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.Tue, 23 May 2023 17:52:44 +0200Action of a cyclic group on the interval poset of a Boolean latticehttps://ask.sagemath.org/question/68630/action-of-a-cyclic-group-on-the-interval-poset-of-a-boolean-lattice/Let $G$ be a cyclic group of order $n$ and $B=B_n$ the Boolean lattice of an $n$-set and $P$ be poset of all intervals of $B$. $G$ acts in a natural way on $B$ by sending a subset $x=(k_1,...,k_m )$ to $gx=(k_1+1,...,k_m+1 )$ (when $g$ is the canonical generator of the cyclic group) where we view the integers mod n in the set $\{(1,...,n)\}$.
This action induces an action of $G$ on the set of all intervals of $B$ by sending an interval $[x,y]$ to $[gx,gy]$.
My question is whether there is an easy way to obtain this action of $G$ on $P$ via Sage.
Here is how far I come with Sage:
p=3
B = Subsets([1,..,p])
def cyc_act(B): return Set(i.mod(p) + 1 for i in B)
BB= posets.BooleanLattice(p, use_subsets=True)
P=BB.intervals_poset()
plot(P)
So I defined the action of G on B and the interval poset, but I am not sure how to continue in a good way.
Thanks for any helpSun, 21 May 2023 22:39:23 +0200https://ask.sagemath.org/question/68630/action-of-a-cyclic-group-on-the-interval-poset-of-a-boolean-lattice/Comment by Max Alekseyev for <p>Let $G$ be a cyclic group of order $n$ and $B=B_n$ the Boolean lattice of an $n$-set and $P$ be poset of all intervals of $B$. $G$ acts in a natural way on $B$ by sending a subset $x=(k_1,...,k_m )$ to $gx=(k_1+1,...,k_m+1 )$ (when $g$ is the canonical generator of the cyclic group) where we view the integers mod n in the set ${(1,...,n)}$.
This action induces an action of $G$ on the set of all intervals of $B$ by sending an interval $[x,y]$ to $[gx,gy]$.
My question is whether there is an easy way to obtain this action of $G$ on $P$ via Sage.
Here is how far I come with Sage:</p>
<pre><code>p=3
B = Subsets([1,..,p])
def cyc_act(B): return Set(i.mod(p) + 1 for i in B)
BB= posets.BooleanLattice(p, use_subsets=True)
P=BB.intervals_poset()
plot(P)
</code></pre>
<p>So I defined the action of G on B and the interval poset, but I am not sure how to continue in a good way.
Thanks for any help</p>
https://ask.sagemath.org/question/68630/action-of-a-cyclic-group-on-the-interval-poset-of-a-boolean-lattice/?comment=68687#post-id-68687Did you try to follow example from documentation? https://doc.sagemath.org/html/en/reference/coercion/sage/categories/action.htmlTue, 23 May 2023 17:52:44 +0200https://ask.sagemath.org/question/68630/action-of-a-cyclic-group-on-the-interval-poset-of-a-boolean-lattice/?comment=68687#post-id-68687