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For a given triple (monad) U : C → C in the category C , we develop a theory of descent for U. We start by introducing the basic constructions associated to a triple: descent data, symmetry operators, and flat connections. The main result of this section asserts that the sets of these objects are bijectively equivalent. Next we construct a monoidal category C (U) such that U is an algebra in C (U) . If C is abelian, we define Amitsur cohomology of U with coefficients in a functor F : C (U)→ D . As an application of this construction, in the case where U is faithfully exact, we describe those morphisms that descend with respect to U. In the last part of the paper we classify all U-forms of a given object C 0 ∈ C . We show that there is a one-to-one correspondence between the set of equivalence classes of U-forms and a certain noncommutative Amitsur cohomology. Let A/B be an extension of associative unitary rings and let C be the category of right B-modules. Then (−)⊗ B A : C → C is a triple which is faithfully exact if and only if the extension A/B is faithfully flat. Specializing our results to this particular setting, we recover faithfully flat descent theory for extensions of (not necessarily commutative) rings.

Solomon's descent algebra is generated by sums of descent classes corresponding to certain hook shapes. This particularly implies that the ring of class functions of any finite symmetric group S n is generated by the irreducible characters corresponding to certain hook partitions of n . As another consequence, a second generating set of Solomon's descent algebra (and of the ring of class functions of S n ) is obtained related to the major index of permutations.

Elements of the hyperoctahedral group Bn can be represented as lists of integers π = π1π2 … πn, where the list of absolute value of the πk's is a permutation of the integers from 1 to n. The descent set of such a π is Des(π) = {kϵ {0, …, n − 1} ¦ πk > πk + 1}, where k = 0, we set π0 = 0. A descent class AS of Bn, is the formal sum of all elements of Bn with a given descent set S ⊆{0, 1, …, n − 1}. In this paper, we shall give a combinatorial proof of a result of Solomon [6] stating that the product, in the group algebra Q[Bn], of two descent classes is a linear combination of descent classes. We refer to the linear span of these descent classes (it is a subalgebra ∑Bn of Q[Bn], as Solomon's Hyperoctahedral descent algebra. We derive from our combinatorial multiplication rule (for ∑ Bn) that there is a family of ideals Jv(n) of ∑ Bn such that J0(n) ⊆- J1(n) … ⊆- Jn(n) , with ∑ Bn − v + 1 ∼- ∑ BnJv(n) . In [4], Garsia and Reutenauer relate a similar decomposition of the multiplicative structure of the descent algebra of the symmetric group Sn, to the action of the symmetric group on the free Lie algebra. It develops that one can introduce a Bn-analog of the free Lie algebra, and give similar results for ∑ Bn.