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.Wed, 02 Dec 2020 10:27:39 +0100Posets with infinite outer automorphism grouphttps://ask.sagemath.org/question/54449/posets-with-infinite-outer-automorphism-group/This is probably very complicated, but is there a quick method to see whether the outer automorphism group of the incidence algebra of a finite connected poset over the rationals is finite using SAGE?
We can assume that the poset is additionally such that the underlying graph of the Hasse quiver is not a tree and such that there is no element x in the poset that is comparable to all other elements (see https://ask.sagemath.org/question/54438/filtering-certain-posets/ ), since in those cases the outer automorphism group is indeed finite or the incidence algebra is heredity (in which case it is rather boring).
Im interested in infinite outer automorphism groups to construct some exotic examples, but such posets seem to be rather rare.Sun, 29 Nov 2020 19:19:10 +0100https://ask.sagemath.org/question/54449/posets-with-infinite-outer-automorphism-group/Comment by FrédéricC for <p>This is probably very complicated, but is there a quick method to see whether the outer automorphism group of the incidence algebra of a finite connected poset over the rationals is finite using SAGE?</p>
<p>We can assume that the poset is additionally such that the underlying graph of the Hasse quiver is not a tree and such that there is no element x in the poset that is comparable to all other elements (see <a href="https://ask.sagemath.org/question/54438/filtering-certain-posets/">https://ask.sagemath.org/question/544...</a> ), since in those cases the outer automorphism group is indeed finite or the incidence algebra is heredity (in which case it is rather boring).</p>
<p>Im interested in infinite outer automorphism groups to construct some exotic examples, but such posets seem to be rather rare.</p>
https://ask.sagemath.org/question/54449/posets-with-infinite-outer-automorphism-group/?comment=54454#post-id-54454Can this be computed using some kind of homology group ?Mon, 30 Nov 2020 09:39:35 +0100https://ask.sagemath.org/question/54449/posets-with-infinite-outer-automorphism-group/?comment=54454#post-id-54454Comment by klaaa for <p>This is probably very complicated, but is there a quick method to see whether the outer automorphism group of the incidence algebra of a finite connected poset over the rationals is finite using SAGE?</p>
<p>We can assume that the poset is additionally such that the underlying graph of the Hasse quiver is not a tree and such that there is no element x in the poset that is comparable to all other elements (see <a href="https://ask.sagemath.org/question/54438/filtering-certain-posets/">https://ask.sagemath.org/question/544...</a> ), since in those cases the outer automorphism group is indeed finite or the incidence algebra is heredity (in which case it is rather boring).</p>
<p>Im interested in infinite outer automorphism groups to construct some exotic examples, but such posets seem to be rather rare.</p>
https://ask.sagemath.org/question/54449/posets-with-infinite-outer-automorphism-group/?comment=54459#post-id-54459Yes it can be computed using graph theoretic data, see theorem 2 of http://www-math.mit.edu/%7Erstan/pubs/pubfiles/5.pdf . By that result, the outer automorphism group should be finite iff r=t. Im not sure whether such functions exists in SAGE already.Mon, 30 Nov 2020 11:18:58 +0100https://ask.sagemath.org/question/54449/posets-with-infinite-outer-automorphism-group/?comment=54459#post-id-54459Answer by FrédéricC for <p>This is probably very complicated, but is there a quick method to see whether the outer automorphism group of the incidence algebra of a finite connected poset over the rationals is finite using SAGE?</p>
<p>We can assume that the poset is additionally such that the underlying graph of the Hasse quiver is not a tree and such that there is no element x in the poset that is comparable to all other elements (see <a href="https://ask.sagemath.org/question/54438/filtering-certain-posets/">https://ask.sagemath.org/question/544...</a> ), since in those cases the outer automorphism group is indeed finite or the incidence algebra is heredity (in which case it is rather boring).</p>
<p>Im interested in infinite outer automorphism groups to construct some exotic examples, but such posets seem to be rather rare.</p>
https://ask.sagemath.org/question/54449/posets-with-infinite-outer-automorphism-group/?answer=54461#post-id-54461Some code for the r-number, not tested.
def r_number(P):
H = P.hasse_diagram()
M = Matroid(H.to_undirected())
b = FreeModule(GF(2), tuple(H.edges(labels=False))).basis()
vecs = [sum(b[e] for e in ci).to_vector() for ci in M.circuits()]
return matrix(vecs).rank()
Remains only to do the same thing for the subspace that defines s.
