While the real purpose of this posting is to test whether my messages to spnetwork are coming through, why not point out a couple typos in this otherwise wonderful paper:

page 2, line 4 from below, not counting formulas: \(J\) should be \(S\) in "if \(x \in J\) and \(x < y\), then \(y \in S\)".

page 3: The formula that describes self-duality should of course be \(\left< EF, G\right> = \left<E\otimes F, \delta G\right>\), not \(\left< EF, G\right> = \left<E\otimes G, \delta G\right>\).

There are also various minor typos but they're easily noticed. Finally, there is the issue that "self-dual" in this paper doesn't mean what "self-dual" usually means. In the Malvenuto-Reutenauer paper, a Hopf algebra is called self-dual if it is endowed with a bilinear form that makes multiplication adjoint to comultiplication; the form is not required to be nondegenerate.

I have not read past page 4 so far and I already know that the Hopf algebra \(\mathbb{Z}\mathbf{D}\) they are defining here is very important. It has a Hopf subalgebra \(\mathbb{Z}\mathbf{DS}\) made of so-called "special double posets" (aka labelled posets), which has a canonical projection onto the Malvenuto-Reutenauer Hopf algebra of permutations, and boasts an easily-described(!) inner product that projects onto the inner product of Malvenuto-Reutenauer. Loic Foissy, in arXiv:1109.1101v3, has also embedded the Malvenuto-Reutenauer Hopf algebra as a Hopf subalgebra of \(\mathbb{Z}\mathbf{DS}\) in two different ways.

#spnetwork arXiv:0905.3508

https://petitions.whitehouse.gov/petition/reverse-policy-which-prohibits-massive-open-online-courses-moocs-including-students-sanctioned/dkpm2cyM

Many probably will have guessed what it's about from the (garbled) URL: the same law that is making Sourceforge and google code block IPs from Iran and Cuba has now forced coursera to do the same. It's a tad more absurd in this case and I hope the waves go higher this time.

There are reasons for the sanctions and none of them applies to educational offerings, plain and simple.

The starting point of this paper ( arXiv:1108.1172 ) is the (relatively well-known) rowmotion operation (also called Panyushev complementation or Fon-der-Flaass action). This is a permutation of the set of order ideals of any given finite poset \( \mathcal P \). When the poset \( \mathcal P \) is an rc-poset (this notion is introduced in Definition 4.5 and roughly means a poset \( \mathcal P \) with a projection map \( \mathcal \pi \) from \( \mathcal P \) to the lattice \( \left< (2,0), (1,1) \right> \subseteq \mathbb{Z}^2 \) such that whenever an element \(p \in \mathcal P\) covers another element \(q\), we have \( \pi(p) - \pi(q) \in \left\lbrace (-1, 1), (1, 1) \right\rbrace \) ), a further permutation of the set of order ideals of \( \mathcal P \) is defined, called promotion. It owes its name to the fact that it generalizes Schützenberger's promotion on skew standard Young tableaux with two rows (albeit this generalization does not seem to extend to tableaux with three or more rows or semistandard tableaux; this might be a question worth further study). Various posets appearing in practice (rectangles, root posets and others) turn out to be rc-posets, often allowing less abstract interpretations of the rowmotion and promotion operators (Sections 3 and 6-8 of the paper).

One of the main results of the paper (Theorem 5.2) claims that for any rc-poset, rowmotion and promotion are conjugate (in the group of permutations of the set of order ideals of the poset). This is based on a lovely group-theoretical lemma (Lemma 5.1 in the paper), which is actually a consequence of an argument in Coxeter group theory (thanks to Nathan Williams for pointing this out).

