Basic Representations of A_{2l}^(2) and D_{l+1}^(2) and the Polynomial Solutions to the Reduced BKP Hierarchies

11 Pages·0.11 MB·English

by  Tatsuhiro Nakajima

TMUP-HEL-9403
January, 1994
(2) (2)
Basic Representations of A and D and
2l l+1
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the Polynomial Solutions to the Reduced BKP Hierarchies
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1 Tatsuhiro NAKAJIMA

1 Department of Physics, Tokyo Metropolitan University
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7 Minami-Ohsawa 1-1, Hachioji, Tokyo 192-03, Japan
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and
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Hirofumi YAMADA
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Department of Mathematics, Tokyo Metropolitan University
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-t Minami-Ohsawa 1-1, Hachioji, Tokyo 192-03, Japan
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r Abstract
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(2) (2)
Basic representations of A and D are studied. The weight vectors are represented
2l l+1
in terms of Schur’s Q-functions. The method to get the polynomial solutions to the
reduced BKP hierarchies is shown to be equivalent to a certain rule in Maya game.
§1 Introduction
Nonlinear integrable systems are important in mathematical physics. Among others
theKP hierarchy has been deeply studied invariousaspects such asrepresentation theory,
algebraic geometry and two dimensional gravity. In 1981 Date et al. introduced a KP
like hierarchy of nonlinear differential equations, which has a symmetry of the infinite
dimensional Lie algebra B = go(∞) and call it the KP hierarchy of B type or BKP
∞
hierarchy for short [1, 2]. Although this hierarchy has nice properties which reflect the
Lie algebra symmetry, it seems that a few researches have been made compared with the
KP hierarchy. This letter is an attempt to give an explicit expression of the weighted
homogeneous solutions to the reduced BKP hierarchies.
We first review some ingredients of the BKP hierarchy and affine Lie algebras for the
following discussions.
Let φ (n ∈ Z) be the neutral free fermion satisfying
n
{φ ,φ } = (−1)mδ . (1)
m n m+n,0
Note that φ2 = 1/2. We denote by B = B ⊕ B the superalgebra generated by the
0 0 1
neutral free fermions. Let F = F ⊕F = B |0i⊕B |0i (resp. F∗ = h0|B ⊕h0|B ) be
0 1 0 1 0 1
the Fock space (resp. dual Fock space), where the vacuum |0i (resp. h0|) is defined by
φ |0i = 0 (n < 0), (resp. h0|φ = 0 (n > 0)). (2)
n n
The vacuum expectation value h0|a|0i(a ∈ B) is uniquely determined by setting h0|1|0i =
1, h0|φ |0i = 0.
0
We construct the realization of F. The normal ordering for the neutral fermions is
defined by
: φ φ := φ φ −h0|φ φ |0i. (3)
i j i j i j
2
Define the Hamiltonian by
1
H(t) = (−1)n+1t φ φ . (4)
2 X X j n −j−n
j≥1,oddn∈Z
Let V = C[t ,t ,t ,...] be a polynomial ring. There is an isomorphism between F and
1 3 5 0
V, defined by
a|0i 7−→ h0|eH(t)a|0i, a ∈ B0. (5)
The subspace
 
B = c : φ φ :;c = 0 if |m−n| >> 0 (6)
∞  X mn m n mn 
m,n∈Z
 
of B admits a structure of a Lie algebra isomorphic to the one dimensional central
0
extension of o(∞). The representation of B on V is defined by the vertex operator
∞
11−q/p 1 ∂
Z(p,q) = exp (pj −qj)t exp−2 (p−j −q−j) −1, (7)
21+q/p  X j X j ∂t 
j:odd   j:odd j
 
