On the Expansion Coefficients of Tau-function of the BKP Hierarchy

0.21 MB·

by  Yoko Shigyo

On the Expansion Coefficients of Tau-function of the
BKP Hierarchy
6 Yoko Shigyo∗
1
0 Department of Mathematics, Tsuda College,
2
Kodaira, Tokyo, 187-8577, Japan
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n Abstract
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2 We study the series expansion of the tau function of the BKP hierarchy applying
v theadditionformulaeoftheBKPhierarchy. Anyformalpowerseriescanbeexpanded
3
in terms of Schur functions. It is known that, under the condition τ(x) 6= 0, a formal
8
0 powerseriesτ(x)isasolutionoftheKPhierarchyifandonlyifitscoefficients ofSchur
2
function expansion are given by the so called Giambelli type formula. A similar result
0
is known for the BKP hierarchy with respect to Schur’s Q-function expansion under
.
1 a similar condition. In this paper we generalize this result to the case of τ(0) = 0.
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6
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: 1 Introduction
v
i
X
The tau function of the KP hierarchy can be expanded in terms of Schur functions χ (x),
r µ
a x = (x ,x ,···) as
1 2
τ(x) = ξ χ (x), (1)
µ µ
µ
X
where µ runs over all partitions. If τ(0) = 1, ξ for an arbitrary partition µ is written in
µ
terms of ξ corresponding to the hook diagram (i|j) as
(i|j)
ξ = det(ξ ) , (2)
(i1···,ik|j1,···,jk) (ir|js) 1≤r,s≤k
where (i ··· ,i |j ,··· ,j ) is the Frobenius’ notation for a partition [7]. This formula is
1 k 1 k
calledtheGiambelli formula since it canbeconsidered astheinterpretation oftheGiambelli
formula for Schur function [7] to ξ . It is known that, under the condition τ(0) = 1, (1) is a
µ
∗
e-mail: [email protected]
1
solution of the KP hierarchy if and only if (2) holds for any partition (i ··· ,i |j ,··· ,j ).
1 k 1 k
This follows from Sato’s theory of the KP hierarchy [17, 4, 16, 11].
A formal power series τ(x), x = (x ,x ,···) can be expanded in terms of Schur’s Q-
1 3
function as
x
τ(x) = ξ Q , (3)
µ µ
2
Xµ (cid:16) (cid:17)
where µ runs over all strict partitions. In this case, if τ(0) = 1, then ξ for a strict partition
µ
µ = (µ ,··· ,µ ) satisfy
1 2l
ξ = Pf(ξ ), (4)
(µ1,···,µ2l) (µi,µj)
where Pf (a ) denotes the Pfaffian of a skew symmetric matrix (a ) (see (21) for more
ij ij
precise definition and notation). Similarly to the KP case, under the condition τ(0) = 1,
(3) is a solution of the BKP hierarchy if and only if (4) holds for any strict partition µ.
This is proved in [3] using the free fermion construction of the tau function.
There are many solutions for which τ(0) = 0. To the best of the author’s knowledge
the formula corresponding to (2) or (4) in this case is not known in general. In this paper
we have found a formula of the form (4) in the case τ(0) = 0, which we call the Giambelli
type formula, and have proved that (3) is a solution to the BKP hierarchy if and only if
the Giambelli type formulae hold. It seems that the KP case is more involved and is now
under investigation.
Let us describe our results more precisely. For a strict partition λ = (λ ,··· ,λ ) of
1 M
length M we consider a function of the form
x x
τ(x) = Q ( )+ ξ Q ( ). (5)
λ µ µ
2 2
|µ|>|λ|
X
Inordertogiveaformulaforξ weintroducenotationforPfaffian. ForsymbolsX ,··· ,X ,
µ 1 N
N ∈ N with X = Λ,Λ(j)(1 ≤ j ≤ M),n, n ∈ N, we denote by (X ,X )(= −(X ,X )) the
i i j j i
(i,j) component of a skew symmetric matrix and by (X ,··· ,X ) the Pfaffian of the
i1 i2m
matrix ((X ,X )) (See Example 1). We define
ir is r,s
(Λ(i),n) = ξ ,
(λ1,···,λˆi,···,λM,n)
(Λ,n) = ξ ,
(λ1,···,λM,n)
(n ,n ) = ξ ,
i j (λ1,···,λM,ni,nj)
(Λ,Λ(i)) = (Λ(i),Λ(j)) = 0.