Let me sketch an alternative proof of this lemma. I am not saying it is fundamentally different from the proof given in the paper (in fact, just as the proof in the paper, it can be used in the proof of Theorem 5.4 to obtain the same element \(D\), so I wouldn't be surprised if it boils down to the same construction), but I think it is slicker by means of being more abstract. We'll prove the following more general fact:

*Lemma 5.1x.* Let \(G\) be a group, and \(g_1,g_2,...,g_n\) be finitely many elements of \(G\). Assume that \(g_ig_j = g_jg_i\) for any two elements \(i\) and \(j\) of \(\left\{1,2,...,n\right\}\) satisfying \(\left|i-j\right| > 1\). Let \(\mathfrak{S}_n\) denote the \(n\)-th symmetric group. Let \(\omega\) and \(\nu\) be elements of \(\mathfrak{S}_n\). Then, the elements \(g_{\omega(1)} g_{\omega(2)} ... g_{\omega(n)}\) and \(g_{\nu(1)} g_{\nu(2)} ... g_{\nu(n)}\) of \(G\) are conjugate.

Lemma 5.1x generalizes Lemma 5.1. This is because under the conditions of Lemma 5.1, we have
\[
g_ig_j = g_i\underbrace{1}_{=(g_ig_j)^2=g_ig_jg_ig_j}g_j = \underbrace{g_ig_i}_{=g_i^2=1}g_jg_i\underbrace{g_jg_j}_{=g_j^2=1} = g_jg_i
\]
for any \(i\) and \(j\) satisfying \(\left|i-j\right| > 1\), so that Lemma 5.1x can be applied, and Lemma 5.1 follows.

Proof of Lemma 5.1x. First, a definition: For any subgroup \(H\) of \(G\), we define a binary relation \(\sim_H\) on \(G\) by letting
\[
a \sim_H b \ \ \ \ \ \Longleftrightarrow\ \ \ \ \ \left(\text{there exists }h\in H\text{ such that }a = hbh^{-1}\right)
\]
for all elements \(a\) and \(b\) of \(G\). This relation \(\sim_H\) is easily seen to be an equivalence relation. It is a generalization of conjugacy, because when \(H = G\), it is simply the relation of being conjugate elements in \(G\). For general \(H\), it refines the conjugacy relation: If two elements \(a\) and \(b\) of \(G\) satisfy \(a\sim_H b\), then \(a\) and \(b\) are conjugate.

For every \(m\in\left\{0,1,...,n\right\}\), let \(H_m\) be the subgroup of \(G\) generated by the elements \(g_1,g_2,...,g_m\). Clearly, \(H_0 \subseteq H_1 \subseteq ... \subseteq H_n\).

We will now prove the following claim:

Claim 1: For any \(m\in\left\{1,2,...,n\right\}\) and any two permutations \(\left(i_1,i_2,...,i_m\right)\) and \(\left(j_1,j_2,...,j_m\right)\) of the list \(\left(1,2,...,m\right)\), we have \(g_{i_1} g_{i_2} ... g_{i_m} \sim_{H_{m-1}} g_{j_1} g_{j_2} ... g_{j_m}\).

Proof of Claim 1: We will prove Claim 1 by induction over \(m\). The induction base case (\(m=1\)) is obvious, since there is only one permutation of the list \((1)\).

Now, for the induction step, let \(M\) be an element of \(\left\{2,3,...,n\right\}\). We will prove Claim 1 for \(m=M\), assuming that it is proven for \(m=M-1\).

Let \(\left(i_1,i_2,...,i_M\right)\) and \(\left(j_1,j_2,...,j_M\right)\) be two permutations of the list \(\left(1,2,...,M\right)\). We need to show that \(g_{i_1} g_{i_2} ... g_{i_M} \sim_{H_{M-1}} g_{j_1} g_{j_2} ... g_{j_M}\).

