Unital versions of the higher order peak algebras
aa r X i v : . [ m a t h . C O ] O c t UNITAL VERSIONS OF THE HIGHER ORDER PEAK ALGEBRAS
MARCELO AGUIAR, JEAN-CHRISTOPHE NOVELLI, JEAN-YVES THIBON
Abstract.
We construct unital extensions of the higher order peak algebras de-fined by Krob and the third author in [Ann. Comb. 9 (2005), 411–430.], andshow that they can be obtained as homomorphic images of certain subalgebras ofthe Mantaci-Reutenauer algebras of type B . This generalizes a result of Bergeron,Nyman and the first author [Trans. AMS 356 (2004), 2781–2824.]. Introduction A descent of a permutation σ ∈ S n is an index i such that σ ( i ) > σ ( i + 1). Adescent is a peak if moreover i > σ ( i ) > σ ( i − descent algebra Σ n . The peak algebra ˚ P n of S n is a subalgebra of its descent algebra, spanned bysums of permutations having the same peak set. This algebra has no unit.Descent algebras can be defined for all finite Coxeter groups [19]. In [2], it is shownthat the peak algebra of S n can be naturally extended to a unital algebra, whichis obtained as a homomorphic image of the descent algebra of the hyperoctahedralgroup B n .The direct sum of the peak algebras turns out to be a Hopf subalgebra of the directsum of all descent algebras, which can itself be identified with Sym , the Hopf algebraof noncommutative symmetric functions [9]. As explained in [5], it turns out that afair amount of results on the peak algebras can be deduced from the case q = − q -identity of [11]. Specializing q to other roots of unity, Krob and the thirdauthor introduced and studied higher order peak algebras in [12]. Again, these arenon-unital, and it is natural to ask whether the construction of [2] can be extendedto this case.We will show that this is indeed possible. We first construct the unital versions ofthe higher order peak algebras by a simple manipulation of generating series. We thenshow that they can be obtained as homomorphic images of the Mantaci-Reutenaueralgebras of type B . Hence no Coxeter groups other than B n and S n are involvedin the process; in fact, the construction is related to the notion of superization, asdefined in [16], rather than to root systems or wreath products. Acknowledgements.
This work started during a stay of the first author at the University Paris-Est Marne-la-Vall´ee.Aguiar is partially supported by NSF grant DMS-0600973. Novelli and Thibon are partially sup-ported by the grant ANR-06-BLAN-0380. The authors would also like to thank the contributors of
Date : November 2, 2018. the MuPAD project, and especially of the combinat part, for providing the development environmentfor their research (see [10] for an introduction to MuPAD-Combinat). Notations and background
Noncommutative symmetric functions.
This article is a continuation of[12]. We will assume familiarity with the notations of [9] and with the main resultsof [12]. We recall a few definitions for the convenience of the reader.The Hopf algebra of noncommutative symmetric functions is denoted by
Sym , orby
Sym ( A ) if we consider the realization in terms of an auxiliary alphabet A . Linearbases of Sym n are labelled by compositions I = ( i , . . . , i r ) of n (we write I (cid:15) n ).The noncommutative complete and elementary functions are denoted by S n and Λ n ,and S I = S i · · · S i r . The ribbon basis is denoted by R I . The descent set of I isDes( I ) = { i , i + i , . . . , i + · · · + i r − } . The descent composition of a permutation σ ∈ S n is the composition I = D ( σ ) of n whose descent set is the descent set of σ .Recall from [8] that for an infinite totally ordered alphabet A , FQSym ( A ) is thesubalgebra of C h A i spanned by the polynomials(1) G σ ( A ) = X std( w )= σ w, that is, the sum of all words in A n whose standardization is the permutation σ ∈ S n .The noncommutative ribbon Schur function R I ∈ Sym is then(2) R I = X D( σ )= I G σ . This defines a Hopf embedding
Sym → FQSym . The Hopf algebra
FQSym isself-dual under the pairing ( G σ , G τ ) = δ σ,τ − (Kronecker symbol). Let F σ := G σ − ,so that { F σ } is the dual basis of { G σ } .The internal product ∗ of FQSym is induced by composition ◦ in S n in the basis F ,that is,(3) F σ ∗ F τ = F σ ◦ τ and G σ ∗ G τ = G τ ◦ σ . Each subspace
Sym n is stable under this operation, and anti-isomorphic to the de-scent algebra Σ n of S n .For f i ∈ FQSym and g ∈ Sym , we have the splitting formula(4) ( f . . . f r ) ∗ g = µ r · ( f ⊗ · · · ⊗ f r ) ∗ r ∆ r g , where µ r is r -fold multiplication, and ∆ r the iterated coproduct with values in the r -th tensor power.2.2. The Mantaci-Reutenauer algebra of level 2.