Then we prove that (5) is a solution to the BKP hierarchy if and only if ξ for µ =
µ
(µ ,··· ,µ ) being a partition of length l is given by the following formula:
1 l
ξ = (Λi,Λ(1),··· ,Λ(M),µ ,··· ,µ ), (6)
µ 1 l
where i = 0,1 is chosen in such a way that i+M +l is even and Λ0 means that Λ is not
inserted there. In the case of λ = ∅ this formula recovers the result of the case of τ(0) = 1.
This result is proved using addition formulae for the tau function of the BKP hierarchy.
In our previous paper [18] we have studied the addition formulae of the BKP hierarchy and
2
proved that the simplest case of the addition formula is equivalent to the BKP hierarchy
itself. In this sense the addition formulae have enough information on the BKP hierarchy.
We derive the relations among {ξ } by expanding addition formula. By solving these
µ
relations we prove (6). It should be mentioned that it is difficult to derive (6) from the
DJKM equation (20), which is used to derive (4).
Recently Giambelli and Jacobi-Trudi type formulaefor the expansion coefficients attract
much attention in relation to the study of 2 dimensional solvable lattice models [1, 12, 6]
and higher genus theta functions [4, 8]. It is interesting to study relations of our results to
such subjects.
Finally, it is known that totality of solutions to the BKP hierarchy is parametrized
by the infinite dimensional orthogonal Grassmann manifold [2, 3]. It should be clarified
whether our result can be proved from this point of view. It is also interesting to study the
relation with cluster algebras [15].
This paper consists of two sections and two appendixes. In section 2, we consider the
addition formulae in the BKP hierarchy. We first review the BKP hierarchy and introduce
Schur’s Q-function. Then we study the expansions of the addition formulae in [18]. In
section 3, we state and prove the main result of this paper. In appendix A, necessary facts
onfree fermion andthe boson-fermion correspondence are reviewed. We prove that a formal
power series of variable x = (x ,x ,···) can be expanded in terms of Schur’s Q-function in
1 3
appendix B.
2 The BKP hierarchy and its addition formulae
2.1 Addition formulae of the BKP hierarchy
In this section, we review the BKP hierarchy and its addition formulae.
Set
α3 α5 ∞
[α] = (α, , ,···), ξ˜(x,k) = x k2n−1, x = (x ,x ,x ,···), y = (y ,y ,y ,···).
o 2n−1 1 3 5 1 3 5
3 5
n=1
X
The BKP hierarchy [2] is a system of non-linear equations for τ(x) given by
dk
e−2ξ˜(y,k)τ(x−y −2[k−1] )τ(x+y +2[k−1] ) = τ(x−y)τ(x+y),
o o
2πik
I
where the integral means taking the coefficient of k−1 in the expansion of the integrand in
n
the series of k. Let α ,··· ,α be parameters. Set y = [α ] and compute the integral
1 n l=1 l
by taking residues, then we have the addition formulae for τ(x) of the BKP hierarchy [18].
P
There are two kinds of addition formulae according as n is odd or even as follows. If n is
odd and greater than or equal to 3, then we have
n
A τ(x)τ(x+2 [α ] )
1...n l o
l=1
X
n n
= (−1)i−1τ(x+2[α ] )A τ(x+2 [α ] ). (7)
i o 1···ˆi···n l o
i=1 l=1,l6=i
X X
3
If n is even and greater than or equal to 4, then we have
n
A τ(x)τ(x+2 [α ] )
1...n l o
l=1
X
n−1 n−1
α
= (−1)i−1 i,nτ(x+2[α ] +2[α ] )A τ(x+2 [α ] ). (8)
α˜ i o n o 1···ˆi···n−1 l o
i,n
i=1 l=1,l6=i
X X
Here A is defined by
1...n
n
α
ij
A = , α˜ = α +α , α = α −α ,
1...n ij i j ij i j
α˜
ij
i<j
Y
andˆi means that i is removed.
