Descent Algebra Research Articles

Cohomology of Coxeter arrangements and Solomon’s descent algebra

We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon algebra of a finite Coxeter group W W and relate it to the descent algebra of W W . As a result, we claim that both the group algebra of W W and the Orlik-Solomon algebra of W W can be decomposed into a sum of induced one-dimensional representations of element centralizers, one for each conjugacy class of elements of W W . We give a uniform proof of the claim for symmetric groups. In addition, we prove that a relative version of the conjecture holds for every pair ( W , W L ) (W, W_L) , where W W is arbitrary and W L W_L is a parabolic subgroup of W W , all of whose irreducible factors are of type A A .

NATURAL ENDOMORPHISMS OF SHUFFLE ALGEBRAS

We show that there exist two natural endomorphism algebras for shuffle bialgebras such as Sh (X), where X is a graded set. One of these endomorphism algebras is a natural extension of the Malvenuto–Reutenauer Hopf algebra and is defined using graded permutations. The other one, the dendriform descent algebra, is a subalgebra of the first defined by mimicking the definition of the descent algebras by convolution from the graded projections in the tensor algebra. We study these algebras for their own, show that they carry bidendriform structures and establish freeness properties, study their generators, dimensions, bases, and also feature their relations to the internal structure of shuffle algebras. As an application of these ideas, we give a new proof of Chapoton's rigidity theorem for shuffle bialgebras.

Spectral properties of descent algebra elements

In this article, we determine the eigenvalues and their corresponding multiplicities of the action on the group algebra of a finite Coxeter group of an element of its descent algebra. Meanwhile, we identify a slight error in the paper of Bergeron, Bergeron, Howlett, and Taylor in a formula for certain structure coefficients in the descent algebra. We provide the correct formula, and give an example which explicitly shows the difference between both formulas.

Mould calculus, polyhedral cones, and characters of combinatorial Hopf algebras

We describe a method for constructing characters of combinatorial Hopf algebras by means of integrals over certain polyhedral cones. This is based on ideas from resurgence theory, in particular on the construction of well-behaved averages induced by diffusion processes on the real line. We give several interpretations and proofs of the main result in terms of noncommutative symmetric and quasi-symmetric functions, as well as generalizations involving matrix quasi-symmetric functions. The interpretation of noncommutative symmetric functions as alien operators in resurgence theory is also discussed, and a new family of Lie idempotents of descent algebras is derived from this interpretation.

On the quiver presentation of the descent algebra of the symmetric group

We describe a presentation of the descent algebra of the symmetric group Sn as a quiver with relations. This presentation arises from a new construction of the descent algebra as a homomorphic image of an algebra of forests of binary trees, which can be identified with a subspace of the free Lie algebra. In this setting we provide a short new proof of the known fact that the quiver of the descent algebra of Sn is given by restricted partition refinement. Moreover, we describe certain families of relations and conjecture that for fixed n∈N the finite set of relations from these families that are relevant for the descent algebra of Sn generates the ideal of relations of an explicit quiver presentation of that algebra.

GROUP BIALGEBRAS AND PERMUTATION BIALGEBRAS

AbstractMalvenuto 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.

Computations for Coxeter arrangements and Solomonʼs descent algebra II: Groups of rank five and six

In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group W acting on the graded components of its Orlik–Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of W. The refined conjecture relates the character above to a decomposition of the regular character of W related to Solomonʼs descent algebra of W. The refined conjecture has been proved for symmetric and dihedral groups, as well as for finite Coxeter groups of rank three and four. In this paper, we prove the conjecture for finite Coxeter groups of rank five and six. The techniques developed and implemented in this paper provide previously unknown decompositions of the regular and Orlik–Solomon characters of the groups considered.

Computations for Coxeter arrangements and Solomonʼs descent algebra: Groups of rank three and four

In recent papers we have refined a conjecture of Lehrer and Solomon expressing the characters of a finite Coxeter group W afforded by the homogeneous components of its Orlik–Solomon algebra as sums of characters induced from linear characters of centralizers of elements of W. Our refined conjecture also relates the Orlik–Solomon characters above to the terms of a decomposition of the regular character of W related to the descent algebra of W. A consequence of our conjecture is that both the regular character of W and the character of the Orlik–Solomon algebra have parallel, graded decompositions as sums of characters induced from linear characters of centralizers of elements of W, one for each conjugacy class of elements of W. The refined conjecture has been proved for symmetric and dihedral groups. In this paper we develop algorithmic tools to prove the conjecture computationally for a given finite Coxeter group. We use these tools to verify the conjecture for all finite Coxeter groups of rank three and four, thus providing previously unknown decompositions of the regular characters and the Orlik–Solomon characters of these groups.

