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*∈S*n*is an index*i*such that*σ (i) > σ (i*+1). A descent
*is a peak if moreoveri >*1 and*σ (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 ˚P*

*n*ofS

*n*is 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 case*q*= −1 of a*q-identity*

J.-C. Novelli (

^{)}

^{·}

^{F. Saliola}

^{·}J.-Y. Thibon

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]. Specializing*q* 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 ofS*n* can be naturally extended
to a unital algebra which is obtained as a homomorphic image of the descent algebra
of the hyperoctahedral group*B**n*. 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 type*B*constructed 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 type*A, 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 type*B* as well, which
yields a simple generating function in terms of noncommutative symmetric functions
of type*B.*

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 alphabet*A. Linear*
**bases of Sym*** _{n}*are labeled by compositions

*I*=

*(i*1

*, . . . , i*

_{r}*)*of

*n*(we write

*In). The*noncommutative complete and elementary functions are denoted by

*S*

*n*and

*Λ*

*n*, and

*S*

*=*

^{I}*S*

_{i}_{1}· · ·

*S*

_{i}*. The ribbon basis is denoted by*

_{r}*R*

_{I}*. The descent set ofI*is Des(I )= {

*i*1

*, i*1+

*i*2

*, . . . , i*1+ · · · +

*i*

*1}. The descent composition of a permutation*

_{r−}*σ*∈S

*is the composition*

_{n}*I*of

*n*whose descent set is the descent set

*D(σ )*of

*σ*.

2.2 The Mantaci–Reutenauer algebra of type*B*

**We denote by MR the free product Sym****Sym 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)*and

*S*

_{n}_{¯}=

*S*

_{n}*(A). We denote by*¯

*F*→ ¯

*F*

*the involutive antiautomorphism which ex-*changes

*S*

*n*and

*S*

*n*

_{¯}. The bialgebra structure is defined by the requirement that the series

*σ*1=

*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

*(f*1· · ·*f*_{r}*)*∗*g*=*μ** _{r}*·

*(f*1⊗ · · · ⊗

*f*

_{r}*)*∗

*r*

*Δ*

^{r}*g,*(2) where

*μ*

*r*is

*r-fold multiplication, andΔ*

*the iterated coproduct with values in the*

^{r}*rth tensor power and the conditions:σ*

_{1}is neutral,

*σ*¯

_{1}is central, and

*σ*¯

_{1}∗ ¯

*σ*

_{1}=

*σ*

_{1}. 2.3 Noncommutative symmetric functions of type

*B*

Noncommutative symmetric functions of type*B* 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, with*σ*˜_{1}grouplike. 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 of

*B*

*n*.

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, for***F* ∈**Sym(A),**

*F** ^{}*=

*F (A*| ¯

*A)*=

*F (A*−

*qA)*¯ |

*q*=−1 (3) called the supersymmetric version, or superization, of

*F*[16]. It is also equal to

*F** ^{}*=

*F*∗

*σ*

_{1}

^{}*.*(4)

Indeed,*σ*_{1}* ^{}*is grouplike, and for

*F*=

*S*

*, the splitting formula gives*

^{I}*(S*

*i*

_{1}· · ·

*S*

*i*

_{r}*)*∗

*σ*

_{1}

*=*

^{}*μ*

*r*

*(S**i*_{1}⊗ · · · ⊗*S**i*_{r}*)*∗

*σ*_{1}* ^{}*⊗ · · · ⊗

*σ*

_{1}

^{}=*S*^{I }*.* (5)
We have

*σ*_{1}* ^{}*= ¯

*λ*

_{1}

*σ*

_{1}=

*Λ*_{¯}_{i}*S*_{j}*.* (6)

The element*σ*¯_{1}is central for the internal product, and

¯

*σ*_{1}∗*F (A,A)*¯ =*F (A, A)*¯ =*F*∗ ¯*σ*_{1}*.* (7)
The basis element*S*˜^{I}**of BSym, where***I*=*(i*_{0}*, i*_{1}*, . . . , i*_{r}*)*is a*B*-composition (that
is,*i*0may be 0), can be embedded as

*S*˜* ^{I}*=

*S*

_{i}_{0}

*(A)S*

^{i}^{1}

^{i}^{2}

^{···}

^{i}

^{r}*(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*≥1*m*_{i}*(λ)*!.

The reverse refinement order on compositions is denoted by. The nonincreasing
rearrangement of a composition is denoted by*I*↓. The refinement order on partitions
is denoted by≺*p*:*λ*≺*p**μ*if*λ*is finer than*μ, that is, each part ofμ*is a sum of parts
of*λ.*

**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.

Let*P** _{n}* be the sequence of partitions of

*n*ordered 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,

*P*_{5}= [11111,2111,311,221,41,32,5]*.* (9)
Now, start with

*e*1* ^{n}*:= 1

*n*!*S*_{1}* ^{n}* (10)

and define by induction

*e** _{λ}*:= 1

*m*

_{λ}*S*

*∗*

^{λ}

*S** _{n}*−

*μ<λ*

*e*_{μ}*.* (11)

**Theorem 3.1 [17] The family***(e*_{λ}*)*_{λ}_{}_{n}*forms a complete system of orthogonal idem-*
**potents for Sym*** _{n}*.