Remark. Notice that the coefficient A in this paper is defined by an inverse of A
1...n 1...n
in [18].
The following theorem has been proved in [18].
Theorem 1 The addition formulae (7) and (8) are equivalent to the BKP hierarchy re-
spectively.
Remark. In [18] we had proved that the case of n = 3 of (7) is equivalent to the BKP
hierarchy instead of the whole addition formulae.
2.2 Addition formulae in terms of the expansion coefficients
First we introduce notation for partitions [7]. A partition λ = (λ ,··· ,λ ) is a non-
1 n
increasing sequence of non-negative integers. The non-zero λ are called parts of λ. We
i
define the length l(λ) of λ as the number of parts of λ and the weight |λ| of λ as the sum
of parts of λ. We say λ = (λ ,··· ,λ ) is a strict partition if λ > ··· > λ ≥ 0. (This
1 2l 1 2n
definition of a strict partition is different from that in [7].) We identify (λ ,··· ,λ ,0)
1 2l−1
and (λ ,··· ,λ ). If l = 0 then λ is considered as ∅.
1 2l−1
To study the series expansion ofthe taufunction of the BKPhierarchy we first introduce
Schur’s Q-function. Schur’s Q- function is defined for strict partitions.
For a non-negative integer r we define the symmetric polynomial q of α = (α ,··· ,α )
r 1 N
by
N
1+tα
q tr = i.
r
1−tα
i
r≥0 i=1
X Y
For r > s ≥ 0, we set
s
Qsym(α) = q q +2 (−1)iq q .
(r,s) r s r+i s−i
i=1
X
4
If r < s, we define Qsym(α) as
(r,s)
sym sym
Q (α) = −Q (α).
(r,s) (s,r)
For any strict partitions λ = (λ ,··· ,λ ), Schur’s Q-function is defined by
1 l
sym sym
Q (α) = Pf Q (α) ,
λ (λi,λj) 1≤i,j≤2n
(cid:16) (cid:17)
where Pf(a ) denotes the Pfaffian of A= (a ) (see (21) for the definition of
ij 1≤i,j≤2n ij 1≤i,j≤2n
Pfaffian in detail). We set x = (αi +···+αi )/i, N ≥ |λ|. It is known that Qsym(α) can
i 1 N λ
uniquely be expressed as a polynomial of x = (x ,x ,x ,···). We denote this polynomial
1 3 5
by Q (x). Then we have the relation
λ
αi +···+αi
Qsym(α) = Q (x), x = 1 N.
λ λ i i
We extend the definition of Q for an arbitrary permutation λ of a strict partition by skew
λ
symmetry. It is possible to expand any formal power series τ(x) of x = (x ,x ,x ,···) as
1 3 5
follows [19] (see appendix B):
x
τ(x) = ξ Q ( ), (9)
µ µ
2
µ
X
∂
ξ = 2−l(µ)Q (∂˜)τ(x) , ∂˜= (∂˜ ,∂˜ ,···), ∂˜ = xi (10)
µ µ 1 3 i
x=0 i
(cid:12)
where µ runs over all strict partitions. For(cid:12)any permutation µ of a strict partition we define
(cid:12)
ξ by Equation (10). Then we have
µ
ξ = sgnσξ ,
(µσ(1),···,µσ(2l)) (µ1,···,µ2l)
for a permutation σ of degree 2l. In general we define ξ , m ≥ 0, ♯{i|m = 0} ≤ 1,
(m1,···,mn) i i
such that it is skew symmetric in the indices. It means, in particular, ξ = 0 if
(m1,···,mn)
m = m for some i 6= j.
i j
In order to study the expansion of the addition formulae the following proposition plays
a key role.