Words and Polynomial Invariants of Finite Groups in Non-Commutative Variables

Let V be a complex vector space with basis {x 1, x 2, . . . , x n } and G be a finite subgroup of GL(V). The tensor algebra T(V) over the complex is isomorphic to the polynomials in the non-commutative variables x 1, x 2, . . . , x n with complex coefficients. We want to give a combinatorial interpretation for the decomposition of T(V) into simple G-modules. In particular, we want to study the graded space of invariants in T(V) with respect to the action of G. We give a general method for decomposing the space T(V) into simple modules in terms of words in a Cayley graph of the group G. To apply the method to a particular group, we require a homomorphism from a subalgebra of the group algebra into the character algebra. In the case of G as the symmetric group, we give an example of this homomorphism from the descent algebra. When G is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions and the number of free generators of the algebras of invariants in terms of those words.

An inductive approach to Coxeter arrangements and Solomon’s descent algebra

In our recent paper (Douglass et al. arXiv:1101.2075 (2011)), we claimed that both the group algebra of a finite Coxeter group W as well as the Orlik---Solomon algebra of W can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements of W, and gave a uniform proof of this claim for symmetric groups. In this note, we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for finite Coxeter groups of rank up to 2.

On the quiver presentation of the descent algebra of type $A$ or $B$

On the quiver presentation of the descent algebra of type $A$ or $B$

Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labeled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products Γ ≀ S n and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon’s multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type B .

The $(1-\mathbb{E})$ -transform in combinatorial Hopf algebras

We extend to several combinatorial Hopf algebras the endomorphism of symmetric functions sending the first power-sum to zero and leaving the other ones invariant. As a “transformation of alphabets”, this is the \((1-\mathbb{E})\)-transform, where \(\mathbb{E}\) is the “exponential alphabet,” whose elementary symmetric functions are \(e_{n}=\frac{1}{n!}\). In the case of noncommutative symmetric functions, we recover Schocker’s idempotents for derangement numbers (Schocker, Discrete Math. 269:239–248, 2003). From these idempotents, we construct subalgebras of the descent algebras analogous to the peak algebras and study their representation theory. The case of WQSym leads to similar subalgebras of the Solomon–Tits algebras. In FQSym, the study of the transformation boils down to a simple solution of the Tsetlin library in the uniform case.

Representation theory of the higher-order peak algebras

The representation theory (idempotents, quivers, Cartan invariants, and Loewy series) of the higher-order unital peak algebras is investigated. On the way, we obtain new interpretations and generating functions for the idempotents of descent algebras introduced in Saliola (J. Algebra 320:3866, 2008).

The Quiver of an Algebra Associated to the Mantaci-Reutenauer Descent Algebra and the Homology of Regular Semigroups

We develop the homology theory of the algebra of a regular semigroup, which is a particularly nice case of a quasi-hereditary algebra in good characteristic. Directedness is characterized for these algebras, generalizing the case of semisimple algebras studied by Munn and Ponizovksy. We then apply homological methods to compute (modulo group theory) the quiver of a right regular band of groups, generalizing Saliola’s results for a right regular band. Right regular bands of groups come up in the representation theory of wreath products with symmetric groups in much the same way that right regular bands appear in the representation theory of finite Coxeter groups via the Solomon-Tits algebra of its Coxeter complex. In particular, we compute the quiver of Hsiao’s algebra, which is related to the Mantaci-Reutenauer descent algebra.