We denote by MR the freeproduct Sym ⋆ Sym of two copies of the Hopf algebra of noncommutative symmetricfunctions [14]. That is, MR is the free associative algebra on two sequences ( S n )and ( S ¯ n ) ( n ≥ Sym as noncommutative symmetricfunctions on two auxiliary alphabets: S n = S n ( A ) and S ¯ n = S n ( ¯ A ). We denote by NITAL VERSIONS OF THE HIGHER ORDER PEAK ALGEBRAS 3 F ¯ F the involutive automorphism which exchanges S n and S ¯ n . The bialgebrastructure is defined by the requirement that the series(5) σ = X n ≥ S n and ¯ σ = X n ≥ S ¯ n are grouplike.The internal product of MR can be computed from the splitting formula and theconditions that σ is neutral, ¯ σ is central, and ¯ σ ∗ ¯ σ = σ .In [15], an embedding of MR in the Hopf algebra BFQSym of free quasi-symmetricfunctions of type B (spanned by colored permutations) is described. Under this em-bedding, left ∗ -multiplication by Λ n = G n n − ... , corresponds to right multiplicationby n n − . . . , B n . This implies that left ∗ -multiplicationby λ is an involutive anti-automorphism of BFQSym , hence of MR .2.3. Noncommutative symmetric functions of type B . The hyperoctahedralanalogue
BSym of Sym , defined in [6], is the right
Sym -module freely generated byanother sequence ( ˜ S n ) ( n ≥
0, ˜ S = 1) of homogeneous elements, with ˜ σ grouplike.This is a coalgebra, but not an algebra. It is endowed with an internal product,for which each homogeneous component BSym n is anti-isomorphic to the descentalgebra of B n . 3. Solomon descent algebras of type B Descents in B n . The hyperoctahedral group B n is the group of signed permu-tations. A signed permutation can be denoted by w = ( σ, ǫ ) where σ is an ordinarypermutation and ǫ ∈ {± } n , such that w ( i ) = ǫ i σ ( i ). If we set w (0) = 0, then, i ∈ [0 , n −
1] is a descent of w if w ( i ) > w ( i + 1). Hence, the descent set of w is a subset D = { i , i + i , . . . , i + i + · · · i r − } of [0 , n − D a so-called type- B composition (a composition whose first part can be zero)( i − , i , . . . , i r − , n − i r − ).For example, if one encodes ǫ as a boolean vector for readability, the signed permu-tation w = (231546 , B composition I = (0 , , , w = (231546 , B composition I = (2 , , D is mappedto ˜ S I := ˜ S i S I ′ by Chow’s anti-isomorphism [6], where I ′ = ( i , . . . , i r ).3.2. Noncommutative supersymmetric functions.