Proposition 1 Let α ,··· ,α be parameters which satisfy |α | > ··· > |α |. Then
1 2n 1 2n
2n
A τ(x+2 [α ] ) = ξ˜ (x)αm1···αm2n. (11)
1···2n i o (m1···,m2n) 1 2n
Xi=1 mi∈XZ,i6=2n,
m2n≥0
˜
If m ≥ 0 for any i and the number of i with m = 0 is at most one, ξ (x) is skew
i i (m1,···,m2n)
symmetric in the indices. If (m ,··· ,m ) is a permutation of a strict partition λ, then
1 2n
ξ˜ (0) = 2l(λ)ξ .
(m1···,m2n) (m1,···,m2n)
Proof. We use free fermions [2, 3] (see appendix A for notation) to prove this proposition.
Let us consider the vertex operator
XB(α) = e n:oddxnαne−2 n:odd∂˜nα−n.
P P
5
The vertex operators satisfy, for |α| > |β|,
1−β/α
X (α)X (β) = e xnαn+ xnβne−2 ∂˜nα−n−2 ∂˜nβ−n. (12)
B B
1+β/α P P P P
We apply this vertex operator to 1 = h0|eHB(x)|0i. Using(12) we get
XB(α1)···XB(α2n)·1 = A1···2ne 2i=n1 k:oddxkαki. (13)
P P
By the boson-fermion correspondence this is equal to
X (α )···X (α )h0|eHB(x)|0i = 2nh0|eHB(x)φ(α )···φ(α )|0i
B 1 B 2n 1 2n
= 2nh0|eHB(x)φ ···φ |0iαm1···αm2n (14)
m1 m2n 1 2n
mXi∈Z
= η (x)αm1 ···αm2n, (15)
(m1,···,m2n) 1 2n
mXi∈Z
where
η (x) = 2nh0|eHB(x)φ ···φ |0i.
(m1,···,m2n) m1 m2n
Therefore
2n
A1···2nτ(x+2 [αi]o) = A1···2ne 2i=n1 k:odd2∂˜kαkiτ(x)
P P
i=1
X
= ξ˜ (x)αm1 ···αm2n,
(m1,···,m2n) 1 2n
mXi∈Z
where
˜ ˜
ξ (x) = η (2∂)τ(x).
(m1,···,m2n) (m1,···,m2n)
If m > ··· > m ≥ 0, it is known that (see [14, 2, 19]):
1 2n
x
h0|eHB(x)φ ···φ |0i = 2−nQ , λ = (m ,··· ,m ).
m1 m2n λ 2 1 2n
(cid:16) (cid:17)
In this case we have, by (10),
ξ˜ (0) = 22nξ , if m 6= 0,
(m1,···,m2n) (m1,···,m2n) 2n (16)
(ξ˜(m1,···,m2n)(0) = 22n−1ξ(m1,···,m2n), if m2n = 0.
By the commutation relations of {φ } if there are no pairs (i,j) such that m + m = 0,
n i j
˜ ˜
then ξ (x) is skew symmetric in the indices. In particular ξ (x) is skew
(m1,···,m2n) (m1,···,m2n)
symmetric if m ≥ 0 for any i and the number of i with m = 0 is at most one. Therefore
i i
similar equations to (16) arevalidfor anarbitrarypermutationof (m ,··· ,m ), m > ··· >
1 k 1
m ≥ 0. Finally in the sum of (15) m ≥ 0 since φ |0i = 0 for n < 0.✷
2n 2n n
Remark. Since m ≥ 0, in the sum in the right hand side of (11) we can put α = 0.
2n 2n
Then a similar expansion is valid for A τ(x + 2n−1[α ] ) and ξ˜ (0) =
1···2n−1 i=1 i o (m1,···,m2n−1)
22n−1ξ for m ≥ 1, 1 ≤ i ≤ 2n−1.