Following [14], define the (left) Zassenhaus idempotents*ζ** _{n}*by the generating se-
ries

*σ*1=:^{←}

*k*≥1

*e*^{ζ}* ^{k}*= · · ·

*e*

^{ζ}^{3}

*e*

^{ζ}^{2}

*e*

^{ζ}^{1}

*.*(12) For example,

*S*1=*ζ*1*,* *S*2=*ζ*2+1

2*ζ*_{1}^{2}*,* *S*3=*ζ*3+*ζ*2*ζ*1+1

6*ζ*_{1}^{3}*,* (13)

*S*4=*ζ*4+*ζ*3*ζ*1+1
2*ζ*_{2}^{2}+1

2*ζ*2*ζ*_{1}^{2}+ 1

24*ζ*_{1}^{4}*,* (14)

*S*5=*ζ*5+*ζ*4*ζ*1+*ζ*3*ζ*2+1

2*ζ*3*ζ*_{1}^{2}+1

2*ζ*_{2}^{2}*ζ*1+1

6*ζ*2*ζ*_{1}^{3}+ 1

120*ζ*_{1}^{5}*,* (15)
*S*_{6}=*ζ*_{6}+*ζ*_{5}*ζ*_{1}+*ζ*_{4}*ζ*_{2}+1

2*ζ*_{4}*ζ*_{1}^{2}+1

2*ζ*_{3}^{2}+*ζ*_{3}*ζ*_{2}*ζ*_{1}
+1

6*ζ*_{3}*ζ*_{1}^{3}+1
6*ζ*_{2}^{3}+1

4*ζ*_{2}^{2}*ζ*_{1}^{2}+ 1

24*ζ*_{2}*ζ*_{1}^{4}+ 1

720*ζ*_{1}^{6}*,* (16)

so that

*ζ*1=*S*1*,* *ζ*2=*S*2−1

2*S*^{11}*,* *ζ*3=*S*3−*S*^{21}+1

3*S*^{111}*,* (17)

*ζ*_{4}=*S*_{4}−*S*^{31}−1
2*S*^{22}+3

4*S*^{211}+1

4*S*^{112}−1

4*S*^{1111}*,* (18)

*ζ*_{5}=*S*_{5}−*S*^{41}−*S*^{32}+*S*^{311}+*S*^{212}−2

3*S*^{2111}−1

3*S*^{1112}+1

5*S*^{11111}*,* (19)

*ζ*6=*S*6−*S*^{51}−*S*^{42}+*S*^{411}−1
2*S*^{33}+1

2*S*^{321}+*S*^{312}−5
6*S*^{3111}
+1

3*S*^{222}−1

6*S*^{2211}+1

2*S*^{213}−1

2*S*^{2121}−2

3*S*^{2112}+13
24*S*^{21111}

−1

6*S*^{1122}+ 1

12*S*^{11211}−1

6*S*^{1113}+1

6*S*^{11121}+ 5

24*S*^{11112}−1

6*S*^{111111}*.* (20)
Note that, in particular,

*S** _{n}*=

*λ**n*

1

*m*_{λ}*ζ*_{λ}_{1}*ζ*_{λ}_{2}· · ·*ζ*_{λ}_{r}*.* (21)
For a composition*I*=*(i*_{1}*, . . . , i*_{r}*), define as usualζ** ^{I}*:=

*ζ*

_{i}_{1}· · ·

*ζ*

_{i}*. Since*

_{r}*ζ*

*≡*

_{n}*S*

*modulo smaller terms in the refinement order on compositions,*

_{n}*ζ*

*≡*

^{I}*S*

*modulo smaller terms. So the*

^{I}*ζ*

*family is unitriangular on the basis*

^{I}*S*

*, so it is a basis of*

^{J}**Sym.**

In the sequel, we shall need a condition for a product*S** ^{I}*∗

*ζ*

*to be zero.*

^{J}**Lemma 3.2 Let**IandJbe two compositions ofn. Then,

*S** ^{I}*∗

*ζ*

*=*

^{J}⎧⎪

⎨

⎪⎩

0 *ifJ*↓ ≺*p**I*↓*,*

*m**I*↓*ζ*^{I}*ifJ↓ =I*↓,

*K*↓=*J*↓*c*_{I J}^{K}*ζ*^{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*_{i}_{1}∗*ζ*^{J}^{(1)}

· · ·

*S*_{i}* _{p}*∗

*ζ*

^{J}

^{(p)}*,* (23)

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

Since*S*_{i}* _{j}* ∗

*ζ*

^{J}*=0 if*

^{(j )}*J*

*is not a composition of*

^{(j )}*i*

*, it follows that*

_{j}*S*

*∗*

^{I}*ζ*

*=0 if*

^{J}*J*↓ ≺

*p*

*I*↓. Moreover, if

*J*↓ =

*I*↓, then

*S** ^{I}*∗

*ζ*

*=*

^{J}*σ*∈S*p*

*(S**i*_{1}∗*ζ**J*_{σ (1)}*)*· · ·*(S**i** _{p}*∗

*ζ*

*J*

_{σ (p)}*)*=

*m*

*I*

*ζ*

^{I}*.*(24)

