Table Of ContentCOMPOUND BASIS ARISING FROM THE BASIC A(1)-MODULE
1
KAZUYAAOKAGE,HIROSHIMIZUKAWAANDHIRO-FUMIYAMADA
8
0
0 Abstract. A new basis for the polynomial ring of infinitely many variables
2 isconstructed whichconsists ofproducts ofSchur functions andQ-functions.
ThetransitionmatrixfromthenaturalSchurfunctionbasisisinvestigated.
n
a
J
8
1 1. Introduction
] This note concerns with realizations of the basic representationof the affine Lie
T algebra of type A(1)(cf. [6]). The most well-known realization is PU, principal,
R 1
untwisted, whose representation space is
.
h FPU =C[t ;j ≥1,odd].
t j
a
m In the context of nonlinear integrable systems, this space appears as that of the
KdV hierarchy. The second one is HU, homogeneous, untwisted, which is on
[
FHU = F(m); F(m)=C[t ;j ≥1]⊗qm.
1 j
v m∈Z
M
8 This space is for the NLS (nonlinear Schr¨odinger) hierarchy and also for the Fock
1
representation of the Virasoro algebra(cf. [5]). The third one is PT, principal,
8
twisted, on FPT which coincides with FPU. And the fourth one is HT, homoge-
2
. nous, twisted, on FHT which is the same as FHU. The Lie algebra of type A(1) is
1 1
0 isomorphictothatoftypeD(2). OnecandiscusstwistedrealizationofA(1)-modules
2 1
8 via this isomorphism.
0 The purpose of this note is to give a weight basis for FHT and compare it with
v: a standard Schur function basis for FHU. We will show that the transition matrix
i has several interesting combinatorial features. This is a detailed version of our
X
announcement [1].
r
a
2. A quick review of realizations
Let us first consider the principal untwisted realization on FPU = C[t ;j ≥
j
1,odd]. To describe a weight basis for this space we need Schur functions and
Schur’s Q-functions in our setting. Let P be the set of all partitions of n and put
n
P = P . For λ∈P , the Schur function S (t) is defined by
n≥0 n n λ
S tm1tm2···
S (t)= χλ 1 2 ,
λ ρm !m !···
1 2
ρ=(1m1X2m2...)∈Pn
where the summation runs over all partitions ρ=(1m12m2...) of n, and χλ is the
ρ
irreducible character of the symmetric group S , indexed by λ and evaluated at
n
the conjugacy class ρ. The Schur functions are the ordinary irreducible characters
ofthe generallineargroups. If the groupelementg haseigenvaluesx ,x ,..., then
1 2
1
2 KAZUYAAOKAGE,HIROSHIMIZUKAWAANDHIRO-FUMIYAMADA
the original irreducible character is recovered by putting p := jt (j ≥ 1), where
j j
p = xj is the j-th power sum of the eigenvalues.
j i≥1 i
The 2-reduction of a polynomial f(t) is to “kill” the even numbered variables
P
t ,t ,..., i.e. ,
2 4
f(2)(t)=f(t)| ∈FPU.
t2=t4=...=0
The2-reducedSchurfunctionsarelinearlydependentingeneral. Howeveralllinear
relations among them are known, and one can choose certain set P′ ⊂ P so that
S(2);λ∈P′ forms a basis for FPU (cf. [2]).
λ
n The spaceoFPU also affords the principal twisted realization. A weight basis
is best described by Schur’s Q-functions. Let SP (resp. OP ) be the set of all
n n
strict (resp. odd) partitions of n and put SP = SP , OP = OP . For
n≥0 n n≥0 n
λ∈SP , the Q-function Q (t) is defined by
n λ
S S
Qλ(t)= 2ℓ(λ)−2ℓ(ρ)+ǫζρλmtm1 1!tmm3 3!······,
1 3
ρ=(1m13Xm3...)∈OPn
where the summation runs overall oddpartitionsρ=(1m13m3...) ofn, ǫ=0 or 1
accordingto thatℓ(λ)−ℓ(ρ) is evenor oddandζλ is the irreduciblespin character
ρ
of S , indexed by λ and evaluated at the conjugacy class ρ. For the Q-functions,
n
we set p := 1jt (j ≥ 1,odd) as the relation with the “eigenvalues”. A more
j 2 j
detailed account is found in [9]. Here we remark the relation of Q-functions and
the P-functions. We define inner product h , i on F(0) by hp ,p i = z (q)δ ,
q λ µ q λ λµ
wherezλ(q)=zλ i≥1(1−qλi)−1. Notethatzλ(−1)cannotbedefinedforλwhich
has even parts. Therefore we have to re-define h , i by setting hp ,p i =
−1 λ µ −1
Q
2−ℓ(λ)z δ . The P-functionsaredualtotheQ-functionswith respectto the inner
λ λµ
product h , i on FPU. For a strict partition λ, we see that P (t) = 2−ℓ(λ)Q (t)
−1 λ λ
(cf. [8]).
