The canonical solutions of the Q-systems and the Kirillov-Reshetikhin conjecture

38 Pages·0.44 MB·English

by  Atsuo Kuniba

THE CANONICAL SOLUTIONS OF THE Q-SYSTEMS
AND THE KIRILLOV-RESHETIKHIN CONJECTURE
2
0
0
ATSUO KUNIBA, TOMOKI NAKANISHI, AND ZENGO TSUBOI
2

n
a Abstract. Westudyaclass ofsystems offunctional equationsclosely
J relatedtovariouskindsofintegrablestatisticalandquantummechanical
8 models. WecallthemthefiniteandinfiniteQ-systemsaccordingtothe
2 numberoffunctionsandequations. ThefiniteQ-systemsappearasthe
thermalequilibriumconditions(theSutherland-Wuequation)forcertain

] statistical mechanical systems. Some infinite Q-systems appear as the
A
relationsofthenormalizedcharactersoftheKRmodulesoftheYangians
Q and the quantum affine algebras. We give two types of power series
. formulae for the unique solution (resp. the unique canonical solution)
h
for a finite (resp. infinite) Q-system. As an application, we reformulate
t
a the Kirillov-Reshetikhin conjecture on the multiplicities formula of the
m KR modules in terms of thecanonical solutions of Q-systems.
[

2
v
1. Introduction
5
4
In the series of works [K1, K2, KR], Kirillov and Reshetikhin studied
1
5 the formal counting problem (the formal completeness) of the Bethe vec-
0 tors of the XXX-type integrable spin chains, and they empirically reached
1
a remarkable conjectural formula on the characters of a certain family of
0
finite-dimensional modules of the Yangian Y(g). Let us formulate it in the
/
h
following way.
t
a
m Conjecture 1.1. Let g be a complex simple Lie algebra of rank n. We
v: set y = (ya)na=1, ya = e−αa for the simple roots αa of g. Let Qm(a)(y) be
(a)
Xi the normalized g-character of the KR module Wm (u) (a = 1,...,n; m =
r 1,2,...; u ∈ C) of the Yangian Y(g); and Qν(y) := (a,m)(Qm(a)(y))νm(a).
a Then, the formula
Q
(a) (a)
P (ν,N)+N
(1.1) ν(y) (1 e−α) = m m (y )mNm(a),
Q − N(a) a
αY∈∆+ N=X(Nm(a))(aY,m)(cid:18) m (cid:19)
∞
m k
(1.2) P(a)(ν,N) = ν(a)min(k,m) N(b)d A min( , )
m k − k a ab d d
b a
k=1 (b,k)
X X
holds. Here, A = (A ) is the Cartan matrix of g, d are coprime positive
ab a
integers such that (d A ) is symmetric, ∆ is the set of all the positive
a ab +
roots of g, and a = Γ(a+1)/Γ(a b+1)Γ(b+1).
b −
(cid:0) (cid:1)
1
2 ATSUO KUNIBA, TOMOKI NAKANISHI, AND ZENGO TSUBOI
Remark 1.2. Due to the Weyl character formula, the series in the RHS of
(1.1) should be a polynomial of y, and its coefficients are identified with
the multiplicities of the g-irreducible components of the tensor product
(a,m)Wm(a)(um(a))⊗νm(a), where um(a) are arbitrary.
N
Remark 1.3. There are actually two versions of Conjecture 1.1. The above
one is the version in [HKOTY] which followed [K1, K2]. In the version in
[KR], the binomial coefficients a are set to be 0 if a< b; furthermore, the
b
equality is claimed, not for the entire series in the both hand sides of (1.1),
(cid:0) (cid:1)
but only for their coefficients of the powers yM “in the fundamental Weyl
chamber”; namely, M = (M )n satisfies
a a=1
n
(1.3) ν(a)mΛ M α P ,
m a− a a ∈ +
(a,m) a=1
X X
where Λ are the fundamental weights and P is the set of the dominant
a +
integral weights of g. So far, it is not proved that the two conjectures are
equivalent. The both conjectures are naturally translated into the ones for
the untwisted quantum affine algebras, which are extendable to the twisted
quantum affine algebras [HKOTT]. In this paper, we refer all these con-
jectures as the Kirillov-Reshetikhin conjecture. More comments and the
current status of the conjecture will be given in Section 5.7.