If*J*↓ ≺*p**I*↓and*J*↓ =*I*↓, then a term in the r.h.s. of (23) is nonzero iff all*J** ^{()}*are
compositions of

*i*

*. 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*=

*(λ*1

*, λ*2

*, . . . , λ*

*k*

*),*

*e*

*λ*= 1

*m*_{λ}*ζ**λ*_{1}· · ·ζ*λ*_{k}*.* (26)
*Proof Lete*^{}* _{λ}*be the right-hand side of (26). We will show that these elements satisfy
the same induction as the

*e*

*(11).*

_{λ}From Lemma3.2we have

*S** ^{λ}*∗

*e*

^{}

*=*

_{λ}*m*

*λ*

*e*

^{}

_{λ}*.*(27)

Now, using (21), we get
*m**λ**e*^{}* _{λ}*=

*S*

*∗*

^{λ}

*S**n*−

*μ*=*λ*

*e*_{μ}^{} =*S** ^{λ}*∗

*S**n*−

*μ<λ*

*e*_{μ}^{} *,* (28)

where the last equality follows again from Lemma3.2. Hence,*e**λ*=*e*^{}* _{λ}*.
Note that, thanks to Lemma3.2, the induction formula for

*e*

*λ*simplifies to

*e**λ*= 1
*m*_{λ}*S** ^{λ}*∗

*S**n*−

*μ*≺*p**λ*

*e**μ* *.* (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*

*. Here, the principal idempotents*

_{n}*e*

*are members of the basis, which leads to a simpler derivation of the representation theory.*

_{λ}We start with a basic lemma, easily derived from the splitting formula (compare
[14, Lemma 3.10]). Recall that the radical of*(Sym*_{n}*,*∗*)*is*R**n*=*R*∩**Sym*** _{n}*, where

*R*

**is the kernel of the commutative image Sym**→

*Sym.*

* Lemma 3.4 Denote by*S(J )

*the set of distinct rearrangements of a compositionJ*.

*LetI*=

*(i*

_{1}

*, . . . , i*

_{r}*)andJ*=

*(j*

_{1}

*, . . . , j*

_{s}*)be two compositions ofn. Then,*

*(i) If(J ) < (I ), thenζ** ^{I}*∗

*ζ*

*=0.*

^{J}*(ii) If(J ) > (I ), thenζ** ^{I}*∗

*ζ*

*∈Vect*

^{J}*ζ*

*:*

^{K}*K*∈S(J ) ∩

*R. More precisely,*

*ζ*

*∗*

^{I}*ζ*

*=*

^{J}*J*_{1}*,...,J*_{r}

|J*k*|=i*k*

J, J1 · · · *J**r*Γ*J*1· · ·Γ*J*_{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 case

*ζ*

*∗*

^{I}*ζ*

*=*

^{J}*m*

_{I}*ζ*

*.*

^{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**

*e**I*= 1

*m*_{I}*ζ*^{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 *P**λ*=**Sym*** _{n}* ∗

*e*

*λ*

contains the *e**I* for *I* ∈S(λ). For *I* ∈S(λ), (i) and (ii) imply that *e**I* ∗*e**λ* is in
Vect*ζ** ^{K}*:

*K*∈S(λ). Hence, this space coincides with

*P*

*λ*. So, we get immediately the explicit decomposition

**Sym*** _{n}*=

*λ**n*

*P**λ**,* *P**λ*=

*I*∈S(λ)

C*e**I**.* (32)

The Cartan invariants

*c** _{λμ}*=dim(e

*μ*∗

**Sym**

*∗*

_{n}*e*

_{λ}*)*(33)

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

*e**μ*∗*e**I*∗*e**λ*=*e**μ*∗*e**I**,* *I*∈S(λ). (34)
By (ii) of Lemma3.4, this is the dimension of the space[*S*^{μ}*(L)*]*λ*spanned by all sym-
metrized products of Lie polynomials of degrees*μ*1*, μ*2*, . . .*formed from*ζ*_{i}_{1}*, ζ*_{i}_{2}*, . . .,*
hence giving back the classical result of Garsia–Reutenauer [10].

3.4 Quiver and*q*-Cartan invariants (Loewy series)

Still relying upon point (ii) of Lemma3.4, we see that*c**λμ*=0 if*λ*is not finer than
(or equal to)*μ. Further, ifμ*is obtained from*λ*by adding up two parts*λ*_{i}*, λ** _{j}*, then

*e*

*∗*

_{μ}*e*

*=0 if*

_{I}*λ*

*=*

_{i}*λ*

*and is a nonzero element of the radical otherwise.*

_{j}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, for*λ* finer than*μ,* *e** _{μ}*∗

*e*

*is nonzero modulo*

_{I}*R*

^{∗}

^{2}iff

*μ*is obtained from

*λ*by summing two distinct parts. More generally,

*e*

*∗*

_{μ}*e*

*is in*

_{I}*R*

^{∗}

*and nonzero modulo*

^{k}*R*

^{∗}

^{k}^{+}

^{1}iff

*(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*

*c**λμ**(q)*=*q*^{(λ)}^{−}* ^{(μ)}* (36)

