Solomon Descent Research Articles

Modular idempotents for the descent algebras of type A and higher Lie powers and modules

The article focuses on four aspects related to the descent algebras of type A. They are modular idempotents, higher Lie powers, higher Lie modules and the right ideals of the symmetric group algebras generated by the Solomon's descent elements. More precisely, we give a construction for the modular idempotents, describe the dimension and character for higher Lie powers and study the structures of the higher Lie modules and the right ideals both in the ordinary and modular cases.

KK-like relations of α′ corrections to disk amplitudes

Inspired by the definition of color-dressed amplitudes in string theory, we define analogous color-dressed permutations replacing the color-ordered string amplitudes by their corresponding permutations. Decomposing the color traces into symmetrized traces and structure constants, the color-dressed permutations define BRST-invariant permutations, which we show are elements of the inverse Solomon descent algebra and we find a closed formula for them. We then present evidence that these permutations encode KK-like relations among the different α′ corrections to the disk amplitudes refined by their MZV content. In particular, the number of linearly independent amplitudes at a given α′ order and MZV content is given by (sums of) Stirling cycle numbers. In addition, we show how the superfield expansion of BRST invariants of the pure spinor formalism corresponding to α′2ζ2 corrections is encoded in the descent algebra.

Eulerian representations for real reflection groups

The Eulerian idempotents, first introduced for the symmetric group and later extended to all reflection groups, generate a family of representations called the Eulerian representations that decompose the regular representation. In Type $A$, the Eulerian representations have many elegant but mysterious connections to rings naturally associated with the braid arrangement. In this paper, we unify these results and show that they hold for any reflection group of coincidental type -- that is, $S_{n}$, $B_{n}$, $H_{3}$ or the dihedral group $I_{2}(m)$ -- by giving six characterizations of the Eulerian representations, including as components of the associated graded of the Varchenko-Gelfand ring $\mathcal{V}$. As a consequence, we show that Solomon's descent algebra contains a commutative subalgebra generated by sums of elements with the same number of descents if and only if $W$ is coincidental. More generally, when $W$ is any finite real reflection group, we give a case-free construction of a family of Eulerian representations described by a flat-decomposition of the ring $\mathcal{V}$.

Subalgebras of Solomon's descent algebra based on alternating runs

The number of alternating runs is a natural permutation statistic. We show it can be used to define some commutative subalgebras of the symmetric group algebra, and more precisely of the descent algebra. The Eulerian peak algebras naturally appear as subalgebras of our run algebras. We also calculate the orthogonal idempotents for run algebras in terms of noncommutative symmetric functions.

Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin Types A, B and D

Let Bt(n) be the number of set partitions of a set of size t into at most n parts and let Bt′(n) be the number of set partitions of {1,…,t} into at most n parts such that no part contains both 1 and t or both i and i+1 for any i∈{1,…,t−1}. We give two new combinatorial interpretations of the numbers Bt(n) and Bt′(n) using sequences of random-to-top shuffles, and sequences of box moves on the Young diagrams of partitions. Using these ideas we obtain a very short proof of a generalization of a result of Phatarfod on the eigenvalues of the random-to-top shuffle. We also prove analogous results for random-to-top shuffles that may flip certain cards. The proofs use the Solomon descent algebras of Types A, B and D. We give generating functions and asymptotic results for all the combinatorial quantities studied in this paper.

Minimal polynomials of Young permutation modules and idempotents

Applying an epimorphism of the Solomon descent algebra onto the subring of the Green ring spanned by the isomorphism classes of Young permutation modules, we determine a basis of primitive orthogonal idempotents which diagonalise the multiplication maps of Young permutation modules. We determine direct sum decompositions of tensor products of hook Young permutation modules, the minimal polynomials of all Young permutation modules, and of the Young module Y(r−1,1).

Computations for Coxeter arrangements and Solomon's descent algebra III: Groups of rank seven and eight

In this paper we extend the computations in parts I and II of this series of papers and complete the proof of a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group W acting on the p-th graded component of its Orlik–Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of W for groups of rank seven and eight. For classical Coxeter groups, these characters are given using a formula that is expected to hold in all ranks.

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.

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.

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.

Cyclotomic Solomon algebras

This paper introduces an analogue of the Solomon descent algebra for the complex reflection groups of type G ( r , 1 , n ) . As with the Solomon descent algebra, our algebra has a basis given by sums of ‘distinguished’ coset representatives for certain ‘reflection subgroups.’ We explicitly describe the structure constants with respect to this basis and show that they are polynomials in r. This allows us to define a deformation, or q-analogue, of these algebras which depends on a parameter q. We determine the irreducible representations of all of these algebras and give a basis for their radicals. Finally, we show that the direct sum of cyclotomic Solomon algebras is canonically isomorphic to a concatenation Hopf algebra.

On the decomposition map for symmetric groups

Let R d be the ?-module generated by the irreducible characters of the symmetric group ${\mathcal{S}}_{d}$ . We determine bases for the kernel of the decomposition map. It is known that R d ? ? F is isomorphic to the radical quotient of the Solomon descent algebra when F is a field of characteristic zero. We show that when F has prime characteristic and I br d is the kernel of the decomposition map for prime p then R d /I br d ? ? F is isomorphic to the radical quotient of the p-modular Solomon descent algebra.