Inordertogivethehomogeneous,twistedrealizationweemployacombinatorics
ofstrictpartitions. Weintroducethefollowingh-abacus. Forexample,theh-abacus
of λ=(11,10,5,3,2)is shown below.
1 (cid:13)3
(cid:13)2
4 (cid:13)5 7
6
8 9 (cid:13)11
(cid:13)10
12 13 15
. . .
. . .
. . .
From this h-abacus of λ we read off a triplet (λhc;λh[0],λh[1]) of partitions.
Firstlyλh[0]=(5,1),astrictpartitionobtainedjustbytakinghalvesofthe circled
positions of the leftmost column.
For obtaining λh[1], we need the following process:
(1) Forthethirdcolumn,thecircledpositionscorrespondtothevacancies”◦”.
(2) For the second column, the circled positions correspond to being occupied
”•”.
COMPOUND BASIS ARISING FROM THE BASIC A(1)-MODULE 3
1
(3) Readthe thirdcolumnfrominfinity tothe position3andconsequentlythe
second column from the position 1 to infinity, and draw the Mayadiagram
... 15 11 7 3 1 5 9 ....
• ◦ • ◦ ◦ • ◦
(4) For each •, count the number of vacancies which are on the left of that •,
and get a partition
λh[1]=(3,1).
Next the h-core λhc is obtained by the following moving and removing:
(1) Remove all circles on the leftmost column.
(2) Move a circle one position up along the second or the third column.
(3) Remove the two circles at the positions 1 and 3 simultaneously.
(4) The “stalemate” determines the partition
λhc =(3).
Note that λhc is always of the form
∆h(m)=(4m−3,4m−7,...,5,1) or ∆h(−m)=(4m−1,4m−5,...,7,3)
for some m ∈ N (∆h(0) = ∅). Let HC be the set of all such λhc’s. In this way
we have a one-to-one correspondence between λ ∈ SP and (λhc;λh[0],λh[1]) ∈
HC×SP ×P with the condition
|λ|=|λhc|+2(|λh[0]|+2|λh[1]|).
By making use of this one-to-one correspondence, we define the linear map η :
FPT →FHT by
η(Q (t))=Q (t)S (t′)⊗qm(λ).
λ λh[0] λh[1]
Here
m(λ)=(number of circles on the second column)−(number of circles on the third column)
and S (t′)=S (t)| for any j ≥1. For any integer m, the set
ν ν tj7→t2j
{η(Q );λ∈SP,m(λ)=m}
λ
forms a basis for F(m) = C[t ;j ≥ 1]⊗qm (cf. [4]). Under the condition m = 0,
j
there is a one-to-one correspondence between the following two sets for any n≥0:
(i) {λ∈SP ;λhc =∅},
2n
(ii) {(µ,ν)∈SP ×P ;n +2n =n}.
n0 n1 0 1
3. Compound basis
We begin with some bijections between sets of partitions. The first one is
φ:P −→ SP ×P
n n0 n1
n0+[2n1=n
defined by λ 7→ (λr,λd). Here the multiplicities m (λr) and m (λd) of i ≥ 1 are
i i
given respectively by
1 m (λ)≡1 (mod 2)
m (λr)= i
i
(0 mi(λ)≡0 (mod 2),
4 KAZUYAAOKAGE,HIROSHIMIZUKAWAANDHIRO-FUMIYAMADA
and
1(m (λ)−1) m (λ)≡1 (mod 2)
m (λd)= 2 i i
i (12(mi(λ)) mi(λ)≡0 (mod 2).
For example, if λ=(5344271), then λr =(521) and λd =(54223). We set
P =φ−1(SP ×P ).
n0,n1 n0 n1
The second bijection is
ψ :P −→ OP ×P
n n1 n2
n1+[2n2=n
defined by ψ(λ) = (λo,λe). Here λo is obtained by picking up the odd parts
of λ, while λe is obtained by taking halves of the even parts. For example, if
λ=(5344271), then λo =(531) and λe =(2417).