*ifλis finer than (or equal to)μandc*_{λμ}*(q)*=*0 otherwise.*

**4 Descent algebras of type****B**

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 kernel*σ*_{1}* ^{}*of the
superization map. Chow’s condition

*Θ(S*˜

_{n}*)*=

*S*

*translates into the obvious equality*

_{n}*σ*

_{1}

*∗*

^{}*σ*1=

*σ*

_{1}

*. The second condition*

^{}*Θ(S*

_{n}*(A))*=

*S*

_{n}*(2A)*amounts to

*σ*_{1}* ^{}*∗

*σ*

_{1}

*=*

^{}*σ*

_{1}

*2*

^{}*,* (37)

which is an easy consequence of the splitting formula
*σ*_{1}* ^{}*∗

*σ*

_{1}

*=*

^{}*(¯λ*

_{1}

*σ*

_{1}

*)*∗

*σ*

_{1}

*=*

^{}*λ*¯

_{1}∗

*σ*

_{1}

^{}*σ*_{1}* ^{}*=

*σ*

_{1}

*2*

^{}(38)

since

*λ*¯1∗*σ*_{1}* ^{}*=

*(λ*1∗

*σ*1

*)(λ*1∗

*λ*1

*)*=

*σ*

_{1}

^{}*.*(39) Here we used the fact that left∗-multiplication by

*λ*¯1is an antiautomorphism. Denot- ing by

*μ*

^{}as in [9] the twisted product

*μ*^{}*(A*⊗*B*⊗*C)*=*(λ*_{1}∗*B)AC,* (40)

we have the following:

* Lemma 4.1 ForF, G*∈

**BSym,**

*σ*_{1}* ^{}*∗

*(F G)*=

*μ*

^{}

*σ*

_{1}

*∗*

^{}*F*

⊗*Δ(G)*

*.* (41)

*Proof*

*σ*_{1}* ^{}*∗

*(F G)*=

*μ*

*(λ*_{1}⊗*σ*_{1}*)*∗*(ΔF ΔG)*

=

*(F ),(G)*

*μ*

*(λ*_{1}∗*F*_{1}*G*_{1}*)*⊗*F*_{2}*G*_{2}

=

*(λ*1∗*G*1*)*

*(λ*1∗*F*1*)F*2

*G*2=*μ*^{}
*σ*_{1}* ^{}*∗

*F*

⊗*Δ(G)*

*.* (42)

This is Chow’s third condition, which completes the characterization of*Θ.*

**4.2 Idempotents in BSym**

Define the elements*ζ** _{n}*∈

**BSym by the generating series**

*σ*

_{1}

*=:*

^{}*e*^{ζ}^{1}*e*^{ζ}^{2}*e*^{ζ}^{3}· · ·

· · ·*e*^{ζ}^{3}*e*^{ζ}^{2}*e*^{ζ}^{1}

*.* (43)

For example, collecting the terms of weights 1, 2, and 3, respectively, we have
*S*_{1}* ^{}*=2ζ1

*,*

*S*

_{2}

*=2ζ2+2ζ*

^{}_{1}

^{2}

*,*

*S*

_{3}

*=2ζ3+2ζ2*

^{}*ζ*

_{1}+2ζ1

*ζ*

_{2}+4

3*ζ*_{1}^{3}*,* (44)
so that

*ζ*1=1

2*S*_{1}^{}*,* *ζ*2=1
2*S*_{2}* ^{}*−1

4*S*^{11}*,* *ζ*3=1
2*S*_{3}* ^{}*−1

4*S*^{21}−1

4*S*^{12}+1
6*S*^{111}*.*

(45)
Note that the elements*ζ**n* are well defined and that they are primitive. We shall use
the notation

*e*^{ζ}^{1}*e*^{ζ}^{2}· · ·

=:*E*^{↑}*(ζ ),*

· · ·*e*^{ζ}^{2}*e*^{ζ}^{1}

=:*E*^{↓}*(ζ ).* (46)

Next, define the elements*ζ*˜*n*∈**BSym by the generating series**

*σ*1=:

*n*≥0

*ζ*˜* _{n}* · · ·

*e*

^{ζ}^{2}

*e*

^{ζ}^{1}

=: ˜*ζE*^{↓}*(ζ ).* (47)

For example,

*S*1=*ζ*1+ ˜*ζ*1*,* *S*2=*ζ*2+1

2*ζ*_{1}^{2}+ ˜*ζ*1*ζ*1+ ˜*ζ*2*,* (48)
*S*_{3}=1

6*ζ*_{1}^{3}+*ζ*_{2}*ζ*_{1}+*ζ*_{3}+ ˜*ζ*_{1}*ζ*_{2}+1

2*ζ*˜_{1}*ζ*_{1}^{2}+ ˜*ζ*_{2}*ζ*_{1}+ ˜*ζ*_{3}*,* (49)
so that

*ζ*˜1=*S*1−1

2*S*_{1}^{}*,* *ζ*˜2=*S*2−1
2*S*_{2}* ^{}*−1

2*S*1*S*^{}_{1}+3

8*S*^{11}*,* (50)

*ζ*˜3=*S*3−1

2*S*2*S*_{1}* ^{}*−1

2*S*1*S*_{2}* ^{}*+3

8*S*1*S*^{11}−1
2*S*_{3}* ^{}*+1

4*S*^{21}+1

2*S*^{12}− 5

16*S*^{111}*.* (51)
Since*σ*1 is grouplike and since*e*^{ζ}* ^{n}* is grouplike for all

*n*≥1, the series

*ζ*˜ is also grouplike.