Random walks on quasisymmetric functions

Conditions are provided under which an endomorphism on quasisymmetric functions gives rise to a left random walk on the descent algebra which is also a lumping of a left random walk on permutations. Spectral results are also obtained. Several important random walks are now realized this way: Stanley's QS-distribution results from endomorphisms given by evaluation maps, a-shuffles result from the ath convolution power of the universal character, and the Tchebyshev operator of the second kind introduced recently by Ehrenborg and Readdy yields traditional riffle shuffles. A conjecture of Ehrenborg regarding the spectra for a family of random walks on ab-words is proven. A theorem of Stembridge from the theory of enriched P-partitions is also recovered as a special case.

A Semigroup Approach to Wreath-Product Extensions of Solomon's Descent Algebras

There is a well-known combinatorial model, based on ordered set partitions, of the semigroup of faces of the braid arrangement. We generalize this model to obtain a semigroup ${\cal F}_n^G$ associated with $G\wr S_n$, the wreath product of the symmetric group $S_n$ with an arbitrary group $G$. Techniques of Bidigare and Brown are adapted to construct an anti-homomorphism from the $S_n$-invariant subalgebra of the semigroup algebra of ${\cal F}_n^G$ into the group algebra of $G\wr S_n$. The colored descent algebras of Mantaci and Reutenauer are obtained as homomorphic images when $G$ is abelian.

The Loewy Length of the Descent Algebra of Type D

In this article the Loewy length of the descent algebra of D2m + 1 is shown to be m + 2, for m ≥ 2, by providing an upper bound that agrees with the lower bound in Bonnafe and Pfeiffer (2006). The bound is obtained by showing that the length of the longest path in the quiver of the descent algebra of D2m + 1 is at most m + 1. To achieve this bound, the geometric approach to the descent algebra is used, in which the descent algebra of a finite Coxeter group W is identified with an algebra associated to the reflection arrangement of W.

Words and polynomial invariants of finite groups in non-commutative variables

Let $V$ be a complex vector space with basis $\{x_1,x_2,\ldots,x_n\}$ and $G$ be a finite subgroup of $GL(V)$. The tensor algebra $T(V)$ over the complex is isomorphic to the polynomials in the non-commutative variables $x_1, x_2, \ldots, x_n$ with complex coefficients. We want to give a combinatorial interpretation for the decomposition of $T(V)$ into simple $G$-modules. In particular, we want to study the graded space of invariants in $T(V)$ with respect to the action of $G$. We give a general method for decomposing the space $T(V)$ into simple $G$-module in terms of words in a particular Cayley graph of $G$. To apply the method to a particular group, we require a surjective homomorphism from a subalgebra of the group algebra into the character algebra. In the case of the symmetric group, we give an example of this homomorphism from the descent algebra. When $G$ is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions of the invariant space in term of those words. Soit V un espace vectoriel complexe de base $\{x_1,x_2,\ldots,x_n\}$ et $G$ un sous-groupe fini de $GL(V)$. L'algèbre $T(V)$ des tenseurs de $V$ sur les complexes est isomorphe aux polynômes à coefficients complexes en variables non-commutatives $x_1, x_2, \ldots, x_n$. Nous voulons donner une décomposition de $T(V)$ en $G$-modules simples de manière combinatoire. Plus particulièrement, nous étudions l'espace gradué des invariants de $T(V)$ sous l'action de $G$. Nous présentons une méthode générale donnant la décomposition de $T(V)$ en modules simples via certains mots dans un graphe de Cayley donné. Pour appliquer la méthode à un groupe particulier, nous avons besoin d'un homomorphisme surjectif entre une sous-algèbre de l'algèbre de groupe et l'algèbre des caractères. Pour le cas du groupe symétrique, nous donnons un exemple de cet homomorphisme qui provient de la théorie de l'algèbre des descentes. Pour le groupe diédral, nous avons une réalisation de l'algèbre des caractères comme une sous-algèbre de l'algèbre de groupe. Dans ces deux cas, nous avons une interprétation des dimensions graduées de l'espace des invariants en terme de ces mots.

A quiver presentation for Solomon's descent algebra

The descent algebra Σ ( W ) is a subalgebra of the group algebra Q W of a finite Coxeter group W, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of W. Thus Σ ( W ) is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct Σ ( W ) as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of S, the set of simple reflections in W. From this construction we obtain some general information about the quiver of Σ ( W ) and an algorithm for the construction of a quiver presentation for the descent algebra Σ ( W ) of any given finite Coxeter group W.