(a)
In[KR,K3],itwasclaimedthatthe (y)’ssatisfyasystemofequations
m
Q
( (a)(y))2 = (a) (y) (a) (y)
Qm Qm−1 Qm+1
(1.4) +(y )m( (a)(y))2 ( (b)(y))Gam,bk.
a Qm Qk
(b,k)
Y
(a)
Here, (y)= 1, and G are the integers defined as
Q0 am,bk
A (δ +2δ +δ ) d /d = 2
ba m,2k−1 m,2k m,2k+1 b a
−
A (δ +2δ +3δ d /d = 3
(1.5) G = − ba m,3k−2 m,3k−1 m,3k b a
am,bk
 +2δm,3k+1+δm,3k+2)

A δ otherwise.
− ab dam,dbk

See (4.22) for the original form of (1.4) in [KR, K3]. The relations (1.4)

and (4.22) are often called the Q-system. The importance of the role of the
Q-system to the formula (1.1) was recognized in [K1, K2, KR], and more
explicitly exhibited in [HKOTY, KN2]. In this paper we proceed one step
furtherin this direction; we studythe equation (1.4) in a moregeneral point
of view, and give a characterization of the special power series solution in
(1.1). For this purpose, we introduce finite and infinite Q-systems, where
the former (resp. the latter) is a finite (resp. infinite) system of equations
for a finite (resp. infinite) family of power series of the variable with finite
(resp. infinite) components. The equation (1.4), which is an infinite sys-
tem of equations with the variable with finite components, is regarded as
CANONICAL SOLUTIONS OF Q-SYSTEMS 3
an infinite Q-system with the specialization of the variable (a specialized
Q-system). We show that every finite Q-system has a uniquesolution which
has the same type of the power series formula as (1.1) (Theorem 2.4). In
contrast, infiniteQ-systemsandtheirspecializations, ingeneral, admitmore
than one solutions. However, every infinite Q-system, or its specialization,
has a unique canonical solution (Theorems 3.7 and 4.2), whose definition is
given in Definition 3.5. The formula (1.1) turns out to be exactly the power
series formula for the canonical solution of (1.4) (Theorem 4.3 and Propo-
sition 4.9). Therefore, one can rephrase Conjecture 1.1 in a more intrinsic
(a)
way as follows (Conjecture 5.5): The family ( (y)) of the normalized g-
m
Q
characters of the KR modules is characterized as the canonical solution of
(1.4). This is the main statement of the paper.
Interestingly, thefinite Q-systems also appear in other types of integrable
statistical mechanical systems. Namely, they appear as the thermal equilib-
rium condition (the Sutherland-Wu equation) for the Calogero-Sutherland
model [S], as well as the one for the ideal gas of the Haldane exclusion sta-
tistics [W]. The property of the solution of the finite Q-systems are studied
in [A, AI, IA] from the point of view of the quasi-hypergeometric functions.
We expect that the study of the Q-system and its variations and extensions
will be useful for the representation theory of the quantum groups, and for
the understanding of the nature of the integrable models as well.
WewouldliketothankV.Chari,G.Hatayama, A.N.Kirillov, M.Noumi,
M. Okado, T. Takagi, and Y. Yamada for very useful discussions. We espe-
cially thank K. Aomoto for the discussion where we recognize the very close
relation between the present work and his work, and also for pointing out
the reference [G] to us.
2. Finite Q-systems
A considerable part of the results in this section can befound in the work
by Aomoto and Iguchi [A, IA]. We present here a more direct approach.
