Domain wall partition functions and KP

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by  O Foda

DOMAIN WALL PARTITION FUNCTIONS AND KP
OFODA,MWHEELERANDMZUPARIC
Abstract. Weobservethatthepartitionfunctionofthesixvertexmodelon
a finite square lattice with domain wall boundary conditions is (a restriction
of) a KP τ function and express it as an expectation value of charged free
9 fermions(uptoanoverallnormalization).
0
0
2
To Professor T Miwa on his 60th birthday.

n
a
0. Introduction
J

5 In [1], Korepin introduced domain wall boundary conditions for the six vertex
1
model on an N×N square lattice, and obtained a set of conditions that determine

the corresponding partition function Z . In [2], Izergin proposed a determinant
] N
h expressionforZ ,andshowedthatitsatisfiesKorepin’sconditions. In[3],Lascoux
N
p
rewroteZ as adeterminantofaproductoftwonon-squarematrices. Inthis note
N
-
h we observe that, given Lascoux’s form, ZN is (a restriction of) a KP τ function
t (for all values of the crossing parameter) and rewrite it as an expectation value of
a
m charged free fermions.
In section 1, we recall basic definitions related to the six vertex model and
[
domainwallboundaryconditions,includingKorepin’sconditions,andinsection2,

1 werecallIzergin’sdeterminantexpressionforZ ,followedbyLascoux’sexpression,
N
v
and observe that the latter is (a restriction of) a KP τ function. In section 3, we
1
proposeanexpectationvalueofKPchargedfreefermionsFfreethatbyconstruction
5 N
2 is a KP τ function, and show that (under suitable restrictions of the KP time
2 variables) Ffree becomes equal to Z (up to an overall normalization). In section
N N
.
1 4, we include a number of remarks.
0
9 1. Domain wall partition functions
0
: In this section, we list a number of basic definitions related to the six vertex
v
i model, domain wall boundary conditions, etc. For details, see [4,5].
X
r 1.1. Oriented lattice lines and rapidity variables. Consider a square lattice
a with N horizontal lines (rows) and N vertical lines (columns) that intersect at N2
points, and assign the i-th horizontal line an orientation from left to right and a
rapidity x , and the j-th vertical line an orientation from bottom to top and a
i
rapidity y , as in Figure 1.
j
1.2. Arrows, weights and a crossing parameter. Assigneachline segmentan
arrowthatcanpointineitherdirection,anddefinethevertexv astheintersection
ij
point of the i-th horizontal line and the j-th vertical line, the four line segments
attached to this intersection point and the arrows on these segments. Assign v a
ij
weight w that depends on the orientations of its arrows, the rapidities x and y
ij i j
and a crossing parameter µ that is the same for all vertices.
2000 Mathematics Subject Classification. Primary82B20, 82B23.
Keywordsandphrases. Domainwallpartitionfunctions. Sixvertexmodel. KP.Freefermions.
1
2 OFODA,MWHEELERANDMZUPARIC
1.3. The six vertex model. Since anarrowonaline segmentcanpointineither
direction, there are 24 = 16 possible (types of) vertices. In the six-vertex model,
the weights of all vertices, except six, are zero [4]. The six vertices with non-zero
weights form three pairs of equal-weight vertices. They are shown in Figure 2.
x
1
x
2
x
N
y y y
1 2 N
Figure 1. A square lattice with oriented lines and rapidity variables.
The line orientations are indicated by white arrows.
x x x
i i i
y y y
j j j
x x x
i i i
y y y
j j j
wa wb wc
ij ij ij
Figure 2. The vertices of the six vertex model. Vertices that are in the same
column share the weight shown below that column. The white arrows indicate
the line orientations which are needed to completely specify the vertices.
In the notation of Figure 2, the weights of the six-vertex model are
(1) wa =sinh(−x +y +µ), wb =sinh(−x +y ), wc =sinhµ
ij i j ij i j ij
that satisfy the Yang-Baxter equations [4].
1.4. Domain wall boundary conditions. When allarrowsonthe left and right
boundaries point inwards, and all arrowson the upper and lower boundaries point
outwards, as in Figure 3, we obtain domain wall boundary conditions. The in-
ner arrows remain free. Different inner arrow orientations lead to different lattice
configurations that together constitute the statistical states of the model.
DW PARTITION FUNCTIONS AND KP 3
1.5. Domain wall partition functions. The weight of a lattice configuration is
the product of the weights w ofits vertices v . The weightedsum overall lattice
ij ij
configurations,on an N×N lattice with fixed domain wall boundary conditions, is
a domain wall partition function Z .
N
(2) Z ≡Z x,y,µ := w (x ,y ,µ)
N N ij i j
  allXallowed Yall
  configurations vertices
1
2
N
1 2 N
Figure3. Alatticeconfigurationwithdomainwallboundaryconditions.
1.6. Korepin’sconditions. In[1],Korepinobtainedfourconditionsthatuniquely
determine Z (x,y,µ). These are
N
1. Z is a symmetric function in the {x} and in the {y} rapidity variables.
N
2. Z is a trigonometric polynomial of degree (N −1) in any rapidity variable.
N
3. Setting x =y +µ, Z satisfies the recursion relation
1 1 N
Z | =
N x1=y1+µ
N N
 sinh(−xi+y1)sinhµ sinh(−x1+yj)
iY=2  jY=2 
Z xˆ ,yˆ ,µ
(N−1) 1 1
 
