computations come from).
An ideal "answer" to this question would give me
- Instructions for how to install this package in Sage (assuming Sage's GAP is new enough, which I think it is)
- How to get a group ring nicely with or without this package, from a given (Sage) group
- How to use `PrimitiveCentralIdempotentsByCharacterTable` from within Sage with this
- Ideally, how to apply a given representation to one of these idempotents
That's a tall order, probably, but I don't use the group theory stuff in Sage too often, so it would save me a lot of time if someone who "just knows" the syntax was able to help out. Thanks!kcrismanWed, 11 Mar 2015 17:50:53 -0500http://ask.sagemath.org/question/26163/Order of Irreducible Representation Characters wrong in Sage?http://ask.sagemath.org/question/9300/order-of-irreducible-representation-characters-wrong-in-sage/I am using Sage to calculate Irreducible Representations (IRR) of symmetry groups of graphs. Most of the time the order of the IRR characters match the order of the conjugacy class representatives. For example the trivial representation characters are the first row of the character table (1,1,1,1,...,1) and the first conjugacy class is ( ), the identity representation. But in the example below for a star graph (one central vertex, the 4 other vertices on spokes from the center) the trivial representation characters are the *last* row in the character table and the conjugacy class representatives have the identity as the *first* element.
Example (star graph).
ct=G.character_table()
print ct
[ 1 -1 1 1 -1]
[ 3 -1 -1 0 1]
[ 2 0 2 -1 0]
[ 3 1 -1 0 -1]
[ 1 1 1 1 1] <--- trivial rep. is last
cc=G.conjugacy_classes_representatives()
print cc
[(), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4)]
^ trivial class is first
Is this a bug or known issue for Sage? How can I get around it so I know the orders of the characters and representatives match up properly?
LouChaosThu, 06 Sep 2012 04:04:45 -0500http://ask.sagemath.org/question/9300/