Since \(\left(i_1,i_2,...,i_M\right)\) is a permutation of \(\left(1,2,...,M\right)\), there exists a unique \(x\in\left\{1,2,...,M\right\}\) such that \(i_x=M\). Consider this \(x\). Since \(x\) is unique with this property, every \(j < x\) satisfies \(i_j < M\). Thus, \(g_{i_1} g_{i_2} ... g_{i_{x-1}} \in H_{M-1}\). Now,
\[
g_{i_1} g_{i_2} ... g_{i_M} = \left(g_{i_1} g_{i_2} ... g_{i_{x-1}}\right) g_{i_x} \left(g_{i_{x+1}} g_{i_{x+2}} ... g_{i_M}\right) \\
= \left(g_{i_1} g_{i_2} ... g_{i_{x-1}}\right) g_{i_x} \left(g_{i_{x+1}} g_{i_{x+2}} ... g_{i_M}\right) \left(g_{i_1} g_{i_2} ... g_{i_{x-1}}\right) \left(g_{i_1} g_{i_2} ... g_{i_{x-1}}\right)^{-1} \\
\sim_{H_{M-1}} g_{i_x} \left(g_{i_{x+1}} g_{i_{x+2}} ... g_{i_M}\right) \left(g_{i_1} g_{i_2} ... g_{i_{x-1}}\right)\ \ \ \ \ \left(\text{since }g_{i_1} g_{i_2} ... g_{i_{x-1}} \in H_{M-1}\right) \\
= g_M \left(g_{i_{x+1}} g_{i_{x+2}} ... g_{i_M}\right) \left(g_{i_1} g_{i_2} ... g_{i_{x-1}}\right)\ \ \ \ \ \left(\text{since }i_x=M\right).
\]
Similarly, we can consider the unique \(y\in\left\{1,2,...,M\right\}\) such that \(j_y=M\), and obtain
\[
g_{j_1} g_{j_2} ... g_{j_M} \sim_{H_{M-1}} g_M \left(g_{j_{y+1}} g_{j_{y+2}} ... g_{j_M}\right) \left(g_{j_1} g_{j_2} ... g_{j_{y-1}}\right).
\]

But the two lists \(\left(i_{x+1}, i_{x+2}, ..., i_M, i_1, i_2, ..., i_{x-1}\right)\) and \(\left(j_{y+1}, j_{y+2}, ..., j_M, j_1, j_2, ..., j_{y-1}\right)\) are two permutations of the list \(\left(1, 2, ..., M-1\right)\) (because they are obtained from the lists \(\left(i_1,i_2,...,i_M\right)\) and \(\left(j_1,j_2,...,j_M\right)\), which are permutations of \(\left(1, 2, ..., M\right)\), by removing the letter \(M\) and cyclic rotation). Hence, we can apply Claim 1 with \(m = M-1\) to these two permutations (by the induction hypothesis), and conclude that
\[
\left(g_{i_{x+1}} g_{i_{x+2}} ... g_{i_M}\right) \left(g_{i_1} g_{i_2} ... g_{i_{x-1}}\right) \sim_{H_{M-2}} \left(g_{j_{y+1}} g_{j_{y+2}} ... g_{j_M}\right) \left(g_{j_1} g_{j_2} ... g_{j_{y-1}}\right).
\]
In other words, there exists a \(t \in H_{M-2}\) such that
\[
\left(g_{i_{x+1}} g_{i_{x+2}} ... g_{i_M}\right) \left(g_{i_1} g_{i_2} ... g_{i_{x-1}}\right) = t \left(g_{j_{y+1}} g_{j_{y+2}} ... g_{j_M}\right) \left(g_{j_1} g_{j_2} ... g_{j_{y-1}}\right) t^{-1}.
\]
Consider this \(t\). By the hypothesis of the lemma, \(g_ig_j = g_jg_i\) for any two elements \(i\) and \(j\) of \(\left\{1,2,...,n\right\}\) satisfying \(\left|i-j\right| > 1\). Thus, \(g_M\) commutes with each of the elements \(g_1,g_2,...,g_{M-2}\). Hence, \(g_M\) commutes with every element of the subgroup of \(G\) generated by \(g_1,g_2,...,g_{M-2}\). In other words, \(g_M\) commutes with every element of \(H_{M-2}\) (since \(H_{M-2}\) is the subgroup of \(G\) generated by \(g_1,g_2,...,g_{M-2}\)). In particular, this yields that \(g_M\) commutes with \(t\) (since \(t\) is an element of \(H_{M-2}\)). In other words, \(g_M t = t g_M\).