where xˆ ,yˆ indicate that x and y are missing from the sets {x} and {y}. 
1 1 1 1
4. Z = sinhµ
1
We refer to [1,5] for details and proofs.
2. Z is (a restriction of) a KP τ function
N
2.1. Izergin’s determinant expression. Workingin terms of the variablesu =
i
e2xi, vi =e2yi, and q =e−2µ, Izergin’s determinant expression is
(3)
Ni,j=1(ui−vj)(qui−vj) 1 N
Z u,v,q =c det
N  N Q1≤i<j≤N(ui−uj)(vj −vi) (cid:20)(ui−vj)(qui−vj)(cid:21)i,j=1
  Q
where
4 OFODA,MWHEELERANDMZUPARIC
N
(4) c =(q−1)N u21v21
N i i
iY=1
The virtue of Izergin’s expression is that it is straightforward to show that it
satisfies Korepin’s conditions [2,5].
2.2. Lascoux’s determinant expression. In [3], Lascoux showed that Izergin’s
expression for Z can be rewritten as
N
N
2N−1
(5) ZN u,v,q =cNdet αi,j(u)βj,k(v,q)
  Xj=1
   i,k=1
where
qj−k+1−qk−1
(6) α (u)=h (u), β (v,q)= e (−v)
i,j −i+j j,k −j+k+N−1
q−1
andh (u)ande (v)arethen-thhomogeneousandthen-thelementarysymmetric
n n
functions, respectively [6]. For a proof that Lascoux’s expression is equivalent to
Izergin’s, see [3]. As we will see shortly, the virtue of Lascoux’s expression is that
it is straightforwardto show that it is (a restriction of) a KP τ function.
2.3. Remarks on notation. In α , β , and other matrix elements, i is the row
i,j i,j
and j is the column index. We will omit writing the dependence on the parameter
q explicitly, as it is always there.
2.4. Partitions. We denote the partition of an integer |λ| into parts λ ≥ λ ≥
1 2
···, by [λ] = [λ ,λ ,··· ,λ ], where L is the number of non-zero parts. mn in
1 2 L
[··· ,mn,···] indicates that [λ] hasn partsof lengthm. Further, m0 indicates that
there are no parts of length m and 0n stands for n parts of length 0.
2.5. Young diagrams. Consider the Young diagram λ of a partition [λ] as in
Figure4. Thecells arelabelledfromtoptobottomandfromleftto right. The cell
withcoordinates(i,j), i,j ≥1 is c . The number ofcells c ,c ,···, onthe main
ij 11 22
diagonal is d.
(1,1)(1,2)(1,3)(1,4)
(2,1)(2,2)(2,3)
(3,1)(3,2)(3,3)
(4,1)(4,2)
Figure 4. The Young diagram λ = [4,3,3,2]≡ [4,32,2]. The labels (i,j) in a
cell c are its coordinates. In this example, d=3.
ij
2.6. Hooks. Givenacellc ,wedefine ahookh asthe unionofc ,the setofall
ij ij ij
cells below it and all cells to the right of it. We say that the hook h has a corner
ij
atc . The diagramonthe lefthandsideofFigure5has6hooks,oneforeachcell.
ij
The right hand side of Figure 5 is the hook corresponding to the cell c .
(2,1)
DW PARTITION FUNCTIONS AND KP 5
2.7. Horizontal a-parts and vertical b-parts. We decomposea hook h into a
ij
horizontala-parta oflength|a |thatconsistsofallcellstotherightofc ,anda
ij ij ij
verticalb-partb oflength|b |thatconsistsofc andallcellsbelowit. Hencethe
ij ij ij
allowed a-part lengths are 0,1,2,···, while the allowed b-part lengths are 1,2,···
The length of h will be indicated by |h | as well, so that
ij ij
(7) |h |=|a |+|b |
ij ij ij
2.8. Frobenius coordinates of λ. Decomposing eachhook h , i=1,··· ,d that
ii
hasacorneratacellonthemaindiagonalofλintothecornercellc ,thesetofm
ii i
cells to the right of c , and n cells below it, we obtain 2d integers {m ,n } which
ii i i j
define the Frobenius coordinates of λ.
[1,3] [0,3]
[1,2] [0,2]
[1,1] [0,1]
Figure 5. On the left, the Young diagram corresponding to λ=[2,2,2] ≡[23].
The labels [a,b] in a cell are the a-part and b-part of the hook associated with
that cell. On the right, the hook corresponding to c in λ. In this hook, the
21
a-part a has length 1, the b-part b has length 2.
21 21
2.9. Expanding Z in terms of Schur functions. To make contact with KP
N
theory and eventually obtain an expression for Z as a vacuum expectation value
N
of free fermions, we start by expanding Lascoux’s expression in terms of Schur
functions. Using (6) and the Cauchy-Binet identity in (5), we obtain
N N
(8) Z u,v =c det[h (u)] det[β (v)]
N N −i+jl i,l=1 jl,k l,k=1
  1≤j1<···X<jN≤2N−1
Rewriting j as j = λ +l, then reversing the order of all rows and all
l l (N+1)−l
columns in the first matrix and transposing it, the sum on the right hand side of
(8) becomes
N N
(9) det[h (u)] det β (v)
λi−i+l i,l=1 λ(N+1)−l+l,k l,k=1
0≤λN≤·X··≤λ1≤N−1 (cid:2) (cid:3)
Finally, using the definition of Schur functions
N
(10) S (u)=det[h (u)]
λ λi−i+l i,l=1
and (6) to write
N
c(N)(v) = det β (v)
λ (cid:20) 8:λlconj9;−lconj+(N+1),k (cid:21)l,k=1
8 9 N
q:λlconj;−lconj−k+N+2−qk−1
(11) = det e 8 9 (−v)
q−1 −:λlconj;+lconj+k−2
 l,k=1
where
(12) l =(N +1)−l
conj
6 OFODA,MWHEELERANDMZUPARIC
we end up with
(13) Z u,v =c c(N)(v)S (u)
N N λ λ
  λ⊆[(XN−1)N]
 
