Schützenberger promotion, studied (for example) in Richard Stanley, Promotion and Evacuation, 2009, is a permutation of the set of all linear extensions of a finite poset. Since one can identify the linear extensions of a poset with saturated chains of order ideals in that poset, this allows one to also view Schützenberger promotion as a permutation of the set of the latter. The famous promotion of standard Young tableaux is a particular case of this.
Striker-Williams promotion, defined in Jessica Striker, Nathan Williams, Promotion and Rowmotion, arXiv:1108.1172v3, Definition 4.13, is a permutation of the set of all order ideals (not saturated chains of order ideals!) of a so-called "rc poset" (which is a poset with a map into $\mathbb Z^2$ satisfying certain conditions, best viewed as a way to draw its Hasse diagram on a grid; see below or §4.2 of Striker-Williams for an exact definition).
Apparently people are considering these two promotions to be closely related. However, the only direct relation I am aware of is Striker-Williams Theorem 4.12, which bijects Schützenberger promotion on standard tableaux on a two-rowed Young diagram with Striker-Williams promotion on a poset which looks like a triangle grid.
Questions:
1. Is this really the only relation? Is promotion of standard Young tableaux of a Young diagram with more than $2$ rows not a (known) case of Striker-Williams promotion?
2. I've seen some kind of promotion on semistandard Young tableaux being mentioned on the internet. Assuming it's not a typo, how is that defined?
Appendix:
Let me define the two notions involved for the sake of completeness. Probably the sources quoted give better definitions...
Definition of Schützenberger promotion: Let $P$ be a finite poset. Let $\mathcal L\left(P\right)$ denote the set of all linear extensions of $P$. We define a map $\partial : \mathcal L\left(P\right)\to \mathcal L\left(P\right)$ as follows:
Let $f \in \mathcal L\left(P\right)$ be a linear extension. We set $p=\left|P\right|$, and we view $f$ as a function $P\to\left\lbrace 1,2,...,p\right\rbrace$, i. e., as a labelling of the elements of $P$ by the numbers $1$, $2$, ..., $p$ (we get this labelling by labelling every element $v\in P$ with the number $\left| \left\lbrace w\in P \ \mid \ f\left(w\right)\leq f\left(v\right) \right\rbrace \right|$). Define a (dynamic) map $g:P\to\mathbb Z$ by $g = f$ (we will be modifying $g$, while $f$ remains static). If $p=0$, do nothing. Else, set $u$ to be the element of $P$ labelled $1$ (that is, the smallest element of $P$ with respect to $g$), and do the following loop:
While there exists an element of $P$ covering $u$:
let $v$ be the smallest (with respect to $g$) among the elements of $P$ covering $u$ (that is, the element $p$ of $P$ covering $u$ with smallest $g\left(p\right)$);
slid the label of $v$ down to $u$ (that is, set $g\left(u\right)$ to be $g\left(v\right)$, accepting that $g$ will temporarily fail to be injective);
set $u = v$.
Endwhile.
After the end of this loop, label $u$ with $p+1$ (that is, set $g\left(u\right) = p+1$), and then subtract $1$ from each label (i. e., replace $g$ by $g-\mathbf{1}$, where $\mathbf{1}$ is the constant function $P\to\mathbb Z,\ p\mapsto 1$).
The resulting $g$ is called the promotion of $f$, and denoted by $\partial f$. (It is more common to call it $f\partial$, so that $\partial$ is seen as a map acting from the right).
Definition of Striker-Williams promotion: Let $P$ be a finite poset. Let $J\left(P\right)$ denote the set of all order ideals of $P$. For every $q\in P$, define a map $t_p : J\left(P\right) \to J\left(P\right)$ as follows: Let $I \in J\left(P\right)$. If $I \bigtriangleup \left\lbrace p\right\rbrace$ (with $\bigtriangleup$ standing for "symmetric difference") is an order ideal of $P$, set $t_p\left(I\right) = I \bigtriangleup \left\lbrace p\right\rbrace$. Otherwise, set $t_p\left(I\right) = I$.
Let $\mathbb Z^2_{\operatorname*{ev}}$ denote the $\mathbb Z$-submodule of $\mathbb Z^2$ spanned by $\left(1,1\right)$ and $\left(2,0\right)$. In other words, let $\mathbb Z^2_{\operatorname*{ev}}$ be the set of all $\left(x,y\right)\in\mathbb Z^2$ for which $x+y$ is even.
Now, let $P$ be a finite rc-poset; this means a poset along with a map $\pi : P \to \mathbb Z^2_{\operatorname*{ev}}$ such that whenever an element $p_1$ of $P$ covers an element $p_2$ of $P$, we have $\pi\left(p_1\right)-\pi\left(p_2\right) \in \left\lbrace \left(-1,1\right), \left(1,1\right) \right\rbrace$. (See §4.2 of Striker-Williams for some good pictures of what this means.)
For every $p\in P$, let $\pi_1\left(p\right)$ denote the first coordinate of $\pi\left(P\right)$. Now, consider the composition of the maps $t_p$ in decreasing order of $\pi_1\left(p\right)$ (the relative order of the $t_p$ for distinct $p$ having the same $\pi_1\left(p\right)$ does not matter). This composition is Striker-Williams promotion.
