Representation theory of the higher-order peak algebras

DOI 10.1007/s10801-010-0223-y

Representation theory of the higher-order peak algebras

Jean-Christophe Novelli·Franco Saliola· Jean-Yves Thibon

Received: 2 July 2009 / Accepted: 11 March 2010 / Published online: 7 April 2010

© Springer Science+Business Media, LLC 2010

Abstract 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 de- scent algebras introduced in Saliola (J. Algebra 320:3866,2008).

Keywords Noncommutative symmetric functions·Peak algebras·Finite dimensional algebras·Descent algebras

1 Introduction

A descent of a permutationσ∈Snis an indexisuch thatσ (i) > σ (i+1). A descent is a peak if moreoveri >1 andσ (i) > σ (i−1). The sums of permutations with a given descent set span a subalgebra of the group algebra, the descent algebraΣ_n. The peak algebra ˚Pn ofSnis a subalgebra of its descent algebra, spanned by sums of permutations having the same peak set. This algebra has no unit.

The direct sum of the peak algebras is a Hopf subalgebra of the direct sum of all descent algebras, which can itself be identified with Sym, the Hopf algebra of noncommutative symmetric functions [11]. Actually, in [7], it was shown that most of the results on the peak algebras can be deduced from the caseq= −1 of aq-identity

J.-C. Novelli (

Institut Gaspard Monge, Université Paris-Est Marne-la-Vallée, 5 Boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France

e-mail:novelli@univ-mlv.fr F. Saliola

e-mail:saliola@gmail.com J.-Y. Thibon

e-mail:jyt@univ-mlv.fr

of [14]. Specializingq to other roots of unity, Krob and the third author introduced and studied higher-order peak algebras in [13]. Again, these are nonunital.

In [1], it has been shown that the peak algebra ofSn can be naturally extended to a unital algebra which is obtained as a homomorphic image of the descent algebra of the hyperoctahedral groupBn. This construction has been extended in [3]. It is shown there that unital versions of the higher-order peak algebras can be obtained as homomorphic images of the Mantaci–Reutenauer algebras of typeB.

Our purpose here is to investigate the representation theory of the unital higher- order peak algebras. The classical case has been worked out in [2]. In this reference, idempotents for the peak algebras were obtained from those of the descent algebras of typeBconstructed in [6].

To deal with the general case, we need a different construction of idempotents. It turns out that the recursive algorithm introduced in [17] for idempotents of descent algebras can be adapted to higher-order peak algebras.

In order to achieve this, we need a better understanding of the idempotents gener- ated by the algorithm of [17]. Interpreting them as noncommutative symmetric func- tions, we find that in typeA, these idempotents are associated with a known family of Lie idempotents, the so-called Zassenhaus idempotents, by the construction of [14].

We then show that similar Lie idempotents can be defined in typeB as well, which yields a simple generating function in terms of noncommutative symmetric functions of typeB.

This being understood, we obtain complete families of orthogonal idempotents for the higher-order peak algebras, which can be described either by recurrence relations as in [17] or by generating series of noncommutative symmetric functions.

Finally, we make use of these idempotents to study the quivers, Cartan invariants, and the Loewy series of the unital higher-order peak algebras.

2 Notations and background

2.1 Noncommutative symmetric functions

We will assume familiarity with the standard notations of the theory of noncommu- tative symmetric functions [11] and with the main results of [3,13]. We recall here only a few essential definitions.

The Hopf algebra of noncommutative symmetric functions is denoted by Sym, or by Sym(A)if we consider the realization in terms of an auxiliary alphabetA. Linear bases of Sym_nare labeled by compositionsI =(i1, . . . , i_r)ofn(we writeIn). The noncommutative complete and elementary functions are denoted bySn andΛn, and S^I=S_i1· · ·S_ir. The ribbon basis is denoted byR_I. The descent set ofI is Des(I )= {i1, i1+i2, . . . , i1+ · · · +i_r−1}. The descent composition of a permutationσ∈S_nis the compositionIofnwhose descent set is the descent setD(σ )ofσ.