Since the summation in (13) is over partitions of at most N parts, and there
are N independent u-variables,and N independent v-variables, there are no terms
in the expansion that trivially vanish due to lack of sufficiently many independent
variables.
2.10. Expanding Z in terms of character polynomials. Next, following [7],
N
we introduce the power sum variables
N
1
(14) t = un, n=1,2,···
n n i
Xi=1
and the polynomials p (t)
n
tn1tn2tn3···
(15) p (t)= 1 2 3
n
n !n !n !···
n1+2n2+X3n3+···=n 1 2 3
where p (t)=0 for i<0. Next, we define the polynomials h (t)
i m,n
n
(16) h (t):=(−1)n p (t)p (−t)
m,n k+m+1 n−k
kX=0
and finally the character polynomials χ (t)
λ
d
(17) χ (t):=det h (t)
λ mi,nj i,j=1
(cid:2) (cid:3)
where {m ,n } are the Frobenius coordinates of λ, and the upper limit d is the
i j
number of cells on the main diagonal. In terms of character polynomials and the
t-variables of (14) (and by slightly abusing notation), we rewrite Z (u,v) as
N
(18) Z t,v =c c(N)(v)χ (t)
N N λ λ
  λ⊆(XN−1)N
 
wherec(N)(v) wasdefinedin(11). We concludethatZ canbe expandedinterms
λ N
of character polynomials, with coefficients that are minors of a determinant such
that (by construction) they satisfy Plu¨cker relations [7]. If the t-variables were all
independent, then Z (t,v) would be identically a KP τ function. We refer to [7]
N
for details and proofs.
2.11. Restricted t-variables. Z (u,v) is a function of N independent u-vari-
N
ables. On the other hand, from the definition of the character polynomials, it
easy to show that Z (t,v) requires in general more than N t-variables. Since the
N
latter are power sums of the N u-variables, they cannot be all independent. Let
us write Zrest(t,v) to emphasise the fact that the t-variables are restricted to be
N
power sums of a (generally) smaller number of u-variables. This restriction spoils
the interpretation of the t-variables as KP time variables, and of Zrest(t,v) as a
N
KP τ function.
DW PARTITION FUNCTIONS AND KP 7
2.12. Free t-variables. WecanformallyconsiderZrest(t,v)asourstartingpoint,
N
takethet-variablestobeindependent,writeitasZfree(t,v)andthinkofZrest(t,v)
N N
as a restriction of Zfree(t,v) obtained by setting the t-variables to power sums of
N
the N u-variables using (14). Viewed this way, Zfree(t,v) is a KP τ function, and
N
Zrest(t,v) = Z (u,v), is a restriction of a KP τ function.
N N
3. Fermionic expression for Z
N
Given that any KP τ function can written as the vacuum expectation value of
exponentialsinneutralbilinearsinKPchargedfermions1[7],wewishtowriteZfree
N
as such. Our plan is to propose a vacuum expectation value Ffree(t,v), where we
N
use the superscript free to emphasise that the t-variables are independent, that by
construction is a KP τ function, then show that Ffree(t,v) = Zfree(t,v). Hence
N N
restricting the t-variables in Ffree(t,v) to be powers sums of N u-variables, one
N
recoversZrest(t,v),andZ (u,v),asanexpectationvalueofchargedfreefermions.
N N
3.1. Charged free fermions. Consider the free fermion operators {ψ ,ψ∗}, n∈
n n
Z, with charges {+1,−1} and energies n. They generate a Clifford algebra over C
defined by the anti-commutation relations
[ψ ,ψ ] =0
m n +