An embedding of
BSym asa sub-coalgebra and sub-
Sym -module of MR can be deduced from [14]. To describeit, let us define, for F ∈ Sym ( A ),(6) F ♯ = F ( A | ¯ A ) = F ( A − q ¯ A ) | q = − (the supersymmetric version of F ). The superization of F ∈ Sym ( A ) can also begiven by(7) F ♯ = F ∗ σ ♯ . M. AGUIAR, J.-C. NOVELLI, J.-Y. THIBON
Indeed, σ ♯ is grouplike, and for F = S I , the splitting formula gives(8) ( S i · · · S i r ) ∗ σ ♯ = µ r [( S i ⊗ · · · ⊗ S i r ) ∗ ( σ ♯ ⊗ · · · ⊗ σ ♯ )] = S I♯ . We have(9) σ ♯ = ¯ λ σ = X Λ ¯ i S j . The element ¯ σ is central for the internal product, and(10) ¯ σ ∗ F = ¯ F = F ∗ ¯ σ . Hence,(11) ¯ σ ∗ σ ♯ = λ ¯ σ =: σ ♭ . The basis element ˜ S I of BSym , where I = ( i , i , . . . , i r ) is a type B -composition,can be embedded as(12) ˜ S I = S i ( A ) S i i ··· i r ( A | ¯ A ) . We will identify
BSym with its image under this embedding.3.3.
A proof that BSym is ∗ -stable. We are now in a position to understand why
BSym is a ∗ -subalgebra of MR . The argument will be extended below to the caseof unital peak algebras.Let F, G ∈ Sym . We want to understand why σ F ♯ ∗ σ G ♯ is in BSym . Using thesplitting formula, we rewrite this as(13) µ [( σ ⊗ F ♯ ) ∗ ∆ σ ∆ G ♯ ] = X ( G ) ( σ G ♯ (1) )( F ♯ ∗ σ G ♯ (2) ) . We now only have to show that each term F ♯ ∗ σ G ♯ (2) is in Sym ♯ . We may assumethat F = S I , and for any G ∈ Sym ,(14) S I♯ ∗ σ G ♯ = X ( G ) µ r [( S ♯i ⊗ · · · ⊗ S ♯i r ) ∗ ( σ G ♯ (1) ⊗ · · · ⊗ σ G ♯ ( r ) )]so that it is sufficient to prove the property for F = S n . Now, σ ♯ ∗ σ G ♯ = (¯ λ σ ) ∗ σ G ♯ = X ( G ) (¯ λ ∗ σ G ♯ (1) )( σ G ♯ (2) )= X ( G ) (¯ σ ∗ λ ∗ σ G ♯ (1) ) · σ · G ♯ (2) (15)Now,(16) λ ∗ σ G ♯ (1) = ( λ ∗ G ♯ (1) )( λ ∗ σ ) = ( λ ∗ G ♯ (1) ) λ , NITAL VERSIONS OF THE HIGHER ORDER PEAK ALGEBRAS 5 since λ is an anti-automorphism. We then get σ ♯ ∗ σ G ♯ = X ( G ) (¯ σ ∗ (( λ ∗ G ♯ (1) ) λ ) · σ · G ♯ (2) = X ( G ) (¯ σ ∗ λ ∗ G ♯ (1) ) · (¯ σ ∗ λ ) σ · G ♯ (2) = X ( G ) (¯ λ ∗ G ♯ (1) ) · σ ♯ · G ♯ (2) (17)Now, the result will follow if we can prove that ¯ λ ∗ G ♯ is in Sym ♯ for any G ∈ Sym .For G = S I ,(18) ¯ λ ∗ S I♯ = λ ∗ ¯ σ ∗ S I ∗ σ ♯ = λ ∗ S I ∗ ¯ σ ∗ σ ♯ = λ ∗ S I ∗ σ ♭ . Since left ∗ -multiplication by λ in an anti-automorphism, we only need to prove that¯ λ ∗ S ♭n is of the form G ♯ . And indeed,¯ λ ∗ S ♭n = X i + j = n λ ∗ (Λ i S ¯ j )= X i + j = n ( λ ∗ S ¯ j )( λ ∗ Λ i )= X i + j = n Λ ¯ j S i = S ♯n . (19)This concludes the proof that BSym is a ∗ -subalgebra of BFQSym .4.