2.2 The Mantaci–Reutenauer algebra of typeB

We denote by MR the free product SymSym of two copies of the Hopf algebra of noncommutative symmetric functions [15]. That is, MR is the free associative al- gebra on two sequences(S_n)and (S_n¯)(n≥1). We regard the two copies of Sym

as noncommutative symmetric functions on two auxiliary alphabets: S_n =S_n(A) andS_n¯=S_n(A). We denote by¯ F → ¯F the involutive antiautomorphism which ex- changesSn andSn_¯. The bialgebra structure is defined by the requirement that the series

σ1=

n≥0

S_n and σ¯1=

n≥0

S_n¯ (1)

are grouplike. The internal product of MR can be computed from the splitting for- mula

(f1· · ·f_r)∗g=μ_r·(f1⊗ · · · ⊗f_r)∗rΔ^rg, (2) whereμr isr-fold multiplication, andΔ^r the iterated coproduct with values in the rth tensor power and the conditions:σ₁is neutral,σ¯₁is central, andσ¯₁∗ ¯σ₁=σ₁. 2.3 Noncommutative symmetric functions of typeB

Noncommutative symmetric functions of typeB were introduced in [9] as the right Sym-module BSym freely generated by another sequence(S˜_n)(n≥0, S˜₀=1) of homogeneous elements, withσ˜₁grouplike. This is a coalgebra, but not an algebra. It is endowed with an internal product for which each homogeneous component BSym_n is antiisomorphic to the descent algebra ofBn.

It should be noted that with this definition, the restriction of the internal product of BSym to Sym is not the internal product of Sym. To remedy this inconvenience, we use a different realization of BSym. We embed BSym as a subcoalgebra and sub-Sym-module of MR as follows. Define, forF ∈Sym(A),

F=F (A| ¯A)=F (A−qA)¯ |q=−1 (3) called the supersymmetric version, or superization, ofF [16]. It is also equal to

F=F ∗σ₁. (4)

Indeed,σ₁is grouplike, and forF=S^I, the splitting formula gives (Si₁· · ·Si_r)∗σ₁=μr

(Si₁⊗ · · · ⊗Si_r)∗

σ₁⊗ · · · ⊗σ₁

=S^I . (5) We have

σ₁= ¯λ₁σ₁=

Λ_¯iS_j. (6)

The elementσ¯₁is central for the internal product, and

¯

σ₁∗F (A,A)¯ =F (A, A)¯ =F∗ ¯σ₁. (7) The basis elementS˜^I of BSym, whereI=(i₀, i₁, . . . , i_r)is aB-composition (that is,i0may be 0), can be embedded as

S˜^I=S_i0(A)S^i1i2···ir(A| ¯A). (8) We will identify BSym with its image under this embedding.

2.4 Other notation

For a partition λ, we denote by m_i(λ) the multiplicity of i in λ and set m_λ :=

i≥1m_i(λ)!.

The reverse refinement order on compositions is denoted by. The nonincreasing rearrangement of a composition is denoted byI↓. The refinement order on partitions is denoted by≺p:λ≺pμifλis finer thanμ, that is, each part ofμis a sum of parts ofλ.

3 Descent algebras of type A

3.1 Principal idempotents

In [17], a recursive construction of complete sets of orthogonal idempotents of de- scent algebras has been described. In [14], one finds a general method for construct- ing such families from an arbitrary sequence of Lie idempotents, as well as many remarkable families of Lie idempotents. It is therefore natural to investigate whether the resulting idempotents can be derived from a (possibly known) sequence of Lie idempotents. We shall show that it is indeed the case.

LetP_n be the sequence of partitions ofnordered in the following way: first, sort them by decreasing length, then, for each length, order them by reverse lexicographic order. We denote this order by≤. For example,

P₅= [11111,2111,311,221,41,32,5]. (9) Now, start with

e1ⁿ:= 1

n!S₁ⁿ (10)

and define by induction

e_λ:= 1 m_λS^λ∗

S_n−

μ<λ

e_μ . (11)

Theorem 3.1 [17] The family(e_λ)_λnforms a complete system of orthogonal idem- potents for Sym_n.

Following [14], define the (left) Zassenhaus idempotentsζ_nby the generating se- ries

σ1=:^←

k≥1

e^ζk= · · ·e^ζ3e^ζ2e^ζ1. (12) For example,

S1=ζ1, S2=ζ2+1

2ζ₁², S3=ζ3+ζ2ζ1+1

6ζ₁³, (13)

S4=ζ4+ζ3ζ1+1 2ζ₂²+1

2ζ2ζ₁²+ 1

24ζ₁⁴, (14)