(19) [ψm∗ ,ψn∗]+ =0  ∀ m,n∈Z
[ψ ,ψ∗] =δ
m n + m,n 
3.2. Vacuum states. We define the vacuum states h0| and |0i by the actions
h0|ψ =ψ |0i=0,
n m
(20)  ∀ m<0, n≥0
h0|ψ∗ =ψ∗|0i=0, 
m n
and we adopt the inner product normalization
(21) h0|0i=1
3.3. Creation and annihilation operators. We refer to the operators that an-
nihilate the vacuum state |0i, ψ |0i = 0, m = −1,−2,..., and ψ∗|0i = 0,
m n
n=0,1,2,...asannihilation operators, andtothe rest,ψ |0i=6 0,m=0,1,2,...,
m
and ψ∗|0i=6 0, n=−1,−2,...as creation operators.
n
3.4. Normal ordering. Thenormal-orderedproductisdefined,asusual,byplac-
ing annihilation operators to the right of creation operators
:ψ ψ∗ : =ψ ψ∗−h0|ψ ψ∗|0i
i j i j i j
3.5. The Heisenberg algebra. Consider the operators H ,
m
(22) H := :ψ ψ∗ :, m∈Z
m j j+m
Xj∈Z
that together with the central element 1 form a Heisenberg algebra
(23) [H ,H ]=mδ , ∀ m,n∈Z
m n m+n,0
1Bythis,wemeanlinearcombinationsoftermsoftheformψmψn∗.
8 OFODA,MWHEELERANDMZUPARIC
3.6. The Hamiltonian. Using H ,m=1,2,3,..., we define the Hamiltonian
m
∞
(24) H(t):= t H
m m
mX=1
3.7. Fermion bilinears. Given a partitionλ, we canassociateto eachhook h a
ij
bilinearψ ψ∗ withnetchargezero. Bothfermionsinthebilineararecreation
|aij| −|bij|
operators,
(25) ψ ψ∗ =−ψ∗ ψ
|aij| −|bij| −|bij| |aij|
and normal ordering is unnecessary.
3.8. Boson-fermion correspondence. It is possible to realise fermionic Fock
space expressions as elements in the polynomial ring C[t ,t ,···] [7]. For the pur-
1 2
posesof this work,allwe needis that factthat the characterpolynomialχ (t) can
λ
be generated as follows
(26) h0|eH(t)ψ−∗b1···ψ−∗bdψad···ψa1|0i=(−1)b1+···+bdχλ(t)
where a < ··· < a and b < ··· < b are the a-part and b-part lengths of the
d 1 d 1
hooks associated with cells c ,c ,···, on the main diagonal of λ, and d is the
11 22
number of these cells. In terms of Frobenius coordinates,a =m and b =n +1.
i i j j
3.9. Partitions and fermion monomials. Consider the fermion monomial con-
sisting of d ψ and d ψ∗ fermions, ψ∗ ···ψ∗ ψ ···ψ , where a < ··· < a
−b1 −bd ad a1 d 1
and b < ··· < b as on the left hand side of (26). The fermions in the monomial
d 1
are ordered according to length and correspond to the a-parts and b-parts of the
partitionλontherighthandsideof(26),withthelongerb-partstotheleftandthe
longer a-parts to the right. We refer to such a monomial associated to a partition
λ as a partition-ordered, or λ-ordered monomial. The partition in Figure 4 has
d = 3, a set of b-parts {4,3,1} and a set of a-parts {3,1,0}. The corresponding
monomial is ψ∗ ψ∗ ψ∗ ψ ψ ψ .
−4 −3 −1 0 1 3
3.10. A fermion expectation value. Consider the vacuum expectation value