Unital versions of the higher order peak algebras R I ((1 − q ) A ), in the special case q = −
1. In [12], this formula wasstudied in the case where q is an arbitrary root of unity, and higher order analogs ofthe peak algebra were obtained. In [2], it was shown that the classical peak algebracan be extended to a unital algebra, which is obtained as a homomorphic image ofthe descent algebra of type B .In this section, we construct unital extensions of the higher order peak algebras.4.2. Let q be a primitive r -th root of unity. All objects introduced below will dependon q (and r ), although this dependence will not be made explicit in the notation.We denote by θ q the endomorphism of Sym defined by(20) ˜ f = θ q ( f ) = f ((1 − q ) A ) = f ( A ) ∗ σ ((1 − q ) A ) . We denote by ˚ P the image of θ q and by P the right ˚ P -module generated by the S n for n ≥
0. Note that ˚ P is by definition a left ∗ -ideal of Sym . Theorem 4.1. P is a unital ∗ -subalgebra of Sym . Its Hilbert series is (21) X n ≥ dim P n t n = 11 − t − t − · · · − t r . M. AGUIAR, J.-C. NOVELLI, J.-Y. THIBON
Proof –
Since the internal product of homogeneous elements of different degrees iszero, it is enough to show that, for any f, g ∈ Sym , σ ˜ f ∗ σ ˜ g is in P . Thanks to thesplitting formula, σ ˜ f ∗ σ ˜ g = µ [( σ ⊗ ˜ f ) ∗ X ( g ) σ ˜ g (1) ⊗ σ ˜ g (2) ]= X ( g ) ( σ ˜ g (1) )( ˜ f ∗ σ ˜ g (2) ) . (22)Thus, it is enough to check that ˜ f ∗ σ ˜ h is in ˚ P for any f, h ∈ Sym . Now,(23) ˜ f ∗ σ ˜ h = f ∗ σ ((1 − q ) A ) ∗ σ ˜ h , and since ˚ P is a Sym left ∗ -ideal, we only have to show that σ ((1 − q ) A ) ∗ σ ˜ h is in˚ P . One more splitting yields σ ((1 − q ) A ) ∗ σ ˜ h = ( λ − q σ ) ∗ σ ˜ h = µ [( λ − q ⊗ σ ) ∗ X ( h ) σ ˜ h (1) ⊗ σ ˜ h (2) ]= X ( h ) ( λ − q ∗ σ ˜ h (1) )( σ ˜ h (2) )= X ( h ) ( λ − q ∗ ˜ h (1) ) λ − q σ ˜ h (2) (24)(since left ∗ -multiplication by λ − q is an anti-automorphism, namely the compositionof the antipode and q degree ). The first parentheses ( λ − q ∗ ˜ h (1) ) are in ˚ P since it is aleft ∗ -ideal. The middle term is σ ((1 − q ) A ), and the last one is in ˚ P by definition.Recall from [12, Prop. 3.5] that the Hilbert series of ˚ P is(25) X n ≥ dim ˚ P n t n = 1 − t r − t − t − . . . − t r . From [12, Lemma 3.13 and Eq. (3.9)], it follows that S n ∈ ˚ P if and only if n ≡ r , so that the Hilbert series of P is(26) X n ≥ dim P n t n = 11 − t − t − . . . − t r . Back to the Mantaci-Reutenauer algebra
The above proofs are in fact special cases of a master calculation in the Mantaci-Reutenauer algebra, which we carry out in this section.
NITAL VERSIONS OF THE HIGHER ORDER PEAK ALGEBRAS 7
The ♯ transform. Let q be an arbitrary complex number or an indeterminate,and define, for any F ∈ MR ,(27) F ♯ = F ∗ σ ( A − q ¯ A ) = F ∗ σ ♯ . Since σ ♯ is grouplike, it follows from the splitting formula that(28) F F ♯ is an automorphism of MR for the Hopf structure. In addition, it is clear from thedefinition that it is also a endomorphism of left ∗ -modules. We refer to it as the ♯ transform .5.2. Definition of the subalgebras.