S5=ζ5+ζ4ζ1+ζ3ζ2+1

2ζ3ζ₁²+1

2ζ₂²ζ1+1

6ζ2ζ₁³+ 1

120ζ₁⁵, (15) S₆=ζ₆+ζ₅ζ₁+ζ₄ζ₂+1

2ζ₄ζ₁²+1

2ζ₃²+ζ₃ζ₂ζ₁ +1

6ζ₃ζ₁³+1 6ζ₂³+1

4ζ₂²ζ₁²+ 1

24ζ₂ζ₁⁴+ 1

720ζ₁⁶, (16)

so that

ζ1=S1, ζ2=S2−1

2S¹¹, ζ3=S3−S²¹+1

3S¹¹¹, (17)

ζ₄=S₄−S³¹−1 2S²²+3

4S²¹¹+1

4S¹¹²−1

4S¹¹¹¹, (18)

ζ₅=S₅−S⁴¹−S³²+S³¹¹+S²¹²−2

3S²¹¹¹−1

3S¹¹¹²+1

5S¹¹¹¹¹, (19)

ζ6=S6−S⁵¹−S⁴²+S⁴¹¹−1 2S³³+1

2S³²¹+S³¹²−5 6S³¹¹¹ +1

3S²²²−1

6S²²¹¹+1

2S²¹³−1

2S²¹²¹−2

3S²¹¹²+13 24S²¹¹¹¹

−1

6S¹¹²²+ 1

12S¹¹²¹¹−1

6S¹¹¹³+1

6S¹¹¹²¹+ 5

24S¹¹¹¹²−1

6S¹¹¹¹¹¹. (20) Note that, in particular,

S_n=

λn

1

m_λζ_λ1ζ_λ2· · ·ζ_λr. (21) For a compositionI=(i₁, . . . , i_r), define as usualζ^I:=ζ_i1· · ·ζ_ir. Sinceζ_n≡S_n modulo smaller terms in the refinement order on compositions,ζ^I ≡S^I modulo smaller terms. So theζ^I family is unitriangular on the basisS^J, so it is a basis of Sym.

In the sequel, we shall need a condition for a productS^I∗ζ^J to be zero.

Lemma 3.2 LetIandJbe two compositions ofn. Then,

S^I∗ζ^J=

⎧⎪

⎨

⎪⎩

0 ifJ↓ ≺pI↓,

mI↓ζ^I ifJ↓ =I↓,

K↓=J↓c_{I J}^Kζ^K otherwise,

(22)

wherec^K_{I J} is the number of ways of unshufflingJ intop=(I )subwords such that J^(l)has sumi_l and whose concatenationJ⁽¹⁾J⁽²⁾· · ·J^(p)isK.

Proof Since the Zassenhaus idempotentsζ_m are primitive, we have, thanks to the splitting formula (2),

S^I∗ζ^J=

J⁽¹⁾,...,J^(p)

S_i1∗ζ^J(1)

· · ·

S_ip∗ζ^J(p)

, (23)

where the sum ranges over all possible ways of decomposingJ into p (possibly empty) subwords.

SinceS_ij ∗ζ^{J(j )}=0 ifJ^{(j )}is not a composition ofi_j, it follows thatS^I∗ζ^J =0 ifJ↓ ≺pI↓. Moreover, ifJ↓ =I↓, then

S^I∗ζ^J =

σ∈Sp

(Si₁∗ζJ_{σ (1)})· · ·(Si_p∗ζJ_{σ (p)})=mIζ^I. (24)

IfJ↓ ≺pI↓andJ↓ =I↓, then a term in the r.h.s. of (23) is nonzero iff allJ⁽⁾are compositions ofi. In that case, we have

S^I∗ζ^J=

J⁽¹⁾,...,J^(p)

ζ^J(1)· · ·ζ^J(p)=

J⁽¹⁾,...,J^(p)

ζ^{J(1)J(2)···J(p)}, (25)

whence the last case.

Theorem 3.3 For all partitionsλ=(λ1, λ2, . . . , λk), eλ= 1

m_λζλ₁· · ·ζλ_k. (26) Proof Lete_λbe the right-hand side of (26). We will show that these elements satisfy the same induction as thee_λ(11).

From Lemma3.2we have

S^λ∗e_λ=mλe_λ. (27)

Now, using (21), we get mλe_λ=S^λ∗

Sn−

μ=λ

e_μ =S^λ∗

Sn−

μ<λ

e_μ , (28)

where the last equality follows again from Lemma3.2. Hence,eλ=e_λ. Note that, thanks to Lemma3.2, the induction formula foreλsimplifies to

eλ= 1 m_λS^λ∗

Sn−

μ≺pλ

eμ . (29)

3.2 A basis of idempotents

As with any sequence of Lie idempotents, we can construct an idempotent basis of Sym_n from the ζ_n. Here, the principal idempotentse_λ are members of the basis, which leads to a simpler derivation of the representation theory.

