# 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/544... ), 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.

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Can this be computed using some kind of homology group ?

( 2020-11-30 09:39:35 +0200 )edit

Yes it can be computed using graph theoretic data, see theorem 2 of http://www-math.mit.edu/%7Erstan/pubs... . By that result, the outer automorphism group should be finite iff r=t. Im not sure whether such functions exists in SAGE already.

( 2020-11-30 11:18:58 +0200 )edit

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Some 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()

more

Thanks, 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.

( 2020-12-01 16:12:29 +0200 )edit
1

( 2020-12-02 08:33:50 +0200 )edit

Thanks again. I will become a sponsor.

( 2020-12-02 10:27:39 +0200 )edit