Now,
\[
g_{i_1} g_{i_2} ... g_{i_M} \sim_{H_{M-1}} g_M \underbrace{\left(g_{i_{x+1}} g_{i_{x+2}} ... g_{i_M}\right) \left(g_{i_1} g_{i_2} ... g_{i_{x-1}}\right)}_{=t \left(g_{j_{y+1}} g_{j_{y+2}} ... g_{j_M}\right) \left(g_{j_1} g_{j_2} ... g_{j_{y-1}}\right) t^{-1}} \\
= \underbrace{g_M t}_{= t g_M} \left(g_{j_{y+1}} g_{j_{y+2}} ... g_{j_M}\right) \left(g_{j_1} g_{j_2} ... g_{j_{y-1}}\right) t^{-1} \\
= t g_M \left(g_{j_{y+1}} g_{j_{y+2}} ... g_{j_M}\right) \left(g_{j_1} g_{j_2} ... g_{j_{y-1}}\right) t^{-1} \\
\sim_{H_{M-1}} g_M \left(g_{j_{y+1}} g_{j_{y+2}} ... g_{j_M}\right) \left(g_{j_1} g_{j_2} ... g_{j_{y-1}}\right)\ \ \ \ \ \left(\text{since }t \in H_{M-2} \subseteq H_{M-1}\right) \\
\sim_{H_{M-1}} g_{j_1} g_{j_2} ... g_{j_M}.
\]

This shows that Claim 1 holds for \(m = M\). This completes the induction step.

Claim 1 is thus proven. Now, we can apply Claim 1 to \(m = n\), \(i_k = \omega(k)\) and \(j_k = \nu(k)\) (since \(\left(\omega(1), \omega(2), ..., \omega(n)\right)\) and \(\left(\nu(1), \nu(2), ..., \nu(n)\right)\) are permutations of the list \(\left(1, 2, ..., n\right)\)). We conclude that \(g_{\omega(1)} g_{\omega(2)} ... g_{\omega(n)} \sim_{H_{n-1}} g_{\nu(1)} g_{\nu(2)} ... g_{\nu(n)}\). Hence, the elements \(g_{\omega(1)} g_{\omega(2)} ... g_{\omega(n)}\) and \(g_{\nu(1)} g_{\nu(2)} ... g_{\nu(n)}\) of \(G\) are conjugate. Lemma 5.1x is proven.

Other arguments that prove Lemma 5.1x can be found in §1 of Bill Casselman's notes "Coxeter elements in finite Coxeter groups" ( http://www.math.ubc.ca/~cass/research/pdf/Element.pdf ) or §3.16 of James E. Humphreys, Reflection groups and Coxeter groups, CUP 1994 (thanks to Nathan for the reference). Both of these sources actually work in a more general setting, where the condition "\(\left|i-j\right| > 1\)" is replaced by "\(i\) and \(j\) are not adjacent vertices of \(T\)", where \(T\) is a given tree with vertex set \(\left\{1,2,...,n\right\}\). My proof can also be adopted to this more general setting (exercise).

*

It is worth pointing out a few typos in the paper (version arXiv:1108.1172v3):

On page 3, in Theorem 2.8, the "\(C_n\)" should be a "\(C_{nm}\)".

On page 8, after the proof of Corollary 4.9, "in the singleton set \(I_{i+1}-I_{i}\)" should be "in the singleton set \(I_{i}-I_{i-1}\)".

On page 10, in the proof of Lemma 5.1, both \(\prod_{j=1}^n\)'s should be \(\prod_{i=1}^n\)'s.

On the first line of page 14, I believe both \(J\)'s in "\(e_{i}+e_{n}\in J\) if and only if \(e_{i}-e_{n}\in J\)" should be "\(I\)"'s.

NOTE: I have *not* read the whole paper (my main interest was in Sections 4 and 5).

#spnetwork arXiv:1108.1172

This is a test of SPNetwork. Sorry, James and Ben! (A very nice paper, by the way.
square brackets:
\[
square-brackets\text{square-brackets} \\
newline
\]
and eqnarray:
\begin{eqnarray*}
blah \\
blah
\end{eqnarray*}
Also, italics and *boldface*.
#no_longer_tagged_spnetwork arXiv:math/0407227