which corresponds to the action of : φ(p)φ(q) : on F , where φ(p) = φ pn. The
0 n∈Z n
P
space of the solutions (τ-functions) to the BKP hierarchy is obtained as the group orbit
through 1 ∈ V under this representation.
We consider the reduction of the BKP hierarchy [3]. For a fixed positive integer r,
r-reduced BKP hierarchy is defined through the specialization of the vertex operator (7)
q = pω, ωr = 1,ω 6= 1. (8)
This specialization is known to be equivalent to restricting the algebra B to the subal-
∞
(2) (2)
gebra A for r = 2l+1, and D for r = 2l+2. The representation space will be denoted
2l l+1
by V . It should be noted that the situation is different between odd r and even r. In the
r
r = 2l + 1 case the specialization means the deletion of the variables t (j ≡ 0 mod r).
j
3
Hence the representation space becomes smaller: V = C[t ;j ≥ 1,odd and j ≡/0(mod
r j
r)]. On the other hand, in the r = 2l + 2 case, since the representation space has no
variables with an even index, the space itself is unchanged: V = V.
r
(2)
In order to compute the weight of a given vector, we write down coroots of A and
2l
(2)
D explicitly by means of the neutral fermion operators:
l+1
(2)
A (r = 2l+1)
2l
α∨ = 2 (−1)n+1 : φ φ : +1,
0 X nr−1 −nr+1
n∈Z
α∨ = (−1)n+i : φ φ : − : φ φ : (1 ≤ i ≤ l−1),
i X (cid:16) nr+i −nr−i nr−(i+1) −nr+(i+1) (cid:17)
n∈Z
α∨ = (−1)n+l : φ φ : .
l X nr+l −nr−l
n∈Z
(2)
D (r = 2l+2)
l+1
α∨ = −2 : φ φ : +1,
0 X nr−1 −nr+1
n∈Z
α∨ = (−1)i : φ φ : − : φ φ : (1 ≤ i ≤ l−1),
i X (cid:16) nr+i −nr−i nr−(i+1) −nr+(i+1) (cid:17)
n∈Z
α∨ = 2 (−1)l : φ φ : .
l X nr+l −nr−l
n∈Z
Let δ = l a α be the fundamental imaginary root of the affine Lie algebra, where
i=0 i i
P
(2) (2)
a are the labels of the corresponding generalized Cartan matrix. ForA and D it reads
i 2l l+1
δ = 2 l−1α +α and δ = l α , respectively.
i=0 i l i=0 i
P P
Finally we recall the basic representation of affine Lie algebras. The irreducible rep-
resentation with highest weight Λ is called the basic representation, where
0
Λ (α∨) = 1, Λ (α∨) = 0 (j 6= 0). (9)
0 0 0 j
The representation on F constructed above turns out to be the basic representation
0
4
(2) (2)
L(Λ ) of A and D [6].
0 2l l+1
§2 Weight Vectors, Q-Functions and Maya Game
It is shown in [6] that a vector
(m ,...m ) := φ ···φ |0i ∈ F , (m > ··· > m ≥ 0) (10)
1 2d m2d m1 0 2d 1
(2) (2)
isaweight vector oftheFockrepresentation ofB . SinceA andD areLiesubalgebras
∞ 2l l+1
of B , the vector (10) is also a weight vector of L(Λ ) of these affine Lie algebras. In
∞ 0
view of the principally specialized character of L(Λ ), we have the following proposition.
0
Proposition 1The basic representation L(Λ ) of A(2)(resp. D(2)) has the basis consisting
0 2l l+1
of the weight vectors
{(m ,...,m );m > ··· > m ≥ 0,m ≡/0(mod2l+1)} (11)
1 2d 2d 1 j
(resp. {(m ,...,m );m > ··· > m ≥ 0}). (12)
1 2d 2d 1
Let us turn to the realization V of the basic representation L(Λ ), where r = 2l+1 for
r 0
(2) (2)
A and r = 2l+2 forD . We see how each weight vector of L(Λ ) is expressed explicitly
2l l+1 0
as a polynomial in V . To this end we recall Schur’s Q-functions [8]. Let X ,...,X be
r 1 n
indeterminates. The Hall-Littlewood symmetric function indexed by the Young diagram
Y = (m ,...,m ) (m ≥ ··· ≥ m ≥ 0) is defined by
1 n n 1
X −qX
Q (X ,...,X ;q) = (q;q) wXmn ···Xm1 i j (13)
Y 1 n Y µj X 1 n Y X −X
j≥1 w∈Sn/SYn  mi<mj i j 
where q is a parameter, µ = #{i;m = j} is the multiplicity of j in Y, (q;q) =
j i k
k (1−qi) and SY is the subgroup consisting of permutation w which does not change
i=1 n
Q
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the diagram Y, i.e., m = m for 1 ≤ i ≤ n. As a specialization q = −1 we get Schur’s
w(i) i
Q-function Q (X ,...,X ). It is obvious from (13) that Q (X ,...,X ) = 0 unless Y is
Y 1 n Y 1 n
a strict Young diagram, i.e., µ = 1 for any j ≥ 1.
j