(m1,···,m2n−1) i P
Expanding addition formulae using Proposition 1 we have
6
Proposition 2 Suppose that τ(x) given by (9) is a solution of the BKP hierarchy. Let
(n ,··· ,n ) and (m ,··· ,m ) be permutations of some strict partition of length l and k
1 l 1 k
respectively. Then we have the following equations.
(i) For any k,l ≥ 1 such that k +l ≥ 3 is odd we have
l
ξ ξ = (−1)i−1ξ ξ
(n1,···,nl) (m1,···,mk) (ni,m1,···,mk) (n1,···,nˆi,···,nl)
i=1
X
k
+ (−1)l+i−1ξ ξ . (17)
(m1,···,mˆi,···,mk) (n1,···,nl,mi)
i=1
X
(ii) For any k,l ≥ 1 such that k +l ≥ 4 is even we have
l−1
ξ ξ = (−1)k+i−1ξ ξ
(n1,···,nl) (m1,···,mk) (ni,nl,m1,···,mk) (n1,···,nˆi,···,nl−1)
i=1
X
k
+ (−1)i−1ξ ξ . (18)
(nl,m1,···,mˆi,···,mk) (n1,···,nl−1,mi)
i=1
X
Proof. We prove (ii). Equation (i) can be proved in a similar way.
Assume that τ(x) is a solution of the BKP hierarchy. We set, in (8), n = k +l and
α = −β ,··· ,α = −β .
l 1 n−1 k
k
Inthefollowing wewriteα byα . Then shift xtox+2 [β ] . Multiplying theresulting
n l j=1 j o
equation by
P
l k
α −β
(−1)k s r,
α +β
s r
s=1r=1
YY
we have
k l
A B τ(x+2 [β ] )τ(x+2 [α ] )
1···l 1···k j o j o
j=1 j=1
X X
l−1 k
α α −β β −α
= (−1)i−1A B i,l i r · r l
1···ˆi···l−1 1···kα˜ α +β β +α
i=1 i,l r=1(cid:18) i r r l(cid:19)
X Y
k l−1
×τ(x+2[α ] +2 [β ] +2[α ] )τ(x+2 [α ] )
i o j o l o j o
j=1 j6=i
X X
k l−1 k
α −β β −α
+ (−1)l+iA B s i r l
1···l−1 1···ˆi···k α +β β +α
s i r l
i=1 s=1 r=1,r6=i
X Y Y
k l−1
×τ(x+2 [β ] +2[α ] )τ(x+2 [α ] +2[β ] ), (19)
j o l o j o i o
j6=i j=1
X X
7
where
k
β −β
i j
B = .
1···k
β +β
i j
i<j
Y
We assume |α | > ··· > |α | > |β | > ··· > |β | > |α |. Set x = 0 and expand (19) using
1 l−1 1 k l
(11), then we get
ξ˜ (0)ξ˜ (0)αn1···αnlβm1···βmk
(n1,···,nl) (m1,···,mk) 1 l 1 k
nsX,mr∈Z
l−1
= (−1)k+i−1 ξ˜ (0)ξ˜ (0)αn1···αnlβm1 ···βmk
(n1,···,nˆi,···,nl−1) (ni,nl,m1,···,mk) 1 l 1 k
Xi=1 nsX,mr∈Z
k
+ (−1)i−1 ξ˜ (0)ξ˜ (0)αn1···αnlβm1 ···βmk.
(n1,···,nl−1,mi) (nl,m1,···,mˆi,···,mk) 1 l 1 k
Xi=1 nsX,mr∈Z
Compare the coefficients of αn1···αnlβm1···βmk, m ,n ≥ 0 and rewrite them in terms of
1 l 1 k i j
ξ using (16). Then some powers of 2 appear in both sides, however they are canceled. Thus
✷
we obtain (18).
The following proposition had been proved in [3].
