 
Summary: THE PEAK ALGEBRA
AND
THE DESCENT ALGEBRAS OF TYPES B AND D
MARCELO AGUIAR, NANTEL BERGERON, AND KATHRYN NYMAN
Abstract. We show the existence of a unital subalgebra Pn of the symmetric group
algebra linearly spanned by sums of permutations with a common peak set, which we
call the peak algebra. We show that Pn is the image of the descent algebra of type B
under the map to the descent algebra of type A which forgets the signs, and also the
image of the descent algebra of type D. The algebra Pn contains a twosided ideal
Pn
which is defined in terms of interior peaks. This object was introduced in previous work
by Nyman [28]; we find that it is the image of certain ideals of the descent algebras
of types B and D introduced in [4] and [9]. We derive an exact sequence of the form
0
Pn Pn Pn2 0. We obtain this and many other properties of the peak
algebra and its peak ideal by first establishing analogous results for signed permuta
tions and then forgetting the signs. In particular, we construct two new commutative
semisimple subalgebras of the descent algebra (of dimensions n and n
