The descent algebra of the symmetric group

(Manfred Schocker)

The descent set of a permutation π in the symmetric group S_{n} is

Des π = { i ≤ n − 1 | iπ > (i + 1)π }.

Due to a remarkable result of Solomon, the linear span D of the sums of descent classes

Δ^{D} = ∑_{Des π=D} π (D ⊆ {1, …, n − 1})

is a sub-ring of the integral group ring of S_{n}. Furthermore, there is a homomorphism of rings mapping D onto the ring of class functions of S_{n}, with kernel equal to the Jacobson radical of D.

As a consequence, the irreducible representations of D are of dimension 1 and indexed by partitions of n — and provide a strong connection to the representation theory of S_{n}. In fact, there is the (certainly vague) dictum that the Specht modules of S_{n} have been shrunk to their linear nuclei here.

Results of Blessenohl and Laue describe the Cartan invariants and the descending Loewy series of D, and allow to determine its quiver and representation type. Also, the S_{n}-representations arising from the primitive idempotents of D achieve a natural realization, namely as the multi-homogenous parts of the tensor algebra over a finite-dimensional vector space V with respect to the Poincaré-Birkhoff-Witt basis, thanks to pioneering work of Garsia and Reutenauer. However, in spite of extensive study of these modules through the last years, there is still a haunting lack of comprehension.

Numerous related results have a different backdrop and link the descent algebra to combinatorial settings such as the theory of quasi-symmetric functions, problems related to permutation statistics, or card shuffling and associated random walks. Further connections to representation theory are provided by Jöllenbeck’s noncommutative character theory of the symmetric group, as well as the huge body of research papers on the algebra of noncommutative symmetric functions introduced by Gelfand et al.

A satisfying overall explanation of the mediating role of the algebra D between the various fields is yet to be discovered. It might be hidden in the depths of its representation theory. I am going to display major results on the descent algebra and its representations, hopefully inviting parts of the audience to introduce their algebraic skills onto the subject.