Proposition 3 [3] A series τ(x) of the form (9) is a solution of the BKP hierarchy if and
only if the coefficients {ξ } satisfy the following equations:
λ
ξ ξ = ξ ξ
(m1,···,mk) (m1,···,mk,n1,n2,n3,n4) (m1,···,mk,n1,n4) (m1,···,mk,n2,n3)
−ξ ξ
(m1,···,mk,n2,n4) (m1,···,mk,n1,n3)
+ξ ξ , (20)
(m1,···,mk,n3,n4) (m1,···,mk,n1,n2)
where k is even and (m ,··· ,m ) and (m ,··· ,m ,n ,n ,n ,n ) are permutations of some
1 k 1 k 1 2 3 4
strict partitions λ and µ respectively.
Corollary 1 Suppose that the coefficients {ξ } satisfy (17) and (18), then τ(x) is a solution
µ
of the BKP hierarchy.
Proof. If m 6= 0, n 6= 0 for any i,j, set l = k + 4 and n = m ,··· ,n = m in (18).
i j 1 1 k k
Then the second summation of (18) becomes zero and three terms of the first summation
✷
are left. Thus we obtain (20). Other cases are similarly proved using (18) or (17).
Therefore the set of equations (17) and (18) are also equivalent to the BKP hierarchy.
3 Expansion coefficients of τ(x)
In this section we study the expansion coefficients of τ(x) in detail.
We first introduce notation and some properties on Pfaffians. Let A= (a ) be a
ij 1≤i,j≤2m
skew-symmetric matrix. Then the Pfaffian Pf(a ) [5] is defined by
ij
Pf(a ) = sgn(i ,...,i )·a a ···a , (21)
ij 1 2m i1,i2 i3,i4 i2m−1,i2m
X
8
where the sum is over all permutations of (1,...,2m) such that
i < i < ··· < i , i < i ,··· ,i < i ,
1 3 2m−1 1 2 2m−1 2m
and sgn(i ,...,i ) is the signature of the permutation (i ,...,i ). In order to describe
1 2m 1 2m
Pf(a ) more conveniently we use some set of symbols X , 1 ≤ i ≤ 2m (see Theorem 3).
ij i
First we set (X ,X ) = a . For any permutation i ,··· ,i of 1,··· ,2m we define
i j ij 1 2m
(X ,··· ,X ) = Pf((X ,X )) .
i1 i2m ik il 1≤k,l≤2m
Then it is skew symmetric in the indices. The Pfaffian has the following expansion:
2m
(X ,··· ,X ) = (−1)j(X ,X )(X ,...,Xˆ ,...,X ).
1 2m 1 j 2 j 2m
j=2
X
For example, in the case of m = 2,
(X ,X ,X ,X ) = (X ,X )(X ,X )−(X ,X )(X ,X )+(X ,X )(X ,X ).
1 2 3 4 1 2 3 4 1 3 2 4 1 4 2 3
A Pfaffian analogue of Plu¨cker relations is known [13]:
R
(−1)r(X ,··· ,X ,X )(X ,··· ,Xˆ ,··· ,X )
i1 iS jr j1 jr jR
r=1
X
S
+ (−1)s(X ··· ,Xˆ ,··· ,X )(X ,··· ,X ,X ) = 0, (22)
i1 is iS j1 jR is
s=1
X
where R and S are odd.
In [3] the following theorem is proved by solving Equations (20).
Theorem 2 [3] A formal power series τ(x) given by (9) satisfying the condition τ(0) = 1
is a solution of the BKP hierarchy if and only if the coefficients {ξ } satisfy
µ
ξ = Pf ξ . (23)
µ (µi,µj) 1≤i,j≤2n
(cid:0) (cid:1)
We study the case of τ(0) = 0. In this case it seems that it is difficult to derive the
formula corresponding to (23) by using Equations (20) only. Our strategy is to use larger
set of Equations (17) and (18). Let λ = (λ ,··· ,λ ) be a strict partition of length M. We
1 M
assume that τ(x) has the following expansion:
x x
τ(x) = Q ( )+ ξ Q ( ). (24)
λ µ µ
2 2
|µ|>|λ|
X
We set ξ = 0 if |µ| ≤ |λ| and µ 6= λ. We consider the following subset of the non-trivial
µ
expansion coefficients in (24),
ξ , n > λ , n 6= λ for any j,
(λ1,···,λˆi,···,λM,n) i j
ξ , n ≥ 1,n 6= λ for any i, (25)
 (λ1,···,λM,n) i
ξ , n > n ≥ 1,n ,n 6= λ for any i.
