Lie idempotents in descent algebras [expository notes]

15 Pages·2007·0.189 MB·English

by  Jean-Yves Thibon

LIE IDEMPOTENTS IN DESCENT ALGEBRAS
JEAN-YVES THIBON
Abstract. Lie idempotents are symmetrizers which project the tensor algebra
onto the free Lie algebra. Almost all known examples turn out to belong to the
descent algebras of symmetric groups. This makes it possible to analyze them
in terms of noncommutative symmetric functions. By extending various classical
techniques of the theory of ordinary symmetric functions, it is then possible to
produce many new examples, and in some sense, to classify all the possibilities.
1. Introduction
Let V be a vector space over some field K of characteristic 0. Let T(V) be its
tensor algebra, andL(V) the free Liealgebra generated by V. We denote by L (V) =
n
L(V)∩V⊗n its homogeneous component of degree n. The group algebra KS acts
n
on the right on V⊗n by
(1) (v ⊗v ⊗···⊗v )·σ = v ⊗v ⊗···⊗v .
1 2 n σ(1) σ(2) σ(n)
This action commutes with the left action of GL(V), and when dimV ≥ n, which
we shall usually assume, these actions are the commutant of each other (Schur-Weyl
duality).
Any GL(V)-equivariant projector Π : V⊗n → L (V) can therefore be regarded as
n n
an idempotent π of KS : Π (v) = v·π . By definition (cf. [25]), such an element is
n n n n
called a Lie idempotent whenever its image is L (V). Then, a homogeneous element
n
P ∈ V⊗n is in L (V) if and only if P π = P .
n n n n n
From now on, we fix a basis A = {a ,a ,...} of V. We identify T(V) with the free
1 2
associative algebra KhAi, and L(V) with the free Lie algebra L(A).
2. The Hausdorff series
Lie idempotents arise naturally in the investigation of the Hausdorff series
(2) H(a ,a ,...,a ) = log(ea1ea2 ···eaN) = H (A)
1 2 N n
n≥0
X
which is known to be a Lie series, i.e., each homogeneous component H (A) ∈
n
L (A). This is known as the Baker-Campbell-Hausdorff (BCH) “formula”. It follows
n
immediately from the characterization of L(A) as the space of primitive elements of
the standard comultiplication of KhAi (Friedrich’s criterion1).
1Thiscriterion,whichappearsonlyasafootnote[11,p. 203]inatextonquantumfieldtheory,is
for us, togetherwith its converse(the Milnor-Mooretheorem, due to Cartier)the “founding act”of
combinatorialHopf algebratheory, whichhas actually little to do with Hopf’s originalmotivations.
1
2 J.-Y. THIBON
Actually, the BCH formula is not a formula at all, only a (very important) prop-
erty, which is the basis of the correspondence between Lie groups and Lie algebras.
However, various applications (including real-world ones) require explicit calculation
of the polynomials H (A). That is where Lie idempotents come into play.
n
Forsmall values of nandN, thiscan bedone by hand, andoneobtains forexample
1 1 1 1 1
H (a ,a ,a ) = a a a + a a a − a a a + a a a + a a a
3 1 2 3 1 1 2 1 1 3 1 2 1 1 2 2 1 2 3
12 12 6 12 3
1 1 1 1
− a a a + a a a + a a a − a a a
1 3 1 1 3 3 2 1 1 2 1 2
6 12 12 6
1 1 1 1 1
− a a a + a a a + a a a − a a a − a a a
2 1 3 2 2 1 2 2 3 2 3 1 2 3 2
6 12 12 6 6
1 1 1 1 1
+ a a a + a a a − a a a − a a a + a a a
2 3 3 3 1 1 3 1 2 3 1 3 3 2 1
12 12 6 6 3
1 1 1 1
+ a a a − a a a + a a a + a a a
3 2 2 3 2 3 3 3 1 3 3 2
12 6 12 12
Already, it might not be obvious, at first sight, that this is indeed a Lie polynomial,
which can be rewritten in the form
1 1 1
H (a ,a ,a ) = [a ,[a ,a ]]+ [[a ,a ],a ]+ [a ,[a ,a ]]
3 1 2 3 1 1 2 1 2 2 1 1 3
12 12 12
1 1 1
+ [[a ,a ],a ]+ [a ,[a ,a ]]+ [[a ,a ],a ]
1 3 3 2 2 3 2 3 3
12 12 12
1 1
+ [a ,[a ,a ]]+ [[a ,a ],a ].