We define(29) ˚ Q = MR ♯ , the image of the ♯ transform. Since the latter is an endomorphism of Hopf algebrasand of left ∗ -modules, ˚ Q is both a Hopf subalgebra of MR and a left ∗ -ideal. When q is a root of unity, its image under the specialization ¯ A = A is the non-unital peakalgebra ˚ P of Section 4.2 (and for generic q , it is Sym ).Let Q be the right ˚ Q -module generated by the S n , for all n ≥
0. Clearly, theidentification ¯ A = A maps Q onto P , the unital peak algebra of Section 4.2. Theorem 5.1. Q is a ∗ -subalgebra of MR , containing ˚ Q as a left ideal.Proof – Let
F, G ∈ MR . As above, we want to show that σ F ♯ ∗ σ G ♯ is in Q . Usingthe splitting formula, we rewrite this as(30) µ [( σ ⊗ F ♯ ) ∗ ∆ σ ∆ G ♯ ] = X ( G ) ( σ G ♯ (1) )( F ♯ ∗ σ G ♯ (2) )and we only have to show that each term F ♯ ∗ σ G ♯ (2) is in ˚ Q . We may assume that F = S I , where I is now a bicolored composition, and for any G ∈ MR ,(31) S I♯ ∗ σ G ♯ = X ( G ) µ r [( S ♯i ⊗ · · · ⊗ S ♯i r ) ∗ ( σ G ♯ (1) ⊗ · · · ⊗ σ G ♯ ( r ) )]so that it is sufficient to prove the property for F = S n or S ¯ n . Now, σ ♯ ∗ σ G ♯ = (¯ λ − q σ ) ∗ σ G ♯ = X ( G ) (¯ λ − q ∗ σ G ♯ (1) )( σ G ♯ (2) )= X ( G ) (¯ λ − q ∗ G ♯ (1) ) · σ ♯ · G ♯ (2) (32) M. AGUIAR, J.-C. NOVELLI, J.-Y. THIBON which is in ˚ Q , since it is a subalgebra and a left ∗ -ideal, and similarly,¯ σ ♯ ∗ σ G ♯ = ( λ − q ¯ σ ) ∗ σ G ♯ = X ( G ) ( λ − q ∗ σ G ♯ (1) )(¯ σ ¯ G ♯ (2) )= X ( G ) ( λ − q ∗ G ♯ (1) ) · ¯ σ ♯ · ¯ G ♯ (2) (33)is also in ˚ Q .The various algebras introduced in this paper and their interrelationships are sum-marized in the following diagram.(34) ˚ Q (cid:15) (cid:15) (cid:15) (cid:15) ⊆ Q (cid:15) (cid:15) (cid:15) (cid:15) ⊆ MR (cid:15) (cid:15) (cid:15) (cid:15) ⊆ BFQSym (cid:15) (cid:15) (cid:15) (cid:15) ˚ P ⊆ P ⊆
Sym ⊆ FQSym
Note that in the special case q = −
1, by the results of Section 3.3, Q n is the(Solomon) descent algebra of B n , Q is isomorphic to BSym , and P is the unitalpeak algebra of [2]. 6. Further developments
Inversion of the generic ♯ transform. For generic q , the endomorphism (27)of MR is invertible; therefore(35) ˚ Q ∼ MR . The inverse endomorphism of MR arises from the transformation of alphabets(36) A ( q ¯ A + A ) / (1 − q ) , which is to be understood in the following sense:(37) σ (cid:18) q ¯ A + A − q (cid:19) := Y k ≥ σ q k +1 ( ¯ A ) σ q k ( A ) . Indeed, σ (cid:18) q ¯ A + A − q (cid:19) ∗ σ ( A − q ¯ A ) = Y k ≥ σ q k +1 ( ¯ A − qA ) σ q k ( A − q ¯ A )= Y k ≥ λ − q k +2 ( A ) σ q k +1 ( ¯ A ) λ − q k +1 ( ¯ A ) σ q k ( A )= σ ( A ) . (38) NITAL VERSIONS OF THE HIGHER ORDER PEAK ALGEBRAS 9