The next two lemmas describe some properties of the elements*ζ*˜* _{n}*and

*ζ*

*.*

_{n}

**Lemma 4.2 The ordered exponentials are exchanged as follows:***λ*1∗*E*^{↓}*(ζ )*=*E*^{↑}*(ζ ).* (52)

*In particular,λ*_{1}∗*ζ** _{i}*=

*ζ*

_{i}*for alli*≥0.

*Proof* *λ*1∗ ·is an antiautomorphism, so the left-hand side is
*λ*1∗*e*^{ζ}^{1}

*λ*1∗*e*^{ζ}^{2}

· · ·*.* (53)

Taking into account (39) and recalling that (43) characterizes the*ζ** _{i}*, we see that if we
set

*ζ*_{i}^{}:=*λ*_{1}∗*ζ*_{i}*,* (54)

then

*σ*_{1}* ^{}*=

*e*^{ζ}^{1}^{}*e*^{ζ}^{2}^{}· · ·

· · ·*e*^{ζ}^{2}^{}*e*^{ζ}^{1}^{}

*,* (55)

so that*ζ*_{i}^{}=*ζ**i*.

* Lemma 4.3 For alln*≥1,

*σ*_{1}* ^{}*∗ ˜

*ζ*

*=0. (56)*

_{n}*Proof By definition,*

*E*^{↑}*(ζ )E*^{↓}*(ζ )*=*σ*_{1}* ^{}*=

*σ*

_{1}

*∗*

^{}*σ*1=

*σ*

_{1}

*∗*

^{}*ζ*˜

*E*

^{↓}

*(ζ )*

=*(λ*1*σ*1*)*∗*ζ*˜*E*^{↓}*(ζ )*

=*μ*

*(λ*_{1}⊗*σ*_{1}*)*∗*ζ*˜*E*^{↓}*(ζ )*⊗ ˜*ζE*^{↓}*(ζ )*

=

*λ*_{1}∗ ˜*ζE*^{↓}*(ζ )ζ*˜*E*^{↓}*(ζ )*

=

*λ*_{1}∗*E*^{↓}*(ζ )*

*(λ*_{1}∗ ˜*ζ )ζ*˜*E*^{↓}*(ζ )*=*E*^{↑}*(ζ )(λ*_{1}∗ ˜*ζ )(σ*_{1}∗ ˜*ζ )E*^{↓}*(ζ )*

=*E*^{↑}*(ζ )*
*σ*_{1}* ^{}*∗ ˜

*ζ*

*E*^{↓}*(ζ ),* (57)

so that*σ*_{1}* ^{}*∗ ˜

*ζ*=1.

* Lemma 4.4 For alln*≥1,

*σ*_{1}* ^{}*∗

*ζ*

*n*=2ζ

*n*

*.*(58)

*Proof We have*

*σ*_{1}* ^{}*∗

*ζ*

*=*

_{n}*(λ*

_{1}

*σ*

_{1}

*)*∗

*ζ*

*=*

_{n}*μ*

*(λ*_{1}⊗*σ*_{1}*)*∗*(ζ** _{n}*⊗1+1⊗

*ζ*

_{n}*)*

=2ζ*n**.* (59)
**Proposition 4.5 Let***I* = *(i*0*, . . . , i**p**)* *be a* *B-composition of* *n, and let* *λ* =
*(λ*_{0}*, . . . , λ*_{k}*)be aB-partition ofn.*

*S*˜* ^{I}*∗ ˜

*ζ*

*λ*0

*ζ*

*λ*1· · ·

*ζ*

*λ*

*=*

_{k}0 *ifλI*↓*,*

*(2*^{p}

*j*≥1*m**j*!*)˜ζ**i*_{0}*ζ**i*_{1}· · ·*ζ**i*_{p}*ifλ*=*I*↓*,* (60)
*wherem**j* *is the multiplicity ofj* *in(i*1*, i*2*, . . . , i**p**)(not countingi*0!).