EDIT: here is a tentative, not tested either
def s_number(P):
H = P.hasse_diagram()
b = FreeModule(GF(2), tuple(H.edges(labels=False))).basis()
vecs = []
for x, y in P.relations():
chains = H.all_paths(x, y, report_edges=True)
if len(chains) <= 1:
continue
v0 = sum(b[e] for e in chains[0]).to_vector()
vecs += [v0 + sum(b[e] for e in ci).to_vector() for ci in chains[1:]]
return matrix(vecs).rank()Mon, 30 Nov 2020 11:35:16 +0100https://ask.sagemath.org/question/54449/posets-with-infinite-outer-automorphism-group/?answer=54461#post-id-54461Comment by klaaa for <p>Some code for the r-number, not tested.</p>
<pre><code>def r_number(P):
H = P.hasse_diagram()
M = Matroid(H.to_undirected())
b = FreeModule(GF(2), tuple(H.edges(labels=False))).basis()
vecs = [sum(b[e] for e in ci).to_vector() for ci in M.circuits()]
return matrix(vecs).rank()
</code></pre>
<p>Remains only to do the same thing for the subspace that defines s.</p>
<p>EDIT: here is a tentative, not tested either</p>
<pre><code>def s_number(P):
H = P.hasse_diagram()
b = FreeModule(GF(2), tuple(H.edges(labels=False))).basis()
vecs = []
for x, y in P.relations():
chains = H.all_paths(x, y, report_edges=True)
if len(chains) <= 1:
continue
v0 = sum(b[e] for e in chains[0]).to_vector()
vecs += [v0 + sum(b[e] for e in ci).to_vector() for ci in chains[1:]]
return matrix(vecs).rank()
</code></pre>
https://ask.sagemath.org/question/54449/posets-with-infinite-outer-automorphism-group/?comment=54510#post-id-54510Thanks again. I will become a sponsor.Wed, 02 Dec 2020 10:27:39 +0100https://ask.sagemath.org/question/54449/posets-with-infinite-outer-automorphism-group/?comment=54510#post-id-54510Comment by FrédéricC for <p>Some code for the r-number, not tested.</p>
<pre><code>def r_number(P):
H = P.hasse_diagram()
M = Matroid(H.to_undirected())
b = FreeModule(GF(2), tuple(H.edges(labels=False))).basis()
vecs = [sum(b[e] for e in ci).to_vector() for ci in M.circuits()]
return matrix(vecs).rank()
</code></pre>
<p>Remains only to do the same thing for the subspace that defines s.</p>
<p>EDIT: here is a tentative, not tested either</p>
<pre><code>def s_number(P):
H = P.hasse_diagram()
b = FreeModule(GF(2), tuple(H.edges(labels=False))).basis()
vecs = []
for x, y in P.relations():
chains = H.all_paths(x, y, report_edges=True)
if len(chains) <= 1:
continue
v0 = sum(b[e] for e in chains[0]).to_vector()
vecs += [v0 + sum(b[e] for e in ci).to_vector() for ci in chains[1:]]
return matrix(vecs).rank()
</code></pre>
https://ask.sagemath.org/question/54449/posets-with-infinite-outer-automorphism-group/?comment=54509#post-id-54509You can accept my answer, maybe ? Also consider sponsoring sage : https://github.com/sponsors/sagemathWed, 02 Dec 2020 08:33:50 +0100https://ask.sagemath.org/question/54449/posets-with-infinite-outer-automorphism-group/?comment=54509#post-id-54509Comment by klaaa for <p>Some code for the r-number, not tested.</p>
<pre><code>def r_number(P):
H = P.hasse_diagram()
M = Matroid(H.to_undirected())
b = FreeModule(GF(2), tuple(H.edges(labels=False))).basis()
vecs = [sum(b[e] for e in ci).to_vector() for ci in M.circuits()]
return matrix(vecs).rank()
</code></pre>
<p>Remains only to do the same thing for the subspace that defines s.</p>
<p>EDIT: here is a tentative, not tested either</p>
<pre><code>def s_number(P):
H = P.hasse_diagram()
b = FreeModule(GF(2), tuple(H.edges(labels=False))).basis()
vecs = []
for x, y in P.relations():
chains = H.all_paths(x, y, report_edges=True)
if len(chains) <= 1:
continue
v0 = sum(b[e] for e in chains[0]).to_vector()
vecs += [v0 + sum(b[e] for e in ci).to_vector() for ci in chains[1:]]
return matrix(vecs).rank()
</code></pre>
https://ask.sagemath.org/question/54449/posets-with-infinite-outer-automorphism-group/?comment=54492#post-id-54492Thanks, I used this to search for posets with infinite outer automorphism group which have periodic coxeter matrix. For n<=7 there seem to be no such posets with n points.Tue, 01 Dec 2020 16:12:29 +0100https://ask.sagemath.org/question/54449/posets-with-infinite-outer-automorphism-group/?comment=54492#post-id-54492