By normalizing the term of degree n in (37), we obtain B n -analogs of the q -Klyachko elements defined in [9]:(39) K n ( q ; A, ¯ A ) := n Y i =1 (1 − q i ) S n (cid:18) q ¯ A + A − q (cid:19) = X I (cid:15) n q I ) R I ( q ¯ A + A ) . This expression can be completely expanded on signed ribbons. From the expressionof R I in FQSym , we have(40) R I ( ¯ A + A ) = X C ( σ )= I G σ ( ¯ A + A )where ¯ A + A is the ordinal sum. If we order ¯ A by(41) ¯ a < ¯ a < . . . < ¯ a k < . . . then, arguing as in [16], we have(42) G σ ( ¯ A + A ) = X std( τ,ǫ )= σ G τ,ǫ so that(43) R I ( ¯ A + A ) = X ρ (J)= I R J where for a signed composition J = ( J, ǫ ), the unsigned composition ρ (J) is definedas the shape of std( σ, ǫ ), where σ is any permutation of shape J .6.2. Replacing ¯ A by q ¯ A , one obtains the expansion of the q -Klyachko elements oftype B :(44) K n ( q ; A, ¯ A ) = X J q bmaj(J) R J where(45) bmaj(J) = 2 maj( ρ (J)) + | ǫ | , where | ǫ | is the number of minus signs in ǫ .For example,(46) K ( q ) = R + q R + q R + q R + q R + q R .K ( q ) = R + q R + q R + q R + q R + q R + q R + q R + q R + q R + q R + q R + q R + q R + q R + q R + q R + q R . (47)This major index of type B is the flag major index defined in [1]. Following [1]and considering the signed composition (where ǫ is encoded as boolean vector forreadability)(48) J = (2 , , , ¯3 , ¯1 , ¯2 , , ¯1 , ,
2) = (2113124122 , we can take the smallest permutation of shape (2 , , , , , , , , , α = 1 5 4 3 2 6 9 8 7 11 10 12 13 16 15 14 18 17 19sign it according to ǫ , which yields(50) 1 5 4 3 ¯2 ¯6 ¯9 ¯8 ¯7 11 10 12 13 16 15 14 18 17 19whose standardized is(51) 8 11 10 9 1 2 5 4 3 6 12 13 14 16 7 15 18 17 19and has shape ρ (J) = (2 , , , , , , , ρ (J) is 55, the numberof minus signs in ǫ is 7, so bmaj(J) = 2 ×
55 + 7 = 117.6.3. The major index of type B can be read directly on signed compositions withoutreference to signed permutations as follows: one can get ρ (J) by first adding theabsolute values of two consecutive parts if the left one is signed and the second oneis not, then remove the signs and proceed as before.A different solution consists in reading the composition from right to left, thenassociate weight 0 (resp. 1) to the rightmost part if it is positive (resp. negative)and then proceed left by adding 2 to the weight if the two parts are of the same signand 1 if not. Finally, add up the product of the absolute values of the parts withtheir weight.For example, with the same J as above we have the following weights:J =(2 , , , ¯3 , ¯1 , ¯2 , , ¯1 , , ·
14 + 1 ·
12 + 1 ·
10 + 3 · · · · · · · c of colors 0, 1, . . . , c −
1: the weight of the rightmost cell is its color and theweight of a part is equal to the sum of the weight of the next part and the uniquerepresentative of the difference of the colors of those parts modulo c belonging to theinterval [1 , c ].6.4. Generators and Hilbert series.