(λ1,···,λM,ni,nj) i j i j i


9
We define the components of Pfaffian as
(Λ(i),n) = ξ ,
(λ1,···,λˆi,···,λM,n)
(Λ,n) = ξ , n ≥ 1,
(λ1,···,λM,n)
(n ,n ) = ξ , n > n ≥ 1,
i j (λ1,···,λM,ni,nj) i j
(Λ,Λ(i)) = (Λ(i),Λ(j)) = 0.
Example 1 (i)Pfaffian (Λ,Λ(1),n ,n ) is expanded as
1 2
(Λ,Λ(1),n ,n ) = (Λ,Λ(1))(n ,n )−(Λ,n )(Λ(1),n )+(Λ,n )(Λ(1),n )
1 2 1 2 1 2 2 1
= −ξ ξ +ξ ξ .
(λ1,···,λM,n1) (λ2,···,λM,n2) (λ1,···,λM,n2) (λ2,···,λM,n1)
(ii)Pfaffian (Λ(1),n ,n ,n ) is expanded as
1 2 3
(Λ(1),n ,n ,n ) = (Λ(1),n )(n ,n )−(Λ(1),n )(n ,n )+(Λ(1),n )(n ,n )
1 2 3 1 2 3 2 1 3 3 1 2
= ξ ξ −ξ ξ
(λ2,···,λM,n1) (λ1,···,λM,n2,n3) (λ2,···,λM,n2) (λ1,···,λM,n1,n3)
+ξ ξ .
(λ2,···,λM,n3) (λ1,···,λM,n1,n2)
Notice that (Λ(i),n) = 0 for n < λ , since λ +n < |λ|.
i j6=i j
Then our main theorem is
P
Theorem 3 Suppose that τ(x) has the expansion (24). Then τ(x) is a solution of the
BKP hierarchy if and only if the coefficients ξ , µ = (µ ,··· ,µ ), l(µ) = k are given by the
µ 1 k
following formulae where the quantities in (25) are arbitrary.
(i) M = 2L−1,
(Λ(1),··· ,Λ(2L−1),µ ,··· ,µ ), if k = 2l−1,
1 2l−1
ξ = (26)
µ ((Λ,Λ(1),··· ,Λ(2L−1),µ1,··· ,µ2l), if k = 2l.
(ii) M = 2L,
(Λ,Λ(1),··· ,Λ(2L),µ ,··· ,µ ), if k = 2l−1,
1 2l−1
ξ = (27)
µ ((Λ(1),··· ,Λ(2L),µ1,··· ,µ2l), if k = 2l.
We first remark that (26) and (27) are trivial equations for quantities in (25).
Whether the quantities {ξ } given by (26) and (27) satisfy ξ = 0 for |µ| ≤ |λ|, µ 6= λ
µ µ
is not very obvious. So let us prove the following lemma.
Lemma 1 The quantities {ξ } given by (26) and (27) satisfy ξ = 0 if |µ| ≤ |λ| and µ 6= λ.
µ µ
Proof. We prove the lemma in the case of the length of λ and µ are odd, that is the first
case of (26). Set λ = (λ ,··· ,λ ) and µ = (µ ,··· ,µ ). If l < L, then
1 2L−1 1 2l−1
ξ = (Λ(1),··· ,Λ(2L−1),µ ,··· ,µ )
(µ1,···,µ2l−1) 1 2l−1
= sgn(i ,··· ,i )(Λ(1),µ )···(Λ(2l−1),µ )(Λ(2l),··· ,Λ(2L−1)).
1 2l−1 i1 i2l−1
X
10