# Obtaining quotient posets of the Boolean lattice via Sage

Let $G$ be a subgroup of the symmetric group $S_n$ that acts in a natural way on the Boolean lattice $B_n$, see for example chapter 5 of the book on algebraic combinatorics by Stanley.

$B_n/G$ for a subgroup $G$ of the symmetric group $S_n$ (that acts naturally on $B_n$) is defined as the poset of orbits under the natural order (that is one orbit $a$ is $\geq$ another orbit $b$ if and only if there exists elements $a' \in a$ and $b' \in b$ such that $a' \geq b'$).

The posets $B_n /G$ are graded of rank $n$, rank-symmetric, rank-unimodal, and Sperner. See theorem 5.8. in the book of Stanley. In this book one can also find open problems about such posets and thus it might be a good class to study in Sage.

For example when $n=3$ and $G$ is generated by the permutation (1,2) then the resulting poset is isomorphic to the product of the chain with 2 elements and the chain with 3 elements.

My question is how one can obtain the quotient poset $B_n /G$ in Sage when one inputs the group G generated by cycles?

The Boolean lattice on n points can be obtained in Sage via B4=posets.BooleanLattice(4) . Im not sure how to obtain a subgroup of the symmetric group acting on $B_n$ in SAGE and the resulting quotient poset.

Also is it possible to obtain the list of all connected posets obtained in this way for a given $n$? (meaning all connected posets of the form $B_n /G$ where $G$ is a subgroup of $S_n$). This will be a huge list but maybe for n<=5 it is possible to obtain it via sage.

If you already have code to define $B_n$ and $G$, please provide it.

Use the simplest example to illustrate the question: lowest $n$ and simplest $G$.