For n ≥
0, let(53) S ± n = S n ( A ) ± S n ( ¯ A ) , and denote by H n the subalgebra of MR generated by the S ± k for k ≤ n . For n ≥ S ± n ) ♯ ≡ (1 ∓ q n ) S ± n mod H n − , so that the ( S ± n ) ♯ such that 1 ∓ q n = 0 form a set of free generators in MR ♯ . Conjecture 6.1. If r is odd, a basis of MR ♯ will be parametrized by colored composi-tions such that parts of color are not ≡ r and parts of color are arbitrary.The Hilbert series is then (55) H r ( t ) = 1 − t r − t + t + · · · + t r ) . NITAL VERSIONS OF THE HIGHER ORDER PEAK ALGEBRAS 11 If r is even, there is the extra condition that parts of color are not ≡ r/ r .The Hilbert series is then (56) H r ( t ) = 1 − t r − t + t + · · · + t r ) + t r/ . For example,(57) H ( t ) = 1 + t + 2 t + 4 t + 8 t + 16 t + 32 t + 64 t + 128 t + O (cid:0) t (cid:1) (58) H ( t ) = 1 + 2 t + 6 t + 17 t + 50 t + 146 t + 426 t + 1244 t + 3632 t + O (cid:0) t (cid:1) (59) H ( t ) = 1 + 2 t + 5 t + 14 t + 38 t + 104 t + 284 t + 776 t + 2120 t + O (cid:0) t (cid:1) If these conjectures are correct, the Hilbert series of the right MR ♯ -modules gen-erated by the S n are respectively(60) 11 − t + t + . . . + t r ) , or(61) 11 − t + t + . . . + t r ) + t r/ . according to whether r is odd or even.The cases r = 1 and r = 2 are easily proved as follows. Assume first that q = 1.Set f = 1 + ( σ +1 ) ♯ = ( σ + λ − )( A − ¯ A ) , (62) g = ( σ − ) ♯ − σ − λ − )( A − ¯ A ) . (63)Then, f = g + 4, so that(64) f = 2 (cid:18) g (cid:19) which proves that the ( S + n ) ♯ can be expressed in terms of the ( S − m ) ♯ .Similarly, for q = −
1, one can express(65) f = X n ≥ ( S +2 n ) ♯ + X n ≥ ( S − n +1 ) ♯ in terms of(66) g = X n ≥ ( S − n ) ♯ + X n ≥ ( S +2 n +1 ) ♯ since, as is easily verified,(67) ( f + 2) = g + 4 , i . e ., f = − (cid:18) g (cid:19) . Apparently, this approach does not work anymore for higher roots of unity. Appendix: monomial expansion of the (1 − q ) -kernel The results of [16, 7] allow us to write down a new expansion of S n ((1 − q ) A ),in terms of the monomial basis of [4]. The special case q = 1 gives back a curiousexpression of Dynkin’s idempotent, first obtained in [3].Let σ be a permutation. We then define its left-right minima set LR( σ ) as thevalues of σ that have no smaller value to their left. We will denote by lr( σ ) thecardinality of LR( σ ). For example, with σ = 46735182, we have LR( σ ) = { , , } ,and lr( σ ) = 3.Let us now compute how S n ((1 − q ) A ) decomposes on the monomial basis M σ (see [4]) of FQSym . Thanks to the Cauchy formula of
FQSym [7], we have(68) S n ((1 − q ) A ) = X σ S σ (1 − q ) M σ ( A ) , where S is the dual basis of M . Given the transition matrix between M and G , weimmediately deduce that(69) S σ = X τ<σ − F τ , where < stands for the right weak order on permutations, so that, for example.(70) S = F + F + F . Thanks to [16], we know that F σ (1 − q ) is either ( − q ) k if Des( σ ) = { , . . . , k } or 0otherwise. Let us define hook permutations of hook k the permutations σ such thatDes( σ ) = { , . . . , k } . Now, S σ (1 − q ) amounts to compute the list of hook permutations smaller than σ . Note that hook permutations are completely characterized by theirleft-right minima. Moreover, if τ is smaller than σ in the right weak order, thenLR( τ ) ⊂ LR( σ ).Hence all hook permutations smaller than a given permutation σ belong to theset of hook permutations with left-right minima in LR( σ ). Since by elementarytranspositions decreasing the length, one can get from σ to the hook permutationwith the same left-right minima and then from this permutation to all the others, wehave: Theorem 7.1.
