2019-06-01 13:23:18 +0200 | received badge | ● Student (source) |
2019-05-30 08:08:03 +0200 | asked a question | 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:
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.) |