Constructing group representations

asked 2019-05-29 20:12:22 +0200

ksankar gravatar image

updated 2020-06-01 11:23:14 +0200

FrédéricC gravatar image

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.)

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