We start with a basic lemma, easily derived from the splitting formula (compare [14, Lemma 3.10]). Recall that the radical of(Sym_n,∗)isRn=R∩Sym_n, where Ris the kernel of the commutative image Sym→Sym.

Lemma 3.4 Denote byS(J )the set of distinct rearrangements of a compositionJ. LetI=(i₁, . . . , i_r)andJ=(j₁, . . . , j_s)be two compositions ofn. Then,

(i) If(J ) < (I ), thenζ^I∗ζ^J=0.

(ii) If(J ) > (I ), thenζ^I∗ζ^J∈Vectζ^K:K∈S(J ) ∩R. More precisely, ζ^I∗ζ^J=

J₁,...,J_r

|Jk|=ik

J, J1 · · · JrΓJ1· · ·ΓJ_r, (30)

where for a compositionKofk,Γ_K:=ζ_k∗ζ^K.

(iii) If(J )=(I ), thenζ^I∗ζ^J =0 only for J∈S(I ), in which caseζ^I ∗ζ^J = m_Iζ^I.

Note that the Γ_K are in the primitive Lie algebra. This follows from the

∗-multiplicativity of the coproduct: Δ(f ∗g)=Δ(f )∗Δ(g), see [11, Proposi- tion 5.5].

Corollary 3.5 The elements

eI= 1

m_Iζ^I, In, (31)

are all idempotents and form a basis of Sym_n. This basis contains in particular the principal idempotentse_λ.

3.3 Cartan invariants

By (iii) of Lemma 3.4, the indecomposable projective module Pλ=Sym_n ∗eλ

contains the eI for I ∈S(λ). For I ∈S(λ), (i) and (ii) imply that eI ∗eλ is in Vectζ^K:K∈S(λ). Hence, this space coincides withPλ. So, we get immediately the explicit decomposition

Sym_n=

λn

Pλ, Pλ=

I∈S(λ)

CeI. (32)

The Cartan invariants

c_λμ=dim(eμ∗Sym_n∗e_λ) (33)

are also easily obtained. The above space is spanned by the

eμ∗eI∗eλ=eμ∗eI, I∈S(λ). (34) By (ii) of Lemma3.4, this is the dimension of the space[S^μ(L)]λspanned by all sym- metrized products of Lie polynomials of degreesμ1, μ2, . . .formed fromζ_i1, ζ_i2, . . ., hence giving back the classical result of Garsia–Reutenauer [10].

3.4 Quiver andq-Cartan invariants (Loewy series)

Still relying upon point (ii) of Lemma3.4, we see thatcλμ=0 ifλis not finer than (or equal to)μ. Further, ifμis obtained fromλby adding up two partsλ_i, λ_j, then e_μ∗e_I=0 ifλ_i=λ_j and is a nonzero element of the radical otherwise.

In [8], it is shown that the powers of the radical for the internal product coincide with the lower central series of Sym for the external product:

R^∗j=γ^j(Sym), (35)

where γ^j(Sym) is the ideal generated by the commutators [Sym, γ^j−1(Sym)].

Hence, forλ finer thanμ, e_μ∗e_I is nonzero modulo R^∗2 iff μis obtained from λ by summing two distinct parts. More generally, e_μ∗e_I is in R^∗k and nonzero moduloR^∗k+1iff(λ)−(μ)=k.

Summarizing, we have the following:

Theorem 3.6 [8,17]

(i) In the quiver of Sym_n, there is an arrowλ→μiffμis obtained fromλby adding two distinct parts.

(ii) Theq-Cartan invariants are given by

cλμ(q)=q^(λ)−(μ) (36)

ifλis finer than (or equal to)μandc_λμ(q)=0 otherwise.