More detailed remarks will be given in Section 2.4.
2.1. Finite Q-systems. Throughout Section 2, let H denote a finite index
set. Let w = (w ) and v = (v ) be complex multivariables, and let
i i∈H i i∈H
G = (G ) be a given complex square matrix of size H . We consider
ij i,j∈H
| |
a holomorphic map CH, v w(v) with
D → 7→
(2.1) w (v) = v (1 v )−Gij,
i i j
−
j∈H
Y
where is some neighborhood of v = 0 in CH. The Jacobian (∂w/∂v)(v) is
D
1 at v = 0, so that the map w(v) is bijective around v = w = 0. Let v(w) be
the inverse map around v = w = 0. Inverting (2.1), we obtain the following
4 ATSUO KUNIBA, TOMOKI NAKANISHI, AND ZENGO TSUBOI
functional equation for v (w)’s:
i
(2.2) v (w) = w (1 v (w))Gij.
i i j
−
j∈H
Y
By introducing new functions
(2.3) Q (w) = 1 v (w),
i i
−
the equation (2.2) is written as
(2.4) Q (w)+w (Q (w))Gij = 1.
i i j
j∈H
Y
From now on, we regard (2.4) as a system of equations for a family
( (w)) of power series of w = (w ) with the unit constant terms
i i∈H i i∈H
Q
(i.e., the constant terms are 1). Here, for any power series f(w) with the
unitconstanttermandanycomplexnumberα,wemeanby(f(w))α C[[w]]
∈
the αth power of f(w) with the unit constant term. We can easily reverse
the procedure from (2.1) to (2.4), and we have
Proposition 2.1. The power series expansion of Q (w) in (2.3) gives the
i
unique family (Q (w)) of power series of w with the unit constant terms
i i∈H
which satisfies (2.4).
Definition 2.2. The following system of equations for a family (Q (w))
i i∈H
ofpowerseriesofwwiththeunitconstanttermsiscalleda(finite)Q-system:
For each i H,
∈
(2.5) (Q (w))Dij +w (Q (w))Gij = 1,
j i j
j∈H j∈H
Y Y
where D = (D ) and G = (G ) are arbitrary complex matrices
ij i,j∈H ij i,j∈H
with detD = 0. The equation (2.4), which is the special case of (2.5) with
6
D = I (I: the identity matrix), is called a standard Q-system.
It is easy to see that there is a one-to-one correspondence between the
solutions of the Q-system (2.5) and the solutions of the standard Q-system
(2.6) Q′i(w)+wi (Q′j(w))G′ij = 1, G′ = GD−1,
j∈H
Y
where the correspondence is given by
(2.7) Q′(w) = (Q (w))Dij,
i j
j∈H
Y
(2.8) Q (w) = (Q′(w))(D−1)ij.
i j
j∈H
Y
Therefore, from Proposition 2.1, we immediately have
Theorem 2.3. There exists a unique solution of the Q-system (2.5), which
is given by (2.8), where (Q′(w)) is the unique solution of the standard
i i∈H
Q-system (2.6).