(27) Ffree(t,v)=h0|eH(t)eX0(N)eX1(N)···eXN(N−)2|0i
N
where
N
(28) X(N) = (−)bd(N) ψ∗ ψ , a=0,1,··· ,N −2
a [a+1,1b−1] −b a
Xb=1
where 10 inside a partition [λ] stands for a trivial, length-0 part, and
(29) d(N) =c(N)/c(N)
λ λ φ
The coefficient c(N) was defined in (11), and the (infinitely many) t-parameters
λ
in (27) are taken as independent variables rather than power sums of the (finitely
many) u-parameters. Since Ffree is an expectation value of exponentials of neutral
N
bilinearsin{ψ ,ψ∗ },whicharegeneratorsofgl(∞),followedbythetimeevolution
n m
operator eH(t), it is by construction a KP τ function [7]. Since all fermions in the
linear combinations of bilinears in (28) are creation operators, there is no need for
normal ordering these terms.
DW PARTITION FUNCTIONS AND KP 9
3.11. Acombinatorial interpretation. TheexponentialsthatappearinFfree in
N
(27) have the following interpretation. Consider a rectangular partition of N rows
and (N −1) columns. For example, for N = 3, see the diagram on the left hand
sideofFigure5. Forthej-thcolumn,withN cellsofa-partlengtha=(N−1)−j,
we have an exponential X(N), 0 ≤ a ≤ (N −2), which is a sum of N terms. The
a
b-th term is of the form (−1)bd(N)ψ∗ ψ , where [h] is the partition corresponding
[h] −b a
to the hook that has a corner at that cell, and a and b are the a-part and b-part
lengths of that hook.
3.12. F is proportional to Z . In the rest of this work, we show that
N N
(30) c c(N)(v)Ffree(t,v)=Zfree(t,v)
N φ N N
3.13. The normalization factor. Expanding the exponentials in (27) and using
h0|0i =1, the character polynomial expansion of Ffree starts with 1. On the other
N
hand, since χ (t) = 1, the character polynomial expansion of Z in (18) starts
φ N
with the t-independent term, c c(N)(v). We choose to keep the definition of the
N φ
vertex weights as in (1), and show that F and Z are equal up to a factor of
N N
c c(N)(v).
N φ
3.14. Example: N = 3. Before we prove (30) for general N, we verify it in the
simple but non-trivialcase of N =3. Omitting the superscripts (3) to simplify the
notation, we have
8 9 3
q:λ4−i;+i−j+1−qj−1
(31) cλ = det q−1 e−8:λ4−i9;−i+j+2(−v)
 i,j=1
X = −d ψ∗ ψ +d ψ∗ ψ −d ψ∗ ψ
0 [1,0,0] −1 0 [1,1,0] −2 0 [1,1,1] −3 0
X = −d ψ∗ ψ +d ψ∗ ψ −d ψ∗ ψ
1 [2,0,0] −1 1 [2,1,0] −2 1 [2,1,1] −3 1
where d was defined in (29). Notice that in this case, the relevant partition is
λ
shownonthe lefthandsideofFigure5,andthe bilinearsinX andX correspond
0 1
to the hooks of the cells in the two columns. This leads to
(32) eX0eX1 =1 − d[1,0,0]ψ−∗1ψ0+d[1,1,0]ψ−∗2ψ0−d[1,1,1]ψ−∗3ψ0
− d ψ∗ ψ +d ψ∗ ψ −d ψ∗ ψ
[2,0,0] −1 1 [2,1,0] −2 1 [2,1,1] −3 1
+ d d −d d ψ∗ ψ∗ ψ ψ
[1,1,0] [2,0,0] [1,0,0] [2,1,0] −2 −1 0 1
 