4 Descent algebras of typeB

4.1 Reinterpretation of some results of Chow

We begin by showing that in our realization of BSym, Chow’s map Θ (see [9, Sect. 3.4]) corresponds to the left internal product by the reproducing kernelσ₁of the superization map. Chow’s conditionΘ(S˜_n)=S_ntranslates into the obvious equality σ₁∗σ1=σ₁. The second conditionΘ(S_n(A))=S_n(2A)amounts to

σ₁∗σ₁= σ₁2

, (37)

which is an easy consequence of the splitting formula σ₁∗σ₁=(¯λ₁σ₁)∗σ₁=λ¯₁∗σ₁

σ₁= σ₁2

(38)

since

λ¯1∗σ₁=(λ1∗σ1)(λ1∗λ1)=σ₁. (39) Here we used the fact that left∗-multiplication byλ¯1is an antiautomorphism. Denot- ing byμas in [9] the twisted product

μ(A⊗B⊗C)=(λ₁∗B)AC, (40)

we have the following:

Lemma 4.1 ForF, G∈BSym,

σ₁∗(F G)=μ σ₁∗F

⊗Δ(G)

. (41)

Proof

σ₁∗(F G)=μ

(λ₁⊗σ₁)∗(ΔF ΔG)

=

(F ),(G)

μ

(λ₁∗F₁G₁)⊗F₂G₂

=

(λ1∗G1)

(λ1∗F1)F2

G2=μ σ₁∗F

⊗Δ(G)

. (42)

This is Chow’s third condition, which completes the characterization ofΘ.

4.2 Idempotents in BSym

Define the elementsζ_n∈BSym by the generating series σ₁=:

e^ζ1e^ζ2e^ζ3· · ·

· · ·e^ζ3e^ζ2e^ζ1

. (43)

For example, collecting the terms of weights 1, 2, and 3, respectively, we have S₁=2ζ1, S₂=2ζ2+2ζ₁², S₃=2ζ3+2ζ2ζ₁+2ζ1ζ₂+4

3ζ₁³, (44) so that

ζ1=1

2S₁, ζ2=1 2S₂−1

4S¹¹, ζ3=1 2S₃−1

4S²¹−1

4S¹²+1 6S¹¹¹.

(45) Note that the elementsζn are well defined and that they are primitive. We shall use the notation

e^ζ1e^ζ2· · ·

=:E^↑(ζ ),

· · ·e^ζ2e^ζ1

=:E^↓(ζ ). (46)

Next, define the elementsζ˜n∈BSym by the generating series

σ1=:

n≥0

ζ˜_n · · ·e^ζ2e^ζ1

=: ˜ζE^↓(ζ ). (47)

For example,

S1=ζ1+ ˜ζ1, S2=ζ2+1

2ζ₁²+ ˜ζ1ζ1+ ˜ζ2, (48) S₃=1

6ζ₁³+ζ₂ζ₁+ζ₃+ ˜ζ₁ζ₂+1

2ζ˜₁ζ₁²+ ˜ζ₂ζ₁+ ˜ζ₃, (49) so that

ζ˜1=S1−1

2S₁, ζ˜2=S2−1 2S₂−1

2S1S₁+3

8S¹¹, (50)

ζ˜3=S3−1

2S2S₁−1

2S1S₂+3

8S1S¹¹−1 2S₃+1

4S²¹+1

2S¹²− 5

16S¹¹¹. (51) Sinceσ1 is grouplike and sincee^ζn is grouplike for alln≥1, the series ζ˜ is also grouplike.

The next two lemmas describe some properties of the elementsζ˜_nandζ_n. Lemma 4.2 The ordered exponentials are exchanged as follows:

λ1∗E^↓(ζ )=E^↑(ζ ). (52)

In particular,λ₁∗ζ_i=ζ_i for alli≥0.

Proof λ1∗ ·is an antiautomorphism, so the left-hand side is λ1∗e^ζ1

λ1∗e^ζ2

· · ·. (53)

Taking into account (39) and recalling that (43) characterizes theζ_i, we see that if we set

ζ_i:=λ₁∗ζ_i, (54)

then

σ₁=

e^ζ1e^ζ2· · ·

· · ·e^ζ2e^ζ1

, (55)

so thatζ_i=ζi.

Lemma 4.3 For alln≥1,

σ₁∗ ˜ζ_n=0. (56)

Proof By definition,

E^↑(ζ )E^↓(ζ )=σ₁=σ₁∗σ1=σ₁∗ζ˜E^↓(ζ )

=(λ1σ1)∗ζ˜E^↓(ζ )

=μ

(λ₁⊗σ₁)∗ζ˜E^↓(ζ )⊗ ˜ζE^↓(ζ )

=

λ₁∗ ˜ζE^↓(ζ )ζ˜E^↓(ζ )

=

λ₁∗E^↓(ζ )

(λ₁∗ ˜ζ )ζ˜E^↓(ζ )=E^↑(ζ )(λ₁∗ ˜ζ )(σ₁∗ ˜ζ )E^↓(ζ )

=E^↑(ζ ) σ₁∗ ˜ζ

E^↓(ζ ), (57)

so thatσ₁∗ ˜ζ =1.