The Frobenius formula gives the relation between Q-functions and power sum sym-
metric functions p (X ,...,X ) = Xm +···+Xm:
m 1 n 1 n
n
QY(X1,...,Xn) = X2l(Y′)zY−′1χYY′(−1)Ypm′j(X1,...,Xn). (14)
Y′ j=1
Here the summation runs over all Young diagrams Y′ = (m′,...,m′ ) consisting of odd
1 n
numbers m′j, l(Y′) is the number of nonzero m′j’s, and zY′ = j≥1jνjνj!, where νj denotes
Q
the multiplicity of j in Y′. Although we do not make explicit, χY (−1) is an integer
Y′
computed easily from the character table of projective representations of the symmetric
group S . Introduce new variables t = 2p (X ,...,X ) for positive odd integers j, so
n j j j 1 n
that theQ-functionturns outto beaweighted homogeneous polynomialof t = (t ,t ,...):
1 3
tν1tν3 ···
Q (t) = χY (−1) 1 3 ∈ V. (15)
Y X Y′ ν !ν !···
Y′ 1 3
Another expression is known for the Q-functions [5]. Let Y = (m ,...,m ) (m >
1 2d 2d
··· > m ≥ 0) be a strict Young diagram, and consider 2d×2d skew symmetric matrix
1
Q (t) . Then we have
(cid:16) (mi,mj) (cid:17)
Q (t) = Pf Q (t) , (16)
Y (cid:16) (mi,mj) (cid:17)
where Pf denotes the Pfaffian. Using anti-commutation relations, it is easily deduced that
1
h0|eH(t)φ φ |0i = Q (t) (m > n ≥ 0) (17)
m n (m,n)
2
and
1
h0|eH(t)φ ···φ |0i = Pf Q (t) . (18)
m2d m1 2d (cid:16) (mi,mj) (cid:17)
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Hence every weight vector of the Fock representation of B is expressed by means of a
∞
single Q-function.
For a polynomial P(t) ∈ V, we denote by P(t′) the polynomial in V obtained from
r
P(t) by putting t = 0 for j = 1,2,.... Our claim is as follows:
jr
Proposition 2 The basic representation on V of A(2)with r = 2l + 1 (resp. D(2)with
r 2l l+1
r = 2l+2) has the basis consisting of weight vectors
{Q (t′);Y = (m ,...,m ),m > ··· > m ≥ 0,m ≡/0(mod2l+1)} (19)
Y 1 2d 2d 1 j
(resp. {Q (t′);Y = (m ,...,m ),m > ··· > m ≥ 0}). (20)
Y 1 2d 2d 1
Maya game is a fermion version of the so called “Nim” defined as follows [4]. Consider
infinitely many cells indexed by Z. Each m ∈ Z is said to be either black or white
according as it is filled with a single particle or is empty. The boundary condition is that
m(resp. −m) is white(resp. black) for sufficiently large m > 0. The vacuum is a state
that m is black for all m < 0 and is white for all m ≥ 0. (Figure 1) In general state some
particles excite and occupy some non-negative numbered cells. The game, played by two,
is to move a particle by turn to a lower level until one completes the vacuum. The player
loses the game if he faces the stalemate, i.e., the vacuum state.
To each weight vector (m ,...,m ) we associate a state of Maya game in a following
1 2d
manner. Every positive cell m is filled with a particle excited from the cell −m . There
j j
is an ambiguity at the cell indexed 0. We choose the convention that 0 is always white.
The state of Maya game such obtained from a weight vector is called a symmetric state.
In the language of Young diagrams the relation between symmetric states and weight
vectors is described as follows. For a weight vector (m ,...,m ), draw the corresponding
1 2d
strict Young diagram Y. Let Y be the shift symmetric Young diagram obtained from
e
Y, which is explained in [8]. Then the rule that associates a state of Maya game with
a Young diagram leads to the symmetric state corresponding to (m ,...,m ) (cf. [4]).
1 2d
(Figure 2) We remark that the transposed Young diagram tY leads to the symmetric state
e
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with black 0.
We now state how to obtain weight vectors of weight λ±δ from a given weight vector
of weight λ. The procedure is described by modifying the rule of Maya game.
(2)
First we consider the case of A . Suppose a weight vector of weight λ is given. Then
2l
one of the following possible moves is permitted to have weight λ−δ.
(1) Move a black by 2l+1 to the right.