The third bijection is called the Glaisher map. Let λ = (λ ,λ ,...) be a strict
1 2
partitionofn. Supposethatλi =2piqi (i=1,2,...),whereqi isodd. Thenanodd
partition λ˜ of n is defined by
m2j−1(λ˜)= 2pi.
qi=2Xj−1,i≥1
For example, if λ=(8,6,4,3,1),then λ˜ =(33,113). This gives a bijection between
SP and OP .
n n
Proposition 3.1. Let (n ,n ) be fixed. Then we have
0 1
ℓ(λ)= (ℓ(λr)+2ℓ(λd))= (ℓ(λo)+ℓ(λe))= (ℓ(λ˜r)+ℓ(λe)),
λX∈Pn λX∈Pn λX∈Pn λX∈Pn
ℓ(λ)= (ℓ(λr)+2ℓ(λd))= (ℓ(λo)+ℓ(λe)),
λ∈PXn0,n1 λ∈PXn0,n1 λ∈PXn0,n1
2ℓ(λd)= 2ℓ(λe)= (ℓ(λo)+ℓ(λe)−ℓ(λr))= (ℓ(λ˜r)+ℓ(λe)−ℓ(λr)),
λX∈Pn λX∈Pn λX∈Pn λX∈Pn
and
2ℓ(λd)= (ℓ(λo)+ℓ(λe)−ℓ(λr)).
λ∈XPn0,n1 λ∈PXn0,n1
Looking at the representation spaces FHU and FHT, we have the following two
natural bases for the space
F(0) =C[t ;j ≥1]
n j n
consistingofthehomogenouspolynomialsofdegreensubjecttodegt =j. Namely
j
we have
(i) {S (t);λ∈P },
λ n
(ii) {Q (t)S (t′);λ∈P }.
λr λd n
For simplicity we write
W (t)=Q (t)S (t′)
λ λr λd
for λ∈P and call the set (ii) the compound basis for F(0) .
n n
COMPOUND BASIS ARISING FROM THE BASIC A(1)-MODULE 5
1
Ourproblemis todetermine the transitionmatrixbetweenthese twobases. Let
A =(a ) be defined by
n λµ
(1) S (t)= a W (t)
λ λµ µ
µX∈Pn
for λ∈P .
n
′
Here we remark the relation between our basis and the Q -functions. Lascoux,
LeclercandThibon(cf. [7])introducedtheQ′-functionsasthebasisforF(0) dual
n
to P-functions with respect to the inner product
hF(t),G(t)i :=F(∂˜)G(t)| ,
0 t=0
where∂˜=( ∂ ,1 ∂ ,1 ∂ ,...). ForastrictpartitionµweseethatQ′ (t)=Q (2t).
∂t1 2∂t2 3∂t3 µ µ
For a partition λ which is not necessarily strict, we see that
′ ′
Q (t)=Q (2t)h (t )
λ λr λd
where h is the complete symmetric function indexed by λd. Therefore the tran-
λd
′
sition from W to Q is essentially given by the Kostka numbers.
λ µ
4. transition matrices
In the previous section, functions are expressed in terms of the “time variables”
t=(t ,t ,...) of the soliton equations. However,for the description and the proof
1 2
ofourformula,itismoreconvenienttousethe“original”variablesofthesymmetric
functions, i.e., the eigenvalues x=(x ,x ,...).
1 2
The definition (1) of a is rewritten as
λµ
Sλ(x,x)= aλµQµr(x)Sµd(x2),
µX∈Pn
where (x,x)=(x ,x ,x ,x ,...) and x2 =(x2,x2,...). Hereafter we will denote
1 1 2 2 1 2
W (x)=Q (x)S (x2), V (x)=P (x)S (x2).
λ λr λd λ λr λd
Also we set the following spaces of symmetric functions
Λ=C[p (x);r ≥1], Γ=C[p (x);r ≥1,odd],
r r
and
Γ′ =C[p (x);r ≥2,even]
r
so that
Λ∼=Γ⊗Γ′.
We have two bases for Λ:
W =(W (x)) and V =(V (x)) .
λ λ λ λ
First we notice the following Cauchy identity.
Proposition 4.1.