+ d d −d d ψ∗ ψ∗ ψ ψ
[1,0,0] [2,1,1] [1,1,1] [2,0,0] −3 −1 0 1
 
+ d d −d d ψ∗ ψ∗ ψ ψ
[1,1,1] [2,1,0] [1,1,0] [2,1,1] −3 −2 0 1
 
 
Considerthesixfermionbilinearsontherighthandsideof(32). Usingtheboson-
fermioncorrespondence,eachofthesebilinearscorrespondstoapartition[λ](which
in the case of bilinears is a single hook), and its coefficient d corresponds to the
[λ]
same partition [λ]. If the same is true for the three fermion quartic terms, then
from (18), F would be proportional to Z . To put the coefficients of the quartic
3 3
terms in the right form, we need to show that
10 OFODA,MWHEELERANDMZUPARIC
(33) d d −d d = −d
[1,1,0] [2,0,0] [1,0,0] [2,1,0] [2,2,0]
(34) d d −d d = d
[1,0,0] [2,1,1] [1,1,1] [2,0,0] [2,2,1]
(35) d d −d d = −d
[1,1,1] [2,1,0] [1,1,0] [2,1,1] [2,2,2]
3.15. Plu¨cker relations. Equations (33–35) are examples of Plu¨cker relations,
whichareidentities involvingbilinearsinn×nminorsofa2n×2nmatrixwithzero
determinant. In this case, n=3. For details, see [8]. Using the notation
(36) c ≡c =det[γ ,γ ,γ ]=|γ ,γ ,γ |,
λ [λ1,λ2,λ3] λ3+1 λ2+2 λ1+3 λ3+1 λ2+2 λ1+3
where γ is the 3-component column vector
b
qb−a+1−qa−1
(37) γ = e (−v) ,
b  q−1 −b+a+2 
a=1,2,3
 
we consider the 6×6 determinant
γ γ γ γ γ γ
(38) µ1 µ2 ν1 ν2 ν3 ν4 =0,
(cid:12)(cid:12) 0 0 γν1 γν2 γν3 γν4 (cid:12)(cid:12)
(cid:12) (cid:12)
and Laplace expand it(cid:12)as the sum of bilinears in 3×3 d(cid:12)eterminants
(39) |γ ,γ ,γ ||γ ,γ ,γ |−|γ ,γ ,γ ||γ ,γ ,γ |
µ1 µ2 ν1 ν2 ν3 ν4 µ1 µ2 ν2 ν1 ν3 ν4
+|γ ,γ ,γ ||γ ,γ ,γ |−|γ ,γ ,γ ||γ ,γ ,γ | = 0
µ1 µ2 ν3 ν1 ν2 ν4 µ1 µ2 ν4 ν1 ν2 ν3
We can use (39) to generate the Plu¨cker relations that we need by suitably
choosing the column vectors γ .
b
3.16. The N =3 Plu¨cker relations. TherequiredPlu¨ckerrelation,definedasin
(41),hasatermd d ,whered =1,andλ=[2,2,0]. From(36)weseethatthe
[λ] [φ] [φ]
bilinear that we need is |γ ,γ ,γ ||γ ,γ ,γ | = |γ ,γ ,γ ||γ ,γ ,γ |.
1+0 2+2 3+2 1+0 2+0 3+0 1 4 5 1 2 3
This dictates the choice of parameters
(40) (µ ,µ ,ν ,ν ,ν ,ν )=(1,4,5,1,2,3)
1 2 1 2 3 4
Using (40) in (39), and the antisymmetry property of determinants, we obtain
(41) |γ ,γ ,γ ||γ ,γ ,γ |−|γ ,γ ,γ ||γ ,γ ,γ |=−|γ ,γ ,γ ||γ ,γ ,γ |
1 4 5 1 2 3 1 3 5 1 2 4 1 3 4 1 2 5
which, suing (29) and (36), is the Plu¨cker relation that we need to prove (33).
Equations(34–35) are obtainedby setting (µ ,µ ,ν ,ν ,ν ,ν ) to (2,4,5,1,2,3),
1 2 1 2 3 4
then to (3,4,5,1,2,3)in (39). Finally, using (26), we obtain
(42) cφh0|eH(t)eX0eX1|0i= cλ(v)χλ(t)
λ⊆X[2,2,2]
The right hand side of (42) is Lascoux’s form expanded as in (18). The above
example shows that the proof of (30) relies on using Plu¨cker relations to simplify
the coefficients of expansions of the exponentials that appear in (27).