|Once more, the peak phenomenon!
The peak set of a permutation \pi in the symmetric group S_n consists of
all 1<i<n such that (i-1)\pi<i\pi>(i+1)\pi. The linear
span P_n of the sums of all
permutations in S_n with a given peak set is a sub-algebra of the
symmetric group algebra, due to Nyman; and the direct sum P of all P_n
is a Hopf sub-algebra of the Solomon descent algebra D, dual to the
Stembridge algebra of peak functions.
In our self-contained approach, there is a particular interest in inner
products in P_n, arising from the ordinary multiplication of permutations.
Peak counterparts of several results on the descent algebra D_n will be
presented, based on the fact that P_n turns out to be a left ideal of D_n.
This includes combinatorial and algebraic characterizations of P_n, the
basics of peak Lie idempotents, and a number of observations on the
structure of P_n and some sub-algebras.
Enough information will be provided to transfer these results to the
setting of Stembridge's peak algebra, by duality.