Let n be an integer. Then (71) S n ((1 − q ) A ) = X σ ∈ S n (1 − q ) lr( σ ) M σ . In the particular case q = 1, we recover a result of [3]:(72) Ψ n = X σ ∈ S nσ (1)=1 M σ , where Ψ n is the first Eulerian idempotent [11, Prop. 5.2]. NITAL VERSIONS OF THE HIGHER ORDER PEAK ALGEBRAS 13
References [1]
R. Adin and
Y. Roichman , The flag major index and group actions on polynomial rings ,Europ. J. Combin, (2001), 431–446.[2] M. Aguiar, N. Bergeron, and
K. Nyman , The peak algebra and the descent algebra of typeB and D , Trans. of the AMS. (2004), 2781–2824.[3]
M. Aguiar and
M. Livernet , The associative operad and the weak order on the symmetricgroup , J. Homotopy and Related Structures, n.1 (2007), 57–84.[4] M. Aguiar and
F. Sottile , Structure of the Malvenuto-Reutenauer Hopf algebra of permu-tations , Adv. in Math. (2005), 225–275.[5]
N. Bergeron, F. Hivert and
J.-Y. Thibon , The peak algebra and the Hecke-Clifford algebrasat q = 0 , J. Combinatorial Theory A (2004), 1–19.[6] C.-O. Chow , Noncommutative symmetric functions of type B , Thesis, MIT, 2001.[7]
G. Duchamp, F. Hivert, J.-C. Novelli , and
J.-Y. Thibon , Noncommutative SymmetricFunctions VII: Free Quasi-Symmetric Functions Revisited , preprint, math.CO/0809.4479.[8]
G. Duchamp, F. Hivert , and
J.-Y. Thibon , Noncommutative symmetric functions VI: freequasi-symmetric functions and related algebras , Internat. J. Alg. Comput. (2002), 671–717.[9] I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh , and
J.-Y. Thibon , Noncommutative symmetric functions , Adv. in Math. (1995), 218–348.[10]
F. Hivert and
N. Thi´ery , MuPAD-Combinat, an open-source package for research in alge-braic combinatorics , S´em. Lothar. Combin. (2004), 70p. (electronic).[11] D. Krob, B. Leclerc , and
J.-Y. Thibon , Noncommutative symmetric functions II: Trans-formations of alphabets , Internal J. Alg. Comput. (1997), 181–264.[12] D. Krob and
J.-Y. Thibon , Higher order peak algebras , Ann. Combin. (2005), 411–430.[13] I.G. Macdonald , Symmetric functions and Hall polynomials , 2nd ed., Oxford UniversityPress, 1995.[14]
R. Mantaci and
C. Reutenauer , A generalization of Solomon’s descent algebra for hyper-octahedral groups and wreath products , Comm. Algebra (1995), 27–56.[15] J.-C. Novelli and
J.-Y. Thibon , Free quasi-symmetric functions of arbitrary level , preprintmath.CO/0405597.[16]
J.-C. Novelli and
J.-Y. Thibon , Superization and ( q, t ) -specialization in combinatorial Hopfalgebras , math.CO/0803.1816.[17] S. Poirier , Cycle type and descent set in wreath products , Disc. Math., (1998), 315–343.[18]
C. Reutenauer , Free Lie algebras , Oxford University Press, 1993.[19]
L. Solomon , A Mackey formula in the group ring of a Coxeter group , J. Algebra, , (1976),255-268.[20] R. P. Stanley , Enumerative combinatorics , vol. 2, Cambridge University Press, 1999.(Aguiar)
Department of Mathematics, Texas A&M University, College Station, TX77843, USA (Novelli, Thibon)
Institut Gaspard Monge, Universit´e Paris-Est Marne-la-Vall´ee,,5 Boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vall´ee cedex 2, France
E-mail address , Marcelo Aguiar: [email protected]
E-mail address , Jean-Christophe Novelli: [email protected] (corresponding author)
E-mail address , Jean-Yves Thibon:, Jean-Yves Thibon: