M Theory Lesson 98

Define the signature of a permutation to be the sequence of signs of differences. For example, for $(132)$ in $S_3$ the signature is $(+-)$. For any $S_n$ there are $2^{n - 1}$ signature types, forming a parity cube. Consider formal sums of elements in a signature class. For example, $(132) + (231)$ is such a sum. In 1976, Solomon [1] showed that the product of two such sums is always a linear combination (over $\mathbb{N}$) of signature sums. A simple example from $S_3$ is

$(--)(+-) = (321)[(132) + (231)] = (231) + (132) = (+-)$

This is called the descent algebra. In [2] Loday and Ronco define a Hopf algebra of binary trees which uses this algebra. M theorists will be more familiar with the alternative polytopes to cubes, namely associahedra and permutohedra. The vertices of these polytopes are related to the sequences

$P_n \rightarrow A_n \rightarrow Q_n$

where $Q_n$ is a basis for the Solomon descent algebra. That is, the associahedra sit in between the permutation labellings of trees with distinct levels and the cubes. For $S_3$ this yields the usual pentagon diagram. The direct sum of all group algebras $k S_n$ may be given a Hopf algebra structure, with the descent algebra as a sub Hopf algebra. Using the sequences above, it is shown that the associahedra also have a (graded) Hopf structure.

[1] L. Solomon, J. Alg. 41 (1976) 255-268
[2] J-L. Loday and M.O. Ronco, Adv. Math. 139 (1998) 293-309