GROUP BIALGEBRAS AND PERMUTATION BIALGEBRAS

Malvenuto and Reutenauer (C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra177 (1995), 967–982) showed how the total symmetric group ring ⊕nZΣn could be made into a Hopf algebra with a very nice structure which admitted the Solomon descent algebra as a sub-Hopf algebra. To do this they replaced the group multiplication by a convolution product, thus distancing their structure from the group structure of Σn. In this paper we examine what is possible if we keep to the group multiplication, and we also consider the question for more general families of groups. We show that a Hopf algebra structure is not possible, but cocommutative and non-cocommutative counital bialgebras can be obtained, arising from certain diagrams of group homomorphisms. In the case of the symmetric groups we note that all such structures are weak in the sense that the dual algebras have many zero-divisors, but structures which respect descent sums can be found.

Generalized Descent Algebras

If A is a subset of the set of reflections of a finite Coxeter group W, we define a sub-ℤ-module of the group algebra ℤW. We discuss cases where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra, the group algebra and, if W is of type B, the Mantaci–Reutenauer algebra.

THE DECOMPOSITION OF LIE POWERS

Let $G$ be a group, $F$ a field of prime characteristic $p$ and $V$ a finite-dimensional $FG$-module. Let $L(V)$ denote the free Lie algebra on $V$ regarded as an $FG$-submodule of the free associative algebra (or tensor algebra) $T(V)$. For each positive integer $r$, let $L^r (V)$ and $T^r (V)$ be the $r$th homogeneous components of $L(V)$ and $T(V)$, respectively. Here $L^r (V)$ is called the $r$th Lie power of $V$. Our main result is that there are submodules $B_1$, $B_2$, ... of $L(V)$ such that, for all $r$, $B_r$ is a direct summand of $T^r(V)$ and, whenever $m \geqslant 0$ and $k$ is not divisible by $p$, the module $L^{p^mk} (V)$ is the direct sum of $L^{p^m} (B_k)$, $L^{p^{m - 1}} (B_{pk})$, ..., $L^1 (B_{p^mk})$. Thus every Lie power is a direct sum of Lie powers of $p$-power degree. The approach builds on an analysis of $T^r (V)$ as a bimodule for $G$ and the Solomon descent algebra.