1 2 3 1 2 3
6 6
TheproblemoffindingasystematicprocedureforexpressingtheHausdorffseriesas
a linear combination of commutators was raised at the Gelfand seminar in the 1940’s,
and Dynkin [9] came up with the following solution (also discovered independently
by Specht [29] and Wever [30]):
Theorem 2.1.
1
(3) θ = [...[[[1,2],3],...,],n]
n
n
is a Lie idempotent. Therefore, expanding H as a linear combination of words, and
n
writing H = H θ , gives the required expression.
n n n
In order to apply this recipe, we need a reasonably efficient way to find the expan-
sion on words
(4) H (A) = c w.
n w
w∈An
X
One can show that H (A) is the image of the homogeneous component E (A) of the
n n
product of exponentials
(5) E(A) = ea1ea2 ···eaN = E (A).
n
n≥0
X
LIE IDEMPOTENTS 3
under an element of KS
n
(6) H (A) = E (A)·φ ,
n n n
where
1 (−1)d(σ)
(7) φ = σ (d(σ) = |Des(σ)|).
n n n−1
σX∈Sn d(σ)
Here, Des(σ) = {i|σ(i) > σ(i +(cid:0)1)} (cid:1)denotes the descent set of σ. This formula is
due to Solomon [26] and independently to Bialynicki-Birula, Mielnik and Pleban´ski
[2]. It can be shown, although no one of these properties is clearly apparent on the
expression (7), that φ is a Lie idempotent. It has been rediscovered many times,
n
and is also known as the (first) Eulerian idempotent.
We can now write the Hausdorff polynomials
(8) H (A) = E (A)φ θ = c ·wθ
n n n n w n
w∈An
X
as linear combinations of commutators wθ . However, these commutators are far
n
from being linearly independent, and one would be interested in an expansion of H
n
on a basis of the free Lie algebra.
One way to achieve this is to use Klyachko’s basis of the free Lie algebra. This
little known basis is obtained froma third Lie idempotent, discovered by A. Klyachko
[15], and originally introduced as the solution of a different problem.
This problem was the following. The character of GL(V) on L (V) is known (Witt
n
has given formulas for the dimensions of its weight spaces), and its expression shows
that as a GL(V)-module, L (V) is isomorphic to the space Γ (V), image of the
n n
idempotent
n−1
1
(9) γ = ωkck,
n
n
k=0
X
wherec = (12...n)isann-cycleofS andω aprimitiventhrootofunity. Klyachko’s
n
idempotent κ is an intertwiner between these two isomorphic representations. This
n
means that for any word w ∈ An, (i) wκ = 0 is w is not primitive (i.e., w = vd for
n
some non-trivial divisor d of n), and (ii) if w is primitive, wcκ = ωwκ .
n n
Hence, applying κ to some set of representatives of circular classes of primitive
n
words, for example to Lyndon words (words which are lexicographically minimal
among their circular shifts), we obtain a basis of L (A).
n
There is a closed formula:
1
(10) κ = ωmaj(σ)σ
n
n
σX∈Sn
where ω = e2iπ/n and maj(σ) = j is the major index of σ.
j∈Des(σ)
Now, applying κ to the Lyndon words appearing in the expansion of E (A) ob-
n n
P
tained fromthe Eulerian idempotent gives an expansion ofH (A) on a basis of L (A),
n n
which can be further expanded in terms of commutators by means of Dynkin’s idem-
potent if needed.
4 J.-Y. THIBON
At this point, we are facing three idempotents, which, admittedly, are given by
rather different formulas.
It may therefore come as a surprise that the element
(11) ϕn(q) = 1 (−1)d(σ) qmaj(σ)−(d(σ2)+1)σ
n n−1
σX∈Sn d(σ)
(cid:20) (cid:21)q
is a Lie idempotent, interpolating between our three examples.
Apart from the from the fact that ϕ (1) = φ , none of these properties is evident.
n n
Nevertheless, one can show that ϕ (0) = θ and ϕ (ω) = κ .
n n n n
The explanation of this strange fact starts with the observation that the three
idempotents do have something in common: they all belong to the descent algebra of
the symmetric group.
3. Solomon’s descent algebra
The descent algebras have been introduced by Solomon [27] for general finite Cox-
eter groups in the following way. Let (W,S) be a Coxeter system. One says that
w ∈ W has a descent at s ∈ S if w has a reduced word ending by s. For W = S
n
and s = (i,i+1), this means that w(i) > w(i+1), whence the terminology. In this
i
case, we rather say that i is a descent of w. Let as above Des(w) denote the descent
set of w, and for a subset E ⊆ S, set
(12) D = w ∈ ZW .
E
Des(w)=E
X
Solomon has shown that the D span a Z-subalgebra of ZW. Moreover
E
(13) DE′DE′′ = cEE′E′′DE
E
X
where the coefficients cE are nonnegative integers.
E′E′′
In the case of S , it is convenient to encode descent sets by compositions of n.
n
If E = {d ,...,d }, we set d = 0, d = n and I = C(E) = (i ,...,i ), where
1 r−1 0 r 1 r
i = d − d . We also say that E is the descent set of I. From now on, we shall
k k k−1
write D instead of D .
I E
It is clear from the defintions that φ and κ are in the descent algebra of S . For
n n n
θ , this is also quite easy to see:
n
2
2θ = 1 2 − 1
2
3
2 3 2
3θ3 = 1 2 3 − 1 3 + 1 2 + 1
 
 
 
LIE IDEMPOTENTS 5
4
3 4 4 3
2 3 4  2 2 3  2
4θ = 1 2 3 4 − 1 3 4 + 1 2 4 + 1 2 3 − 1 4 + 1 3 + 1 2 − 1
4  
 
D4     D1111
   
   
| {z }     |{z}
D13  
 
| {z } D112
That is, | {z }
n−1
(14) nθ = (−1)kD .
n 1kn−k
k=0
X
4. Noncommutative symmetric functions
The algebra of ordinary symmetric functions Sym can be regarded as the free as-
sociative and commutative algebra over an infinite sequence (h ) of homogeneous
n n≥1
generators (h is of degree n), so that its linear bases in degree n are naturally la-
n
belled by partitions of n (e.g., products h = h ···h of complete homogeneous
µ µ1 µr
functions).
Similarly, the algebra Sym of noncommutative symmetric functions is the free
associative (but noncommutative) algebra over an infinite sequence (S ) of homo-
n n≥1
geneous generators, endowed with a natural homomorphism S 7→ h (commutative
n n
image).
Thus, linear bases of the homogeneous component Sym of Sym are labelled by
n
compositions of n, exactly as those of the descent algebra Σ of S . It is then
n n
tempting to define a correspondence between them. There is a natural way to do
this. Noncommutative symmetric functions can be realized in terms of an auxiliary
(ordered) alphabet A = {a ,a ,...} by setting
1 2
(15) S (A) = a a ···a ,
n i1 i2 in
i1≤iX2≤...≤in
that is, the sum of nondecreasing words, or, otherwise said, words with no descent.
Then, obviously,
(16) SI = S S ···S
i1 i2 ir
is the sum of words whose descent set is contained in Des(I) (the descents of a word
are defined as for permutations as those i such that w > w , and are similarly
i i+1
encoded as compositions C(w) of n). Introducing the noncommutative ribbon Schur
functions
(17) R (A) = w
I
C(w)=I
X
6 J.-Y. THIBON
whose commutative image are indeed the skew Schur functions indexed by ribbon
diagrams, we have
(18) SI = R
J
J≤I
X
where J ≤ I is the reverse refinement order, which means that Des(J) ⊆ Des(I).
The map α : D → R appears therefore as a natural choice for a correspondence
I I
Σ → Sym .
n n
This choice is not only natural, it is canonical. Indeed, Solomon had proved that
the structure constants of his descent algebra were the same as the decompositon
coefficients of certain tensor products of representations of S . Precisely, if we set
n
(19) BI = D ∈ Σ
I n
J≤I
X
and
(20) BIBJ = bIJBK ,
K
K
X
then, the Kronecker products of the characters βI of S , induced by the trivial
n
representations of the parabolic subgroups S ×···×S , decompose as
i1 ir
(21) βIβJ = bIJβK .
K
K
X
Such products of induced characters can be calculated by Mackey’s formula, whence
the title of Solomon’s paper [27].
So, why is the above correspondence canonical? This is because it is compatible
with the Frobenius characteristic map, from S -characters to symmetric functions.
n
Indeed, ch(βI) = h , the commutative image of SI.
I
The Frobenius characteristic map allows one to define the internal product ∗ on
symmetric functions, by setting h ∗ h = ch(βλβµ). We can now do the same
λ µ
on noncommutative symmetric functions, using the descent algebras instead of the
character rings.
For technical reasons, we want our correspondence to be an anti-isomorphism. We
set
(22) SI ∗SJ = bJISK .
K
K
X
This is because we want to interpret permutations as endomorphisms of tensor alge-
bras: if f (w) = wσ, then f ◦f = f .
σ σ τ τσ
5. Lie idempotents as noncommutative symmetric functions
The first really interesting question about noncommutative symmetric functions is
perhaps “what are the noncommutative power sums?”. Indeed, the answer to this
question is far from being unique.
LIE IDEMPOTENTS 7
If one starts from the classical expression
tk
(23) σ (X) = h (X)tn = exp p ,
t n k
k
( )
n≥0 k≥1
X X
one can choose to define noncommutative power sums Φ by the same formula
k
tk
(24) σ (A) = S (A)tn = exp Φ ,
t n k
k
( )
n≥0 k≥1
X X
but a noncommutative version of the Newton formulas
(25) nh = h p +h p +···+p
n n−1 1 n−2 2 n
which are derived by taking the logarithmic derivative of (23) leads to different non-
commutative power-sums Ψ inductively defined by
k
(26) nS = S Ψ +S Ψ +···+Ψ .
n n−1 1 n−2 2 n
A bit of computation reveals then that
n−1
(27) Ψ = R −R +R −··· = (−1)kR ,
n n 1,n−1 1,1,n−2 1k,n−k
k=0
X
which is analogous to the classical expression of p as the alternating sum of hook
n
Schur functions. Therefore, in the descent algebra, Ψ correponds to Dynkin’s ele-
n
ment, nθ .
n
The Φ can also be expressed on the ribbon basis without much difficulty, and one
n
finds
(−1)l(I)−1
(28) Φ = R
n n−1 I
|I|=n l(I)−1
X
(cid:0) (cid:1)
so that Φ corresponds to nφ .
n n
The case of Klyachko’s idempotent is even more interesting, but to explain it, we
shall need the (1−q)-tranform, to be defined below.
6. The (1−q)-transform
In its classical (commutative) version, the (1−q)-transform ϑ is the algebra en-
q
domorphism of Sym defined on the power sums by ϑ (p ) = (1 − qn)p . In λ-ring
q n n
notation, which is particularly convenient for dealing with such transformations, it
reads f(X) 7→ f((1−q)X). One has to pay attention to the abuse of notation in
using the same minus sign for the λ-ring and for scalars, though these operations are
quite different. That is, ϑ maps p to 2p if n is odd, and to 0 otherwise. Thus,
−1 n n
ϑ (f(X)) = f((1−q)X) is not the same as f((1+1)X) = f(2X).