Lemma 4.4 For alln≥1,

σ₁∗ζn=2ζn. (58)

Proof We have

σ₁∗ζ_n=(λ₁σ₁)∗ζ_n=μ

(λ₁⊗σ₁)∗(ζ_n⊗1+1⊗ζ_n)

=2ζn. (59) Proposition 4.5 Let I = (i0, . . . , ip) be a B-composition of n, and let λ = (λ₀, . . . , λ_k)be aB-partition ofn.

S˜^I∗ ˜ζλ0ζλ1· · ·ζλ_k =

0 ifλI↓,

(2^p

j≥1mj!)˜ζi₀ζi₁· · ·ζi_p ifλ=I↓, (60) wheremj is the multiplicity ofj in(i1, i2, . . . , ip)(not countingi0!).

Proof The splitting formula yields S˜^I∗ ˜ζλ₀ζλ₁· · ·ζλ_k =μp

Si₀⊗S_i

1⊗ · · · ⊗S_i

p

∗

ζ˜_α0ζ^α(0)⊗ · · · ⊗ ˜ζ_αpζ^α(p)

=

S_i0∗ ˜ζ_α

0ζ^α(0) S_i

1∗ ˜ζ_α

1ζ^α(1)

· · · S_i

p∗ ˜ζ_α

pζ^α(p)

. (61) By (4) and Lemma4.3, a summand is zero if anyα_i>0 fori≥1, so that

S˜^I∗ ˜ζλ₀ζλ₁· · ·ζλ_k=

Si₀∗ ˜ζ_α0ζ^α(0) S_i

1∗ζ^α(1)

· · · S_i

r ∗ζ^α(p)

. (62) If a term is nonzero in this equation, thenλI↓. This proves the first case. By (4) and Lemma4.4,S_i∗ζ_i=2ζi, which proves the second case.

We are now in a position to give an explicit formula for the idempotents of [17].

Theorem 4.6 For allB-partitionsλ=(λ0, λ1, . . . , λk)ofn, define the elementseλ∈ BSym_nrecursively by the formula

e_λ= 1

2^k

jmj!S˜^λ∗

S_n−

μ<λ

e_μ , (63)

wheremj is the multiplicity ofj in(λ1, . . . , λk)(not countingλ0!). Then e_λ= 1

jmj!ζ˜_λ0ζ_λ1· · ·ζ_λk. (64)

Proof Lete_λbe the right-hand side of the above equation. By Proposition4.5, S˜^λ∗e_λ = ˜S^λ∗ 1

jm_j!ζ˜_λ0ζ_λ1· · ·ζ_λk=

2^k

j

m_j! e_λ. (65)

By (47),S_n=

e_λ, where the sum ranges over allB-partitionsλofn. Together with the above and Proposition4.5, we have

2^k

j

m_j! e_λ= ˜S^λ∗e_λ= ˜S^λ∗

S_n−

μ=λ

e_μ = ˜S^λ∗

S_n−

μ<λ

e_μ . (66)

Since thee_λ satisfy the same induction as theeλ, they are equal.

Theorem 4.7 [17] The family(e_λ)whereλruns overB-partitions ofnforms a com- plete system of orthogonal idempotents for BSym_n.

5 Idempotents in the higher order peak algebras

Letq be a primitiverth root of unity. We denote byθq the endomorphism of Sym defined by

f˜=θq(f )=f

(1−q)A

=f (A)∗σ1

(1−q)A

. (67)

We denote by ˚P^(r)the image ofθq and byP^(r)the right ˚P^(r)-module generated by S_n forn≥0. Note that ˚P^(r)is by definition a left∗-ideal of Sym. Forr=2, ˚P⁽²⁾ is the classical peak ideal, andP⁽²⁾is the unital peak algebra. For generalr, ˚P^(r)is the higher-order peak algebra of [13], andP^(r)is its unital extension defined in [3].

These objects depend only onrand not on the choice of the primitive root of unity.

Bases ofP^(r)can be labeled byr-peak compositionsI=(i0;i1, . . . , ip)with at most one parti₀divisible byr.