(2) Replace the pair of white cells (k,2l+1−k) by black for 1 ≤ k ≤ l.
(2)
For the case D the moves are
l+1
(1) Replace the white l+1 by black.
(2) Move a black at a positive multiple of l+1 to the right.
(3) Once the cell n(2l +2) becomes black for an n ≥ 1, instead of that single black,
the black pairs (k,n(2l+2)−k) are also permitted for 1 ≤ k < n(2l+2).
The inverse moves give weight vectors of weight λ+δ from one of weight λ.
§3 Polynomial solutions to the reduced BKP hierarchies
In this section we characterize the maximal weight vectors of the basic representation
(2) (2)
L(Λ ) of the affine Lie algebras A and D and, as a consequence, obtain the weighted
0 2l l+1
homogeneous polynomial solutions to the corresponding reduced BKP hierarchy.
(2) (2)
By definition the weight λ of L(Λ ) of A or D is said to be maximal if λ + δ is
0 2l l+1
not a weight. It is known that the totality of the maximal weights of L(Λ ) is given by
0
the Weyl group orbit through the highest weight Λ , and hence each maximal weight
0
is of multiplicity 1 [7]. Since the Weyl group can be seen as a subgroup of the group
(2)
corresponding to the affine Lie algebra, the maximal weight vectors of L(Λ ) of A and
0 2l
(2)
D are polynomial solutions to the (2l+1)-reduced and (2l+2)-reduced BKP hierarchy,
l+1
respectively [3].
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As we have shown in section 2, for each weight vector in L(Λ ) we can associate a
0
symmetric state of Maya game. We have also given a description by means of Maya game
how to obtain weight vectors of weight λ±δ for a given weight vector of weight λ. The
weight λ is maximal if the corresponding weight vectors is at stalemate according to the
rule.
(2)
Let us give a more precise description. For A the weight vector (m ,...,m ) is a
2l 1 2d
maximal weight vector if each m cannot be moved by 2l+1 to the left and there are no
j
(2)
such pairs (m ,m ) = (k,2l+1−k) for 1 ≤ k ≤ l. For D , (m ,...,m ) is a maximal
i j l+1 1 2d
weight vector if there are no m such that m ≡ 0 (mod l + 1) and there are no pairs
j j
(2)
(m ,m ) such that m + m = n(2l + 2) for n ≥ 1. In particular, for the case A the
i j i j 2
following set covers the maximal weight vectors:
{φ,(1,4,7,...,3n−2),(2,5,8,...,3n−1);n ≥ 1}.
(2)
For D the maximal weight vectors of degree up to 12 are
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{φ,(1),(2),(3),(1,2),(1,3),(5),(2,3),(6),(1,5),(1,2,3),(7),(1,6),
(2,5),(1,2,5),(2,7),(3,6),(1,9),(3,7),(1,3,6),(5,6),(2,10),(5,7),
(1,2,9),(1,5,6),(2,3,7)}.
If one uses the language of Young diagrams, the above criterion is stated in simpler words.
Theorem Schur’s Q-function Q (t′) is a maximal weight vector of L(Λ ) of A(2)(resp.
Y 0 2l
(2)
D ) if the shift symmetric diagram Y of Y has no hooks of length divisible by 2l + 1
l+1
e
(resp. 2l+2).
(2) (2)
In both cases, A and D , one recovers the (non-reduced) BKP hierarchy by taking
2l l+1
the limit l → ∞. Hence, as a corollary to the above theorem, we see that the Q-function
Q (t) of any strict Young diagram Y solves the BKP hierarchy [9].
Y
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References
[1] E. Date, M. Kashiwara and T. Miwa, Proc. Japan Acad. 57A (1981) 387
[2] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Physica 4D (1982) 343
[3] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Publ. RIMS. Kyoto Univ. 18 (1982)
1077
[4] S. Hitotumatu, Mathematics of Nim (Morikita, 1968, in Japanese)
[5] P. N. Hoffman and J. F. Humphreys, Projective Representations of the Symmetric
Groups (Oxford, 1992)
[6] M. Jimbo and T. Miwa, Publ. RIMS. Kyoto Univ. 19 (1983) 943
[7] V. Kac, Infinite Dimensional Lie Algebras, 3rd ed. (Cambridge, 1990)
[8] I. G. Macdonald, Symmetric Functions and Hall Polynomials (Oxford, 1979)
[9] Y.-C. You, Adv. Ser. in Math. Phys. 7 (World Scientific, 1989) 449
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