*Proof The splitting formula yields*
*S*˜* ^{I}*∗ ˜

*ζ*

*λ*

_{0}

*ζ*

*λ*

_{1}· · ·

*ζ*

*λ*

*=*

_{k}*μ*

*p*

*S**i*_{0}⊗*S*^{}_{i}

1⊗ · · · ⊗*S*_{i}^{}

*p*

∗

*ζ*˜_{α}^{}_{0}*ζ*^{α}* ^{(0)}*⊗ · · · ⊗ ˜

*ζ*

_{α}

^{}

_{p}*ζ*

^{α}

^{(p)}=

*S*_{i}_{0}∗ ˜*ζ*_{α}^{}

0*ζ*^{α}^{(0)}*S*_{i}^{}

1∗ ˜*ζ*_{α}^{}

1*ζ*^{α}^{(1)}

· · ·
*S*_{i}^{}

*p*∗ ˜*ζ*_{α}^{}

*p**ζ*^{α}^{(p)}

*.* (61)
By (4) and Lemma4.3, a summand is zero if any*α*_{i}*>*0 for*i*≥1, so that

*S*˜* ^{I}*∗ ˜

*ζ*

*λ*

_{0}

*ζ*

*λ*

_{1}· · ·

*ζ*

*λ*

*=*

_{k}*S**i*_{0}∗ ˜*ζ*_{α}^{}_{0}*ζ*^{α}^{(0)}*S*_{i}^{}

1∗*ζ*^{α}^{(1)}

· · ·
*S*_{i}^{}

*r* ∗*ζ*^{α}^{(p)}

*.* (62)
If a term is nonzero in this equation, then*λI*↓. This proves the first case. By (4)
and Lemma4.4,*S*_{i}* ^{}*∗

*ζ*

*=2ζ*

_{i}*i*, 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*=

*(λ*0

*, λ*1

*, . . . , λ*

*k*

*)ofn, define the elementse*

*λ*∈

**BSym**

_{n}*recursively by the formula*

*e** _{λ}*= 1

2^{k}

*j**m**j*!*S*˜* ^{λ}*∗

*S** _{n}*−

*μ<λ*

*e*_{μ}*,* (63)

*wherem**j* *is the multiplicity ofj* *in(λ*1*, . . . , λ**k**)(not countingλ*0!*). Then*
*e** _{λ}*= 1

*j**m**j*!*ζ*˜_{λ}_{0}*ζ*_{λ}_{1}· · ·*ζ*_{λ}_{k}*.* (64)

*Proof Lete*^{}* _{λ}*be the right-hand side of the above equation. By Proposition4.5,

*S*˜

*∗*

^{λ}*e*

_{λ}^{}= ˜

*S*

*∗ 1*

^{λ}

*j**m** _{j}*!

*ζ*˜

_{λ}_{0}

*ζ*

_{λ}_{1}· · ·

*ζ*

_{λ}*=*

_{k}2^{k}

*j*

*m** _{j}*!

*e*

^{}

_{λ}*.*(65)

By (47),*S** _{n}*=

*e*^{}* _{λ}*, where the sum ranges over all

*B*-partitions

*λ*of

*n. 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 the*e*_{λ}^{} satisfy the same induction as the*e**λ*, 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**

Let*q* be a primitive*rth root of unity. We denote byθ**q* **the endomorphism of Sym**
defined by

*f*˜=*θ**q**(f )*=*f*

*(1*−*q)A*

=*f (A)*∗*σ*1

*(1*−*q)A*

*.* (67)

We denote by ˚*P** ^{(r)}*the image of

*θ*

*q*and by

*P*

*the right ˚*

^{(r)}*P*

*-module generated by*

^{(r)}*S*

*for*

_{n}*n*≥0. Note that ˚

*P*

*is by definition a left∗-ideal of Sym. For*

^{(r)}*r*=2, ˚

*P*

*is the classical peak ideal, and*

^{(2)}*P*

*is the unital peak algebra. For general*

^{(2)}*r, ˚P*

*is the higher-order peak algebra of [13], and*

^{(r)}*P*

*is its unital extension defined in [3].*

^{(r)}These objects depend only on*r*and not on the choice of the primitive root of unity.

Bases of*P** ^{(r)}*can be labeled by

*r-peak compositionsI*=

*(i*0;

*i*1

*, . . . , i*

*p*

*)*with at most one part

*i*

_{0}divisible by

*r.*

5.1 The radical

By definition,*P**n** ^{(r)}*is a∗-subalgebra of Sym

_{n}**. The radical of Sym**

*consists of those elements whose commutative image is zero (see [14], Lemma 3.10). The radical of*

_{n}*P*

*n*

*is therefore spanned by the*

^{(r)}*S*

_{i}_{0}·

*θ*

_{q}*(S*

*−*

^{I}*S*

^{I}^{}

*)*such that

*I*

^{}is a permutation of

*I*. Indeed, the quotient of

*P*

*n*

*by the span of those elements is a semisimple commuta- tive algebra, the∗*

^{(r)}**-subalgebra of Sym**

*spanned by the*

_{n}*p*

*λ*(λ

*n) such that at most*one part of

*λ*is multiple of

*r. This special part will be denoted byλ*

_{0}.

