TITLE: Outer automorphism group of posets
QUESTION [5 upvotes]: Let $X$ be a finite poset (we can assume it is connected) and $A_K(X)$ the incidence algebra of $X$ over a field $K$.
The following result is well known, see for example corollary 7.3.7 in the book "Incidence algebras" by Spiegel and O'Donnell.

Theorem: If there exists an element $x \in X$ such that every element is comparable to $x$, then every automorphism of $A_K(X)$ is the composition of an inner automorphism of $A_K(X)$ and an automorphism of $X$.

Note that that theorem includes many posets, for example any poset with a global maximum or minimum.
In particular, this gives an explicit description of the outer automorphism group of $A_K(X)$, which is always finite for such posets $X$.

Question: Is a classification of finite posets $X$ such that the outer automorphism group of $A_K(X)$ is always finite for any field $K$ known? In particular, what is any easy example such that the outer automorphism group can be infinite?

There is a more general result in the book (Theorem 7.3.6.) but it uses multiplicative automorphisms and I do not have a feeling how many there are up to inner automorphisms.

REPLY [2 votes]: Theorem 2 of the paper http://www-math.mit.edu/~rstan/pubs/pubfiles/5.pdf states the following. Let $H$ denote the Hasse diagram of $X$, considered as a graph. Let $r$ denote the dimension  of the mod 2 cycle space $V$ of $H$, and let $t$ denote the dimension of the subspace of $V$ generated by cycles consisting of two unrefinable (or saturated) chains of $X$ with the same endpoints. Then $\mathrm{Aut}(A_K(X))$ is isomorphic to a semidirect product of $(K^*)^{r-t}$ by $\mathrm{Aut}(X)$, where $K^*$ is the multiplicative group of $K$.
The smallest example where $r-t>0$ is the four-element "butterfly" poset, with elements $x,y,z,w$ and cover relations $x<z, x<w, y<z,y<w$. Note also that if $P$ has a $\hat{0}$ or $\hat{1}$, then $r=t$