CANONICAL SOLUTIONS OF Q-SYSTEMS 5
2.2. Power series formulae. In what follows, we use the binomial coeffi-
cient in the following sense: For a C and b Z ,
≥0
∈ ∈
a Γ(a+1)
(2.9) = ,
b Γ(a b+1)Γ(b+1)
(cid:18) (cid:19) −
where the RHS means the limit value for the singularities. We set :=
(Z )H. For D, G in (2.5) and ν = (ν ) CH, we define two pNower
≥0 i i∈H
∈
series of w,
(2.10) Kν (w) = K(D,G;ν,N)wN, wN = wNi,
D,G i
N∈N i∈H
X Y
(2.11) Rν (w) = R(D,G;ν,N)wN
D,G
N∈N
X
with the coefficients
P +N
i i
(2.12) K(D,G;ν,N) = ,
N
i∈H(N)(cid:18) i (cid:19)
Y
1 P +N 1
i i
(2.13) R(D,G;ν,N) = det F − ,
ij
(cid:16)H(N) (cid:17)i∈YH(N) Ni(cid:18) Ni−1 (cid:19)
where we set H(N) = i H N = 0 for each N ,
i
{ ∈ | 6 } ∈N
(2.14) P = P (D,G;ν,N) := ν (D−1) N (GD−1) ,
i i j ji j ji
− −
j∈H j∈H
X X
(2.15) F = F (D,G;ν,N) := δ P +(GD−1) N ,
ij ij ij j ij j
and det is a shorthand notation for det . In (2.12) and (2.13),
H(N) i,j∈H(N)
det and mean 1; namely, Kν (w) and Rν (w) are power series with
∅ ∅ D,G D,G
the unit constant terms. It is easy to check that the both series converge for
w < γγQi/(γ +1)γi+1 , where γ = (GD−1) and zz = exp(zlogz) with
| i| | i i | i − ii
the principal branch π < Im(logz) π chosen.
− ≤
Now we state our main results in this section.
Theorem 2.4 (Power series formulae). Let (Q (w)) be the unique solu-
i i∈H
tion of (2.5). For ν CH, let Qν (w) := (Q (w))νi. Then,
∈ D,G i∈H i
(2.16) Qν (w) = Kν (w)/QK0 (w),
D,G D,G D,G
(2.17) Qν (w) = Rν (w).
D,G D,G
ThepowerseriesformulaeforQ (w)areobtainedasspecialcasesof(2.16)
i
and (2.17) by setting ν = (ν ) as ν = δ .
j j∈H j ij
One may recognize that the first formula (2.16) is analogous to the for-
mula (1.1), where the denominator K0 (w) in (2.16) corresponds to the
D,G
Weyl denominator in the LHS of (1.1). As mentioned in Section 1, the for-
mula(1.1) is interpreted as theformalcompleteness of the XXX-type Bethe
vectors. In the same sense, the second formula (2.17) is analogous to the
6 ATSUO KUNIBA, TOMOKI NAKANISHI, AND ZENGO TSUBOI
formal completeness of the XXZ-type Bethe vectors in [KN1, KN2]. See
Section 2.4 for more remarks.
Example 2.5. Let H = 1. Then, (2.5) is an equation for a single power
| |
series Q(w),
(2.18) (Q(w))D +w(Q(w))G = 1,
where D =0 and G are complex numbers, and the series (2.11) reads as
6
ν ∞ Γ((ν +NG)/D)( w)N
(2.19) Rν (w) = − .
D,G D Γ((ν +NG)/D N +1)N!
N=0 −
X
The equation (2.18) and the power series formula (2.19) are well known and
have a very long history since Lambert (e.g. [B, pp. 306–307]).
Example 2.6. Consider the case G = O in (2.5),
(2.20) (Q (w))Dij +w = 1.
j i
j∈H
Y
This is easily solved as
(2.21) Q (w) = (1 w )(D−1)ij,
i j
−
j∈H
Y
and, therefore,
(2.22) QνD,O(w) = (1−wi) j∈Hνj(D−1)ji = (1−wi)−Pi(D,O;ν,N),
i∈H P i∈H
Y Y
where N is arbitrary. Using the binomial theorem
∈ N
∞
β+N
(2.23) (1 x)−β−1 = xN,
− N
N=0(cid:18) (cid:19)
X
one can directly check that
P 1+N
(2.24) Qν (w) = i− i wNi = Rν (w),
D,O N i D,O
N∈N i∈H(N)(cid:18) i (cid:19)
X Y
(2.25) Qν (w) = j∈H(1−wi) j∈Hνj(D−1)ji−1 = KDν,O(w).
D,O (1P w )−1 K0 (w)
Q j∈H − i D,O
2.3. Proof of Theorem 2.4 Qand basic formulae. Theorem 2.4 is re-
gardedasaparticularlyniceexampleofthemultivariableLagrangeinversion
formula (e.g. [G]) where all the explicit calculations can be carried through.