We denote by*P*_{n}* ^{(r)}*the subset of partitions of

*n*with at most one part divisible by

*r.*

The simple*P**n** ^{(r)}*-modules, and the principal idempotents, can therefore be labeled
by

*P*

_{n}*.*

^{(r)}5.2 An induction for the idempotents

Define a total order*<*on*P*_{n}* ^{(r)}*as follows: sort the partitions by decreasing length and
sort partitions of the same length by reverse lexicographic order. For example,

*P*_{5}* ^{(2)}*= [11111,2111,311,41,23,5]

*,*(68)

*P*

_{7}

*= [1111111,211111,31111,4111,2311,511,331,61,43,25,7]*

^{(2)}*.*(69) Now, set

*e*_{1}^{(r)}*n* := 1

*n*!*S*_{1}* ^{n}* (70)

and define by induction

*e*_{λ}* ^{(r)}*:= 1

*m*

*λ*

*T** ^{λ}*∗

*S** _{n}*−

*μ<λ*

*e*^{(r)}_{μ}*,* (71)

where*T** _{m}*=

*R*

*if*

_{m}*r*|

*m,T*

*=*

_{m}*R*

_{r}*i*

*j*if

*rm*and

*m*=

*ir*+

*j*with 0

*< j < r, andT*

*=*

^{λ}*T*

*λ*

_{0}

*T*

*λ*

_{1}· · ·

*T*

*λ*

*for*

_{p}*λ*=

*(λ*0;

*λ*1

*, . . . , λ*

*p*

*)*∈

*P*

_{n}*. It follows from [13, Corollary 3.17]*

^{(r)}and from the definition of*P** ^{(r)}*that

*T*

*∈*

^{λ}*P*

*. We want to prove that*

^{(r)}*(e*

^{(r)}

_{λ}*)*

*λ*∈*P*_{n}* ^{(r)}*is a complete system of orthogonal idempotents
for

*P*

*.*

^{(r)}To this aim, we introduce the sequence of (left) Zassenhaus idempotents*ζ**n** ^{(r)}*of
level

*r*as the unique solution of the equation

*σ*1=

*p*≥0

*ζ*_{pr}* ^{(r)}*
←

*i*≥1,r

*i*

*e*^{ζ}^{i}^{(r)}*.* (72)

Note that*ζ*_{n}* ^{(r)}*=

*ζ*

*n*for

*n <*2r.

For example, for*r*=2,

*ζ*_{1}* ^{(2)}*=

*S*

_{1};

*ζ*

_{2}

*=*

^{(2)}*S*

_{2}−1

2*S*^{11}; *ζ*_{3}* ^{(2)}*=

*S*

_{3}−

*S*

^{21}+1

3*S*^{111}*,* (73)
*ζ*_{4}* ^{(2)}*=

*S*

_{4}−

*S*

^{31}+1

2*S*^{211}−1

8*S*^{1111}*,* (74)

*ζ*_{5}* ^{(2)}*=

*S*

_{5}−

*S*

^{41}+1

2*S*^{311}−*S*^{23}+*S*^{221}−1

2*S*^{2111}+1
2*S*^{113}

−1

2*S*^{1121}+1

5*S*^{11111}; (75)

and for*r*=3,

*ζ*_{1}* ^{(3)}*=

*S*1;

*ζ*

_{2}

*=*

^{(3)}*S*2−1

2*S*^{11}; *ζ*_{3}* ^{(3)}*=

*S*3−

*S*

^{21}+1

3*S*^{111}*,* (76)

*ζ*_{4}* ^{(3)}*=

*S*4−

*S*

^{31}−1 2

*S*

^{22}+3

4*S*^{211}+1

4*S*^{112}−1

4*S*^{1111}*,* (77)

*ζ*_{5}* ^{(3)}*=

*S*5−

*S*

^{41}−

*S*

^{32}+

*S*

^{311}+

*S*

^{212}−2

3*S*^{2111}−1

3*S*^{1112}+1

5*S*^{11111}*,* (78)

*ζ*_{6}* ^{(3)}*=

*S*

_{6}−

*S*

^{51}−

*S*

^{42}+

*S*

^{411}+

*S*

^{312}−2

3*S*^{3111}+1

3*S*^{222}−1
6*S*^{2211}

−2

3*S*^{2112}+3

8*S*^{21111}−1

6*S*^{1122}+ 1

12*S*^{11211}+ 5

24*S*^{11112}−1

9*S*^{111111}*.* (79)
Define now for*λ*=*(λ*0;*λ*1*, . . . , λ*_{k}*)*∈*P*_{n}* ^{(r)}*,

*e*^{}^{(r)}* _{λ}* := 1

*m*

_{λ}*ζ*

_{λ}

^{(r)}0 *ζ*_{λ}^{(r)}

1 · · ·*ζ*_{λ}^{(r)}

*k* *.* (80)

We will show that*e*^{}^{(r)}* _{λ}* =

*e*

^{(r)}*for all*

_{λ}*λ*∈

*P*

_{n}*. We begin with two lemmas.*

^{(r)}**Lemma 5.1**

*Δ*
*ζ*_{n}^{(r)}

=

1⊗*ζ*_{n}* ^{(r)}*+

*ζ*

_{n}*⊗1*

^{(r)}*ifrn,*

*n/r*

*i*=0*ζ*_{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*≥0*ζ*_{rp}* ^{(r)}*is grouplike. If we define new elements

*Y*

*p*by

*σ*1=:

→
*p*

*e*^{Y}* ^{rp}*
←

*r**i*

*e*^{Y}^{i}*,* (82)

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

*p*≥0*ζ**rp** ^{(r)}*. Then

*ζ*

_{i}*=*

^{(r)}*Y*

*i*if

*ri. 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*≥0*ζ*_{rp}* ^{(r)}*is

grouplike.

