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

4 Responses so far »

  1. 1

    Matti Pitkanen said,

    Dear Kea,

    I am a little bit confused. I would guess that signature sum is group algebra homomorphism: signature sum for a product of group algebra elements (sums of group elements) is product of sums of signature sums.

    I am however puzzled since the formula $(–)(+-) = (321)[(132) + (231)] = (231) + (132) = (+-)$
    does not fit with this expectation. I would have expectd (–)[(+-)+(-+)] on left hand side.


  2. 2

    Kea said,

    Matti, let’s do an example from S4:


    = (4123+3124+2134)(4312+4213+3214)
    = 2431 + 3421 + 1432 + 4321 + 1423 + 1324 + 3412 + 2413 + 2314

    = (+–) + (—) + (+-+)

    Sorry for the abuse of notation using signatures to represent class sums…

  3. 3

    David Speyer said,

    If you are interested in this material, you might like to look at Nathan Reading’s paper


    where he puts both of the examples you include into a more general context of lattice congruences. There isn’t a categorical side to his theory … yet!

  4. 4

    Kea said,

    Thanks for the link, David.

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