Here, we present the most direct calculation based on the multivariable
residue formula (the Jacobi formula in [G, Theorem 3]).
We first remark that
CANONICAL SOLUTIONS OF Q-SYSTEMS 7
Lemma 2.7. Let G′ = GD−1. For each ν CH, let ν′ CH with ν′ =
∈ ∈ i
ν (D−1) . Then,
j∈H j ji
(P2.26) QνD,G(w) = QνI,′G′(w),
(2.27) Kν (w) = Kν′ (w), Rν (w) = Rν′ (w).
D,G I,G′ D,G I,G′
Proof. The equality (2.26) is due to Theorem 2.3. The ones (2.27) follow
from the fact P (D,G;ν,N) = P (I,G′;ν′,N).
i i
By Lemma 2.7, we have only to prove Theorem 2.4 for the standard case
D = I. Recall that (Proposition 2.1) Qν (w) = (1 v (w))νi, where
I,G i∈H − i
v = v(w) is the inverse map of (2.1). Thus, Theorem 2.4 follows from
Q
Proposition 2.8 (Basic formulae). Let v = v(w) be the inverse map of
(2.1). Then, the power series expansions
w ∂v
(2.28) det j i (w) (1 v (w))νi−1 = Kν (w),
H vi ∂wj − i I,G
(cid:16) (cid:17)iY∈H
(2.29) (1 v (w))νi = Rν (w)
− i I,G
i∈H
Y
hold around w = 0.
Proof. The first formula (2.28). We evaluate the coefficient for wN in the
LHS of (2.28) as follows:
∂v
Res (w) (1 v (w))νi−1(v (w))−1(w )1−Ni−1 dw
i i i
w=0∂w −
iY∈Hn o
= Res (1 v )νi−1(v )−1 v (1 v )−Gij −Ni dv
i i i j
v=0 − −
iY∈Hn (cid:16) jY∈H (cid:17) o
= Res (1 v )−Pi(I,G;ν,N)−1(v )−Ni−1 dv
i i
v=0 −
iY∈Hn o
P (I,G;ν,N)+N
i i
= = K(I,G;ν,N),
N
i∈H(cid:18) i (cid:19)
Y
where we used (2.23) to get the last line. Thus, (2.28) is proved.
The second formula (2.29). By a simple calculation, we have
v ∂w
j i
det (v) (1 v )= det δ +( δ +G )v
i ij ij ij i
H wi ∂vj − H −
(2.30) (cid:16) (cid:17)iY∈H (cid:16) (cid:17)
= d v ,
J i
J⊂H i∈J
X Y
where d := det ( δ +G ), and the sum is taken over all the subsets J
J J ij ij
−
of H. Therefore, the LHS of (2.29) is written as
w ∂v
(2.31) det j i (w) d (1 v (w))νi−1v (w)θ(i∈J) .
J i i
H vi ∂wj −
(cid:16) (cid:17)JX⊂H iY∈Hn o
8 ATSUO KUNIBA, TOMOKI NAKANISHI, AND ZENGO TSUBOI
By a similar residue calculation as above, the coefficient for wN of (2.31) is
evaluated as (θ(true)= 1 and θ(false) = 0)
d Res (1 v )−Pi(I,G;ν,N)−1(v )−Ni+θ(i∈J)−1 dv
J i i
v=0 −
JX⊂H iY∈Hn o
P (I,G;ν,N)+N θ(i J)
i i
= d − ∈
J
N θ(i J)
J⊂H(N) i∈H(N)(cid:18) i− ∈ (cid:19)
X Y
1 P +N 1
i i
= d N (P +N ) −
J i i i
N N 1
(cid:18)J⊂H(N) i∈J i∈H(N)\J (cid:19)i∈H(N) i(cid:18) i− (cid:19)
X Y Y Y
1 P +N 1
i i
= det δ (P +N )+( δ +G )N −
ij j j ij ij j
H(N)(cid:16) − (cid:17)i∈YH(N) Ni(cid:18) Ni−1 (cid:19)
=R(I,G;ν,N).