−1 q=−1
In [17], a consistent definition of ϑ (F) = F((1−q)A) has been introduced as
q
follows. One first defines the complete symmetric functions S ((1 − q)A) via their
n
8 J.-Y. THIBON
generating series [17, Def. 5.1]
(29) σ ((1−q)A) := tnS ((1−q)A) = σ (A)−1σ (A),
t n −qt t
n≥0
X
and then ϑ is defined as the ring homomorphism such that
q
(30) ϑ (S ) = S ((1−q)A).
q n n
It can then be shown [17, Thm. 4.17] that
(31) F((1−q)A) = F(A)∗σ ((1−q)A).
1
For generic q, ϑ is an automorphism, and its inverse is the 1/(1−q)-transform
q
←
A
(32) σt = σtqn(A).
1−q
(cid:18) (cid:19) n≥0
Y
Computing the image of S , one arrives at
n
A
(33) K (q) = (q) S = qmaj(I)R (A).
n n n I
1−q
(cid:18) (cid:19) |I|=n
X
Hence, Klyachko’s element nκ = K(ω) is the specialization of this expression at
n
q = ω. This is puzzling: the commutative image of (33) is a Hall-Littlewood function
(Q˜′ , precisely), andthespecializationofsuch functions atrootsofunity areknown.
(1n)
In this case, one gets the power sum p .
n
7. Hopf algebras enter the scene
At this point, we cansee that thecommutative imagesof ourthree Lieidempotents
are the same: 1p . The symmetric functions 1p have two significant properties: (i)
n n n n
they are idempotent for the internal product, and (ii) they are primitive elements for
the coproduct of Sym.
Identifying f ⊗g with f(X)g(Y), the coproduct of Sym can be defined by ∆f =
f(X +Y), which gives for complete functions
n
(34) ∆h = h ⊗h ,
n i n−i
i=0
X
and similarly, Sym can be endowed with a bialgebra structure, defined by
n
(35) ∆S = S ⊗S
n i n−i
i=0
X
(with the convention S = 1).
0
One may therefore suspect that our Lie idempotents might be primitive elements
of Sym. That this is true can be verified directly for each of them, but we have the
following much stronger result [17]. Let us say that an element e of an algebra is
quasi-idempotent if e2 = c·e for some (possibly 0) scalar c.
LIE IDEMPOTENTS 9
Theorem 7.1. Let F = α(π) be an element of Sym , where π ∈ Σ .
n n
(i) The following assertions are equivalent:
1. π is a Lie quasi-idempotent;
2. F is a primitive element for ∆;
3. F belongs to the Lie algebra L(Ψ) generated by the Ψ .
n
(ii) Moreover, π is a Lie idempotent iff F − 1Ψ is in the Lie ideal [L(Ψ), L(Ψ)].
n n
Thus, Lie idempotents are essentially the same thing as “noncommutative power
sums” (up to a factor n), and we shall from now on identify both notions: a Lie
idempotent in Sym is a primitive element whose commutative image is p /n.
n n
8. A one parameter family of Lie idempotents
Theorem 7.1 suggests a recipe for constructing new examples. Start from a known
family, e.g., Dynkin elements, and take its image by a bialgebra automorphism, e.g.,
ϑ−1. The result is then automatically a sequence of Lie idempotents. In the case
q
under consideration, we get
(36) ϕn(q) = 1−qnΨn A = 1 (−1)ℓ(I)−1 qmaj(I)−(ℓ(2I)) RI(A).
n 1−q n n−1
(cid:18) (cid:19) |I|=n
X ℓ(I)−1
(cid:20) (cid:21)q
The obtention of the closed form in the r.h.s. requires a fair amount of calculation,
but the fact that ϕ (q) is a Lie idempotent is automatic. This being granted, it is
n
not difficult to show that
1 1 1
(37) ϕ (0) = Ψ , ϕ (ω) = K (ω) , ϕ (1) = Φ .
n n n n n n
n n n
Other Hopf automorphisms, like the noncommutative analogs of the transformation
1−t
(38) f(X) −→ f X
1−q
(cid:18) (cid:19)
used in the theory of Macdonald polynomials, lead to other families of Lie idem-
potents. It is not always possible, however, to obtain such a clean closed form for
them.