5.1 The radical

By definition,Pn^(r)is a∗-subalgebra of Sym_n. The radical of Sym_nconsists of those elements whose commutative image is zero (see [14], Lemma 3.10). The radical of Pn^(r)is therefore spanned by theS_i0·θ_q(S^I−S^I)such thatIis a permutation ofI. Indeed, the quotient ofPn^(r)by the span of those elements is a semisimple commuta- tive algebra, the∗-subalgebra of Sym_nspanned by thepλ (λn) such that at most one part ofλis multiple ofr. This special part will be denoted byλ₀.

We denote byP_n^(r)the subset of partitions ofnwith at most one part divisible byr.

The simplePn^(r)-modules, and the principal idempotents, can therefore be labeled byP_n^(r).

5.2 An induction for the idempotents

Define a total order<onP_n^(r)as follows: sort the partitions by decreasing length and sort partitions of the same length by reverse lexicographic order. For example,

P₅⁽²⁾= [11111,2111,311,41,23,5], (68) P₇⁽²⁾= [1111111,211111,31111,4111,2311,511,331,61,43,25,7]. (69) Now, set

e₁^(r)n := 1

n!S₁ⁿ (70)

and define by induction

e_λ^(r):= 1 mλ

T^λ∗

S_n−

μ<λ

e^(r)_μ , (71)

whereT_m=R_mifr|m,T_m=R_rijifrmandm=ir+jwith 0< j < r, andT^λ= Tλ₀Tλ₁· · ·Tλ_p forλ=(λ0;λ1, . . . , λp)∈P_n^(r). It follows from [13, Corollary 3.17]

and from the definition ofP^(r)thatT^λ∈P^(r). We want to prove that(e^(r)_λ )

λ∈P_n^(r)is a complete system of orthogonal idempotents forP^(r).

To this aim, we introduce the sequence of (left) Zassenhaus idempotentsζn^(r)of levelras the unique solution of the equation

σ1=

p≥0

ζ_pr^(r) ← i≥1,ri

e^ζi(r). (72)

Note thatζ_n^(r)=ζnforn <2r.

For example, forr=2,

ζ₁⁽²⁾=S₁; ζ₂⁽²⁾=S₂−1

2S¹¹; ζ₃⁽²⁾=S₃−S²¹+1

3S¹¹¹, (73) ζ₄⁽²⁾=S₄−S³¹+1

2S²¹¹−1

8S¹¹¹¹, (74)

ζ₅⁽²⁾=S₅−S⁴¹+1

2S³¹¹−S²³+S²²¹−1

2S²¹¹¹+1 2S¹¹³

−1

2S¹¹²¹+1

5S¹¹¹¹¹; (75)

and forr=3,

ζ₁⁽³⁾=S1; ζ₂⁽³⁾=S2−1

2S¹¹; ζ₃⁽³⁾=S3−S²¹+1

3S¹¹¹, (76)

ζ₄⁽³⁾=S4−S³¹−1 2S²²+3

4S²¹¹+1

4S¹¹²−1

4S¹¹¹¹, (77)

ζ₅⁽³⁾=S5−S⁴¹−S³²+S³¹¹+S²¹²−2

3S²¹¹¹−1

3S¹¹¹²+1

5S¹¹¹¹¹, (78)

ζ₆⁽³⁾=S₆−S⁵¹−S⁴²+S⁴¹¹+S³¹²−2

3S³¹¹¹+1

3S²²²−1 6S²²¹¹

−2

3S²¹¹²+3

8S²¹¹¹¹−1

6S¹¹²²+ 1

12S¹¹²¹¹+ 5

24S¹¹¹¹²−1

9S¹¹¹¹¹¹. (79) Define now forλ=(λ0;λ1, . . . , λ_k)∈P_n^(r),

e^(r)_λ := 1 m_λζ_λ^(r)

0 ζ_λ^(r)

1 · · ·ζ_λ^(r)

k . (80)

We will show thate^(r)_λ =e^(r)_λ for allλ∈P_n^(r). We begin with two lemmas.

Lemma 5.1

Δ ζ_n^(r)

=

1⊗ζ_n^(r)+ζ_n^(r)⊗1 ifrn, n/r

i=0ζ_ir^(r)⊗ζ_n^(r)_−ir ifr|n.