* Lemma 5.2 Letλ*=

*(λ*0;

*λ*1

*, . . . , λ*

*k*

*)*∈

*P*

_{n}

^{(r)}*and*

*I*=

*(i*0

*, i*1

*, . . . , i*

*p*

*)be anr-peak*

*composition ofn. Then,*

*T** ^{I}*∗

*ζ*

_{λ}

^{(r)}0 *ζ*_{λ}^{(r)}

1 · · ·*ζ*_{λ}^{(r)}

*k* =

0 *ifI*↓*< λ,*

*m**I**ζ*_{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* for*λ*∈*P**n** ^{(r)}*. If

*F*

*=*

^{I}*F*_{i}_{1}· · ·*F*_{i}* _{p}* with each

*F*

_{i}*∈*

_{j}**Sym**

_{i}*, the splitting formula and (81) yield*

_{j}*F** ^{I}*∗

*ζ*

*=*

^{(r)λ}*λ** ^{(1)}*∨···∨

*λ*

*=*

^{(p)}*(λ*

_{1}

*,...,λ*

_{k}*)*

*ra*

_{1}+···+

*ra*

*=*

_{p}*λ*

_{0}

*p*
*j*=1

*F**i** _{j}*∗

*ζ*

_{ra}

^{(r)}*j**ζ*^{(r)λ}

*(j )*

*,* (84)

where*λ*0is the part (possibly 0) of*λ*that is divisible by*r, and whereα*∨*β*denotes
the partition obtained by reordering the concatenation of the partitions*α*and*β.*

Observe that since at most one*i**j* is divisible by*r*, a product in the above summa-
tion is 0 if at least two of the partitions*λ*^{(1)}*, . . . , λ** ^{(p)}*are empty. If

*(I ) > (λ), then*

*k*≤

*p*−2, so this hypothesis is always satisfied. Thus,

*F** ^{I}*∗

*ζ*

*=0 if*

^{(r)λ}*(I ) > (λ).*(85)

Suppose*I*↓ ≤*λ. Then, by the definition of the order,(I )*≥*(λ). Hence, forI*
such that*I*↓ ≤*λ*and*(I ) > (λ), the result follows by takingF** ^{I}*=

*T*

*in (85). So suppose instead that*

^{I}*(I )*=

*(λ). By definition,T*

*=*

^{I}*T*

*i*

_{1}· · ·

*T*

*i*

*, where*

_{p}*T*

*m*=

*R*

*m*if

*r*|

*m*and

*T*

*=*

_{m}*R*

_{r}*i*

*j*if

*m*=

*ir*+

*j*with 0

*< j < r. SinceR*

*can be written as a linear combination of*

_{J}*S*

*for which*

^{K}*J*is a refinement of

*K, it follows thatT*

*is equal to*

^{I}*S*

*plus a linear combination of*

^{I}*S*

*with*

^{K}*(K) > (I ). By takingF*

*=*

^{I}*S*

*in (85), it follows that*

^{K}*T*

*∗*

^{I}*ζ*

*=*

^{(r)λ}*S*

*∗*

^{I}*ζ*

*.*

^{(r)λ}It remains to show that, for*I* and*λ*of the same length,*S** ^{I}*∗

*ζ*

*=*

^{(r)λ}*m*

*I*

*ζ*

*if*

^{(r)λ}*I*↓ =

*λ*and is 0 otherwise. If

*λ*contains no part divisible by

*r, then it follows from*(84) that if

*S*

*∗*

^{I}*ζ*

*=0, we must have*

^{(r)λ}*I*↓ =

*λ, in which caseS*

*∗*

^{I}*ζ*

*=*

^{(r)λ}*m*

_{I}*ζ*

*. Suppose instead that*

^{(r)λ}*λ*contains a part that is divisible by

*r. Then each decomposi-*tion

*(λ*

^{(1)}*, . . . , λ*

^{(p)}*)*in (84) contains at least one

*λ*

*= ∅. Thus, if*

^{(j )}*I*contains no part that is divisible by

*r, thenS*

*∗*

^{I}*ζ*

*=0. Otherwise, the part of*

^{(r)λ}*I*that is divisible by

*r*is bounded by

*λ*

_{0}. This implies that

*λ*≤

*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*=

*(λ*

_{0};

*λ*

_{1}

*, . . . , λ*

_{k}*)*∈

*P*

_{n}*,*

^{(r)}*e*

^{(r)}*=*

_{λ}*e*

^{}

^{(r)}*:= 1*

_{λ}*m**λ*

*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*∈*P**n*^{(r)}*e*^{}^{(r)}* _{λ}* . Hence, by
Lemma5.2,

*m*_{λ}*e*^{}^{(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 elements*e*^{}^{(r)}* _{λ}* and

*e*

_{λ}*satisfy the same induction equation (71).*

^{(r)}