Generalized Descent Algebras

AbstractIf 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.

A Solomon descent theory for the wreath products $G\wr\mathfrak S_n$

We propose an analogue of Solomon's descent theory for the case of a wreath product G/G n , where G is a finite abelian group. Our construction mixes a number of ingredients: Mantaci-Reutenauer algebras, Specht's theory for the representations of wreath products, Okada's extension to wreath products of the Robinson-Schensted correspondence, and Poirier's quasisymmetric functions. We insist on the functorial aspect of our definitions and explain the relation of our results with previous work concerning the hyperoctaedral group.

Differential operator specializations of noncommutative symmetric functions

Let K be any unital commutative Q -algebra and z = ( z 1 , … , z n ) commutative or noncommutative free variables. Let t be a formal parameter which commutes with z and elements of K. We denote uniformly by K 《 z 》 and K 〚 t 〛 《 z 》 the formal power series algebras of z over K and K 〚 t 〛 , respectively. For any α ⩾ 1 , let D [ α ] 《 z 》 be the unital algebra generated by the differential operators of K 《 z 》 which increase the degree in z by at least α − 1 and A t [ α ] 《 z 》 the group of automorphisms F t ( z ) = z − H t ( z ) of K 〚 t 〛 《 z 》 with o ( H t ( z ) ) ⩾ α and H t = 0 ( z ) = 0 . First, for any fixed α ⩾ 1 and F t ∈ A t [ α ] 《 z 》 , we introduce five sequences of differential operators of K 《 z 》 and show that their generating functions form an N CS (noncommutative symmetric) system [W. Zhao, Noncommutative symmetric systems over associative algebras, J. Pure Appl. Algebra 210 (2) (2007) 363–382] over the differential algebra D [ α ] 《 z 》 . Consequently, by the universal property of the N CS system formed by the generating functions of certain NCSFs (noncommutative symmetric functions) first introduced in [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218–348, MR1327096; see also hep-th/9407124], we obtain a family of Hopf algebra homomorphisms S F t : N Sym → D [ α ] 《 z 》 ( F t ∈ A t [ α ] 《 z 》 ) , which are also grading-preserving when F t satisfies certain conditions. Note that the homomorphisms S F t above can also be viewed as specializations of NCSFs by the differential operators of K 《 z 》 . Secondly, we show that, in both commutative and noncommutative cases, this family S F t (with all n ⩾ 1 and F t ∈ A t [ α ] 《 z 》 ) of differential operator specializations can distinguish any two different NCSFs. Some connections of the results above with the quasi-symmetric functions [I. Gessel, Multipartite P-partitions and inner products of skew Schur functions, in: Contemp. Math., vol. 34, 1984, pp. 289–301, MR0777705; C. Malvenuto, C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (3) (1995) 967–982, MR1358493; Richard P. Stanley, Enumerative Combinatorics II, Cambridge University Press, 1999] are also discussed.

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.

Modular Lie powers and the Solomon descent algebra

Let V be an r-dimensional vector space over an infinite field F of prime characteristic p, and let L n (V) denote the nth homogeneous component of the free Lie algebra on V. We study the structure of L n (V) as a module for the general linear group GL r (F) when n=pk and k is not divisible by p and where r≥n. Our main result is an explicit 1-1 correspondence, multiplicity-preserving, between the indecomposable direct summands of L k (V) and the indecomposable direct summands of L n (V) which are not isomorphic to direct summands of V ⊗ n . Our approach uses idempotents of the Solomon descent algebras, and in addition a correspondence theorem for permutation modules of symmetric groups.

Cyclic Descents and P-Partitions

Louis Solomon showed that the group algebra of the symmetric group $$\mathfrak{S}_{n}$$ n has a subalgebra called the descent algebra, generated by sums of permutations with a given descent set. In fact, he showed that every Coxeter group has something that can be called a descent algebra. There is also a commutative, semisimple subalgebra of Solomon's descent algebra generated by sums of permutations with the same number of descents: an descent algebra. For any Coxeter group that is also a Weyl group, Paola Cellini proved the existence of a different Eulerian subalgebra based on a modified definition of descent. We derive the existence of Cellini's subalgebra for the case of the symmetric group and of the hyperoctahedral group using a variation on Richard Stanley's theory of P-partitions.

Lie Elements and Knuth Relations

AbstractA coplactic class in the symmetric group consists of all permutations in with a given Schensted Q-symbol, and may be described in terms of local relations introduced by Knuth. Any Lie element in the group algebra of which is constant on coplactic classes is already constant on descent classes. As a consequence, the intersection of the Lie convolution algebra introduced by Patras and Reutenauer and the coplactic algebra introduced by Poirier and Reutenauer is the direct sum of all Solomon descent algebras.

Descent representations and multivariate statistics

Combinatorial identities on Weyl groups of typesAAandBBare derived from special bases of the corresponding coinvariant algebras. Using the Garsia-Stanton descent basis of the coinvariant algebra of typeAAwe give a new construction of the Solomon descent representations. An extension of the descent basis to typeBB, using new multivariate statistics on the group, yields a refinement of the descent representations. These constructions are then applied to refine well-known decomposition rules of the coinvariant algebra and to generalize various identities.