(81)

Proof This means that ζ_n^(r) are primitive if r |n and that the generating series

p≥0ζ_rp^(r)is grouplike. If we define new elementsYpby

σ1=:

→ p

e^Yrp ←

ri

e^Yi, (82)

the standard argument showing that the Zassenhaus elements are primitive shows as well that all theYi are primitive. Now identify the first product in the right-hand side with the generating series

p≥0ζrp^(r). Thenζ_i^(r)=Yi ifri. Since the exponential of a primitive element is grouplike and a product of grouplike series is group-like, both products in the right-hand side above are grouplike. By identification,

p≥0ζ_rp^(r)is

grouplike.

Lemma 5.2 Letλ=(λ0;λ1, . . . , λk)∈P_n^(r) and I =(i0, i1, . . . , ip)be anr-peak composition ofn. Then,

T^I∗ζ_λ^(r)

0 ζ_λ^(r)

1 · · ·ζ_λ^(r)

k =

0 ifI↓< λ,

mIζ_i^(r)

0 ζ_i^(r)

1 · · ·ζ_i^(r)

k ifI↓ =λ. (83)

Proof To simplify the notation, we letζ^(r)λ=ζ_λ^(r)

0 ζ_λ^(r)

1 · · ·ζ_λ^(r)

k forλ∈Pn^(r). IfF^I=

F_i1· · ·F_ip with eachF_ij∈Sym_ij, the splitting formula and (81) yield

F^I∗ζ^(r)λ=

λ⁽¹⁾∨···∨λ^(p)=(λ₁,...,λ_k) ra₁+···+ra_p=λ₀

p j=1

Fi_j∗ζ_ra^(r)

jζ^(r)λ

(j )

, (84)

whereλ0is the part (possibly 0) ofλthat is divisible byr, and whereα∨βdenotes the partition obtained by reordering the concatenation of the partitionsαandβ.

Observe that since at most oneij is divisible byr, a product in the above summa- tion is 0 if at least two of the partitionsλ⁽¹⁾, . . . , λ^(p)are empty. If(I ) > (λ), then k≤p−2, so this hypothesis is always satisfied. Thus,

F^I∗ζ^(r)λ=0 if(I ) > (λ). (85)

SupposeI↓ ≤λ. Then, by the definition of the order,(I )≥(λ). Hence, forI such thatI↓ ≤λand(I ) > (λ), the result follows by takingF^I=T^I in (85). So suppose instead that(I )=(λ). By definition,T^I=Ti₁· · ·Ti_p, whereTm=Rmif r|mandT_m=R_rijifm=ir+jwith 0< j < r. SinceR_Jcan be written as a linear combination ofS^K for whichJ is a refinement ofK, it follows thatT^I is equal to S^Iplus a linear combination ofS^Kwith(K) > (I ). By takingF^I=S^K in (85), it follows thatT^I∗ζ^(r)λ=S^I∗ζ^(r)λ.

It remains to show that, forI andλof the same length,S^I∗ζ^(r)λ=mIζ^(r)λ if I↓ =λand is 0 otherwise. Ifλcontains no part divisible byr, then it follows from (84) that ifS^I∗ζ^(r)λ=0, we must haveI↓ =λ, in which caseS^I∗ζ^(r)λ=m_Iζ^(r)λ. Suppose instead thatλcontains a part that is divisible byr. Then each decomposi- tion(λ⁽¹⁾, . . . , λ^(p))in (84) contains at least oneλ^{(j )}= ∅. Thus, ifI contains no part that is divisible byr, thenS^I∗ζ^(r)λ=0. Otherwise, the part ofI that is divisible byr is bounded byλ₀. This implies thatλ≤I↓. Since we began by assuming that I↓ ≤λ, it follows thatI↓ =λ, and the result follows from (84) as before.

Theorem 5.3 For all partitionsλ=(λ₀;λ₁, . . . , λ_k)∈P_n^(r), e^(r)_λ =e^(r)_λ := 1

mλ

ζ_λ^(r)

0 ζ_λ^(r)

1 · · ·ζ_λ^(r)

k . (86)

Proof From the definition of ζ_m^(r) it follows that S_n =

λ∈Pn^(r)e^(r)_λ . Hence, by Lemma5.2,

m_λe^(r)_λ =T^λ∗e_λ=T^λ∗

S_n−

μ=λ

e^(r)_μ =T^λ∗

S_n−

μ<λ

e^(r)_μ .

Thus, the elementse^(r)_λ ande_λ^(r)satisfy the same induction equation (71).