1
= W (x)V (y).
(1−x y )2 λ λ
i j
i,j≥1 λ∈P
Y X
6 KAZUYAAOKAGE,HIROSHIMIZUKAWAANDHIRO-FUMIYAMADA
Proof. We compute
W (x)V (y)= Q (x)S (x2)P (y)S (y2)
λ λ λr λd λr λd
λ∈P λ∈P
X X
= Q (x)P (y) S (x2)S (y2).
µ µ ν ν
µ∈SP ν∈P
X X
Taking the inner products h , i and h , i on Λ, we obtain
−1 0
1+x y
i j
Q (x)P (y)= ,
µ µ
1−x y
i j
µ∈SP i,j
X Y
and
1
S (x2)S (y2)= .
ν ν 1−x2y2
ν∈P i,j i j
X Y
We have
1
W (x)V (y)= .
λ λ (1−x y )2
i j
λ∈P i,j
X Y
(cid:3)
By a standard argument, we have
Corollary 4.2.
hW (x),V (x)i =δ .
λ µ −1 λµ
Theorem 4.3. The matrix A is integral.
n
Proof. We have
1
W (x)V (y)= = S (x,x)S (y).
λ λ (1−x y )2 λ λ
i j
λ∈P i,j λ∈P
X Y X
Taking the inner product h , i with S (y), we obtain
0 µ
S (x,x)= hW (x)V (y),S (y)i
λ µ µ λ 0
µ∈P
X
= hV (y),S (y)i W (x).
µ λ 0 µ
µ∈P
X
Thus we know
a =hV (y),S (y)i .
λµ µ λ 0
The numbers g defined by
µrν
P (y)= g S (y)
µr µrν ν
ν∈P
X
are called the Stembridge coefficients and are known to be non-negative integers.
Also one finds the following formula in [3].
S (y2)= δ(ξ)cµd S (y),
µp ξ[0],ξ[1] ξ
ξ∈P
X
COMPOUND BASIS ARISING FROM THE BASIC A(1)-MODULE 7
1
whereδ(ξ) is the 2-signofξ, (ξ[0],ξ[1])is the 2-quotientofξ (cf. [10]) andcµd
ξ[0],ξ[1]
is the Littlewood-Richardson coefficient. Hence
V (y)=P (y)S (y2)
µ µr µp
= δ(ξ)g cµd S (y)S (y)
µrν ξ[0],ξ[1] ν ξ
ν,ξ
X
= δ(ξ)g cλ cµd S (y).
µrν νξ ξ[0],ξ[1] λ
λ ν,ξ
X X
Therefore
a = δ(ξ)g cλ cµd
λµ µrν νξ ξ[0],ξ[1]
ν,ξ
X
is an integer. (cid:3)
Example 4.4.
(3,∅) (21,∅) (1,1)
(3) 1 0 1
A = (21) 1 1 0
3
(13) 1 0 −1
(4,∅) (31,∅) (∅,2) (∅,12) (2,1)
(4) 1 0 1 0 1
(31) 1 1 −1 0 1
A = (22) 0 1 1 1 0
4
(14) 1 0 0 1 −1
(212) 1 1 0 −1 −1
As for the columns corresponding to (µ,∅) with µ∈SP , entries are non-negative
n
integers. The submatrix consisting of these columns will be denoted by Γ . The
n
entries of Γ are the Stembridge coefficients, whose combinatorialnature has been
n
known ([11], [8]).
Here we recall the definition of decomposition matrices for the p-modular repre-
sentations of the symmetric group S . Let p be a fixed prime number. A par-
n
tition λ = (λ ,λ ,··· ,λ ) is said to be p-regular of there are no parts satisfying
1 2 ℓ
λ = λ = ··· = λ ≥ 1. Note that a 2-regular partition is nothing but a
i i+1 i+p−1
strictpartition. Thesetofp-regularpartitionsofnisdenotedbyPr(p).Apartition
n
ρ = (1m12m2···) is said to be p-class regular if mp = m2p = ··· = 0. Note that a
2-classregular partition is nothing but an odd partition. The set of p-class regular
partitions of n is denoted by Pc(p). The p-Glaisher map λ 7→ λ˜ is defined in a
n
natural way. This gives a bijection between Pr(p) and Pc(p). For λ ∈ Pr(p), we
n n n
define the Brauer-Schur function B(p)(t) indexed by λ as follows