Thus, (2.29) is proved.
This completes the proof of Theorem 2.4.
Example 2.9. We say that the map w(v) in (2.1) is lower-triangular if the
matrix G is strictly lower-triangular with respect to a certain total order
ij
in H (i.e., G = 0 for i j). Let w(v) be a lower-triangular map. Then,
ij
≺ (cid:22)
v ∂w G v
j i ij j
(2.32) det (v) = det δ + = 1.
ij
H wi ∂vj H 1 vj
(cid:16) (cid:17) (cid:16) − (cid:17)
Thus, the formula (2.28) is simplified as
(2.33) (1 v (w))νi−1 = Kν (w).
− i I,G
i∈H
Y
This type of formulae has appeared in [K1, K2, HKOTY].
Letusisolatethecaseν = 0from(2.28), together withtheformula(2.30),
for the later use:
Corollary 2.10 (Denominator formulae).
w ∂v
(2.34) K0 (w) = det j i (w) (1 v (w))−1,
I,G H vi ∂wj − i
(cid:16) (cid:17)iY∈H
−1
(2.35) K0 (w) = det δ (1 v (w))+G v (w) .
I,G H ij − i ij i
n (cid:16) (cid:17)o
From (2.35) and the first formula of Theorem 2.4, we obtain
Corollary 2.11.
(2.36) Qν (w) = g Kν+δJ(w),
I,G J I,G
J⊂H
X
JJ
(2.37) g := sgn det δ G det G ,
J J′⊂H (cid:18)J′J′(cid:19)i∈J,j∈J′ ij − ij i∈J,j∈J′ ij
|JX′|=|J| (cid:0) (cid:1)
CANONICAL SOLUTIONS OF Q-SYSTEMS 9
where δ = (θ ) , θ = 1 if i J and 0 otherwise, and J = H J.
J i i∈H i
∈ \
From Corollary 2.11, one can easily reproducethe second formula of The-
orem 2.4. We leave it as an exercise for the reader.
2.4. Remarks onrelatedworks. i)The formal completeness of the Bethe
vectors. In [K1, K2, HKOTY, KN1, KN2, KNT], the formal completeness
of the XXX/XXZ-type Bethe vectors are studied. In the course of their
analysis, several power series formulae in this section appeared in special-
ized/implicit forms. For example, Lemma 1 in [K1] is a special case of
(2.33), Theorem 4.7 in [KN2] is a special case of Proposition 2.8, etc. From
the current point of view, however, the relation between these power series
formulae and the underlying finite Q-systems was not clearly recognized
therein. As a result, these power series formulae and the infinite Q-systems
were somewhat abruptly combined in the limiting procedure to obtain the
power series formula for the infinite Q-systems. We are going to straighten
out this logical entangle, and make the logical structure more transparent
by Theorem 2.4 and the forthcoming Theorems 3.10, 4.3, Proposition 4.9,
and Conjecture 5.5.
ii) The ideal gas with the Haldane statistics and the Sutherland-Wu equa-
tion. The series Kν (w) has an interpretation of the grand partition func-
D,G
tion of the ideal gas with the Haldane exclusion statistics [W]. The finite
Q-systemappearedin[W]asthethermalequilibriumconditionforthedistri-
butionfunctionsofthesamesystem. Seealso[IA]foranotherinterpretation.
