The peak algebra and the descent algebras of types B and D

We show the existence of a unital subalgebra $\mathfrak{P}_n$ 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 $\mathfrak{P}_n$ 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 $\mathfrak{P}_n$ contains a two-sided ideal $\overset{\circ}{\mathfrak{P}}_n$ which is defined in terms of interior peaks. This object was introduced in previous work by Nyman (2003); we find that it is the image of certain ideals of the descent algebras of types B and D. We derive an exact sequence of the form $0\to\overset{\circ}{\mathfrak{P}}_n \to\mathfrak{P}_n\to\mathfrak{P}_{n-2}\to 0$. We obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs. In particular, we construct two new commutative semisimple subalgebras of the descent algebra (of dimensions $n$ and $\lfloor\frac{n}{2}\rfloor+1)$ by grouping permutations according to their number of peaks or interior peaks. We discuss the Hopf algebraic structures that exist on the direct sums of the spaces $\mathfrak{P}_n$ and $\overset{\circ}{\mathfrak{P}}_n$ over $n\geq 0$ and explain the connection with previous work of Stembridge (1997); we also obtain new properties of his descents-to-peaks map and construct a type B analog.