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In this paper, we study a special case of the first triangular representation of the symmetric groups when n=6 over a field K of characteristic p.

Let m,n ∈ ℕ. We define [Formula: see text] to be the set of (n+m)-braids of the sphere whose associated permutation lies in the subgroup Sn× Smof the symmetric group Sn+mon n+m letters. In a previous paper [13], we showed that if n ≥ 3, then there exists the following generalisation of the Fadell–Neuwirth short exact sequence: [Formula: see text] where [Formula: see text] is the group homomorphism (defined for all n ∈ ℕ) given geometrically by forgetting the last m strings. In this paper we study the splitting of this short exact sequence, as well as the existence of a cross-section for the fibration [Formula: see text] of the quotients of the corresponding configuration spaces.Our main results are as follows: if n = 1 (respectively, n = 2) then the homomorphism p*and the fibration p admit (respectively, do not admit) a section. If n = 3, then p*and p admit a section if and only if m ≡ 0,2 (mod 3). If n ≥ 4, we show that if p*and p admit a section then m ≡ ε1(n - 1)(n - 2) - ε2n(n - 2) (mod n(n - 1)(n - 2)), where ε1,ε2∈ {0,1}.Finally, we show that [Formula: see text] is generated by two of its torsion elements.

Let i>Tn be the space of fully-grown i>n-trees and let i>Vn and i>Vn′ be the representations of the symmetric groups Σi>n and Σi>n+1 respectively on the unique non-vanishing reduced integral homology group of this space. Starting from combinatorial descriptions of i>Vn and i>Vn′, we establish a short exact sequence of {\bb Z}\Sigma_{n+1}-modules, giving a description of i>Vn′ in terms of i>Vn and i>Vi>n+1. This short exact sequence may also be deduced from work of Sundaram. Modulo a twist by the sign representation, i>Vn is shown to be dual to the Lie representation of Σi>n, Liei>n. Therefore we have an explicit combinatorial description of the integral representation of Σi>n+1 on Liei>n and this representation fits into a short exact sequence involving Liei>n and Liei>n+1.

In this paper we prove two conjectures on partitions with certain conditions. A motivation for this is given by a problem in the modular representation theory of the covering groups Sn, of the finite symmetric groups S,n in characteristic 5 . One of the conjectures (Conjecture B below) has been open since 1974, when it was stated by the first author in his memoir [A3]. Recently the second and third author (jointly with A. 0. Morris) arrived at essentially the same conjecture from a completely different direction. Their paper [BMO] was concerned with decomposition matrices of Sn, in characteristic 3. A basic difficulty for obtaining similar results in characteristic 5 (or larger) was the lack of a class of partitions which would be "natural" character labels for the modular characters of these groups. In this connection two conjectures were stated (Conjectures A and B * below), whose solutions would be helpful in the characteristic 5 case. One of them, Conjecture B *, is equivalent to the old Conjecture B mentioned above. Conjecture A is concerned with a possible inductive definition of the set of partitions which should serve as the required labels. In ? 1 we give a brief description of the groups Sn and their representations, leading up to Conjectures A and B * as they were formulated in [BMO]. That section also presents the background for Conjecture B as stated in [A3] and the equivalence of Conjectures B and B * is explained. Sections 2 and 3 are devoted to the proof of Conjecture B, and ?4 to the proof of Conjecture A. 1. THE CONJECTURES AND THEIR BACKGROUND For facts concerning the general representation theory of finite groups needed in the following, the reader is referred to [F, NT]. In 191 1 Schur [S1] proved that the finite symmetric groups S, have covering groups S, of order 2IS, I = 2 * n ! This means that there is an exact sequence

We show the existence of a unital subalgebra 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 this algebra 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 peak algebra contains a two sided ideal which is defined in terms of interior peaks. This object was introduced in previous work by Nyman; we find that it is the image of certain ideals of the descent algebras of types B and D introduced in previous work of N. Bergeron et al. We derive an exact sequence involving the peak ideal and the peak algebras of degrees $n$ and $n-2$. 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 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 these spaces over $n\geq 0$ and explain the connection with previous work of Stembridge; we also obtain new properties of his descents-to-peaks map and construct a type B analog., 45 pages