The one variable case (2.18) also appeared in [S] as the thermal equilibrium
conditionforthedistributionfunctionoftheCalogero-Sutherlandmodel. As
an application of our second formula in Theorem 2.4, we can quickly repro-
duce the “cluster expansion formula” in [I, Eq. (129)], which was originally
calculated by the Lagrange inversion formula, as follows:
(2.38)
∂
logQ (w) = Rν (w)
i ∂νi I,G ν=0
h i
1 P (I,G;0,N)+N 1
= det F (I,G;0,N) j j − wN,
jk
NX∈N Hj,k(N6=i) j∈YH(N) Nj(cid:18) Nj −1 (cid:19)
where Q (w) isthesolutionof(2.4). TheSutherland-Wuequationalso
i i∈H
{ }
playsan importantrolefortheconformalfieldtheoryspectra. (See[BS]and
the references therein.)
iii) Quasi-hypergeometric functions. The series Kν (w) is a special ex-
D,G
ample of the quasi-hypergeometric functions by Aomoto and Iguchi [AI];
when G′ are all integers, it reduces to a general hypergeometric function of
ij
Barnes-Mellin type. A quasi-hypergeometric function satisfies a system of
fractional differential equations and a system of difference-differential equa-
tions [AI]. It also admits an integral representation [A]. In particular, the
integralrepresentation forKν (w)reducestoasimpleform([A,Eq.(2.30)],
I,G
10 ATSUO KUNIBA, TOMOKI NAKANISHI, AND ZENGO TSUBOI
[IA, Eq. (89)]); in our notation,
1
(2.39) Kν (w) = tνi−1f (w,t)−1 dt,
I,G (2π√ 1)|H| i i
− Z niY∈H o
(2.40) f (w,t) := t 1+w tGij,
i i− i j
j∈H
Y
where the integration is along a circle around t = 1 starting from t = 0
i i
for each t . We see that f (w,t) = 0 is the standard Q-system (2.4). The
i i
integral (2.39) is easily evaluated by the Cauchy theorem as [A, eq. (2.32)]
(2.41) Kν (w) = Qν (w)/det(δ Q (w)+G (1 Q (w))),
I,G I,G H ij i ij − j
where Q (w) is the solution of (2.4). The formula (2.41) reproduces a
i i∈H
{ }
version of the Lagrange inversion formula (the Good formula [G, Theorem
2]), and it is equivalent to the formulae (2.16), (2.30), and (2.34).
3. Infinite Q-systems
3.1. Infinite Q-systems. Throughout Section 3, let H be a countable
infinite index set. We fix an increasing sequence of finite subsets of H,
H H H such that limH = H. The result below does not de-
1 2 L
pen⊂donth⊂e·ch··o⊂ice ofthesequen−c→e H ∞ . Anaturalchoice isH = Nand
{ L}L=1
H = 1,...,L . However, we introduce this generality to accommodate
L
{ }
the situation we encounter in Section 4 (cf. (4.1)).
Let w = (w ) be a multivariable with infinitely many components.
i i∈H
For each L N, let w = (w ) be the submultivariable of w. The field
C[[w ]] of th∈e power sLeries ofiwi∈HoLver C is equipped with the standard X -
L L L
adic topology, where X is the ideal of C[[w ]] generated by w ’s (i H ).
L L i L
For L < L′, there is a natural projection pLL′ : C[[wL′]] C[[wL]] suc∈h that
→
pLL′(wi) = wi if i HL and 0 if i HL′ HL. A power series f(w) of w
is an element of th∈e projective limi∈t C[[w]\] = limC[[w ]] of the projective
L
system ←−
(3.1) C[[w ]] C[[w ]] C[[w ]]
1 2 3
← ← ← ···
with the induced topology. Let p bethe canonical projection p :C[[w]]
L L
C[[w ]], and f (w ) be the Lth projection image of f(w) C[[w]]; name→ly,
L L L
∈
f (w )= p (f(w)) and f(w)= (f (w ))∞ .
L L L L L L=1
Here are some basic properties of power series which we use below:
(i) We also present a power series f(w) as a formal sum
(3.2) f(w) = a wN, a C,
N N
∈
N∈N
X
(3.3) = N = (N ) N Z , all but finitely many N are zero ,
i i∈H i ≥0 i
N { | ∈ }