9. The iterated q-bracketing and its diagonalization
There is another one-parameter family of Lie idempotents, for which no closed
expression is known, but which is of fundamental importance.
The reproducing kernel of ϑ , S ((1−q)A) is easily seen to be the image under α
q n
of the iterated q-bracketing operator
(39) S ((1−q)A) = (1−q)α([[···[1,2] ,3] ,...,n] ) ,
n q q q
a natural q-analog of Dynkin’s idempotent. For generic q, this is not an idempotent
at all, but an automorphism.
10 J.-Y. THIBON
The most important property of ϑ is its diagonalization [17, Thm. 5.14]: there is
q
a unique family of Lie idempotents π (q) with the property
n
(40) ϑ (π (q)) = (1−qn)π (q).
q n n
Moreover, ϑ is semi-simple, and its eigenvalues in Sym , the nth homogeneous
q n
component of Sym, are p (1−q) = (1−qλi) where λ runs over the partitions of
λ i
n. The projectors on the corresponding eigenspaces are the maps F 7→ F ∗πI(q) [17,
Q
Sec. 3.4].
Here are the first values of π (q):
n
Ψ Ψ 1 1−q
π (q)=Ψ , π (q)= 2 , π (q)= 3 + [Ψ ,Ψ ] ,
1 1 2 2 3 3 6 (1+2q) 2 1
Ψ 1 (1−q)(2q+1) 1 (1−q)2
π (q)= 4 + [Ψ ,Ψ ]+ [[Ψ ,Ψ ],Ψ ] ,
4 4 12 (1+q+2q2) 3 1 24 (1+q+2q2) 2 1 1
Ψ 1 (1−q)(3q2+2q+1) 1 (1−q)(q+2)
π (q)= 5 + [Ψ ,Ψ ]+ [Ψ ,Ψ ]
5 5 20 (2q3+q2+q+1) 4 1 30 (2q2+2q+1) 3 2
1 (1−q)2(4q3+7q2+7q+2)
+ [[Ψ ,Ψ ],Ψ ]
60 (2q2+q+2)(2q3+q2+q+1) 3 1 1
1 (1−q)2(4q2+9q+7)
− [[Ψ ,Ψ ],Ψ ]
120 (q2+3q+1)(2q2+1+2q) 1 2 2
1 (1−q)3(2q5+2q4+q3+5q2+9q+6)
+ [[[Ψ ,Ψ ],Ψ ],Ψ ] .
120 (2q3−q2+q+3)(2q2+q+2)(2q3+q2+q+1) 2 1 1 1
The idempotents π (q) have interesting specializations. The easiest one is q = 1:
n
Ψ
n
(41) π (1) = .
n
n
This has the strange consequence that, for any Lie idempotent F ∈ Sym ,
n n
F ((1−q)A) Ψ
n n
(42) lim = .
q→1 1−qn n
Next, we have, for ω a primitive n-th root of unity,
(43) π (ω) = κ .
n n
Again, a curious consequence is that
A
(44) lim (1−qn)F ( ) = κ ,
n n
q→ω 1−q
for any Lie idempotent F ∈ Sym .
n n
To describe the next specialization, we need to introduce a new family of non-
commutative power sums. The noncommutative power sums of the third kind Z are
n
defined by
Z Z
(45) σ (A) = exp(Z t) exp( 2 t2) ... exp( n tn) ...
t 1
2 n
The Fer-Zassenhauss formula (cf. [31]) shows that every Z is a Lie element. It is
n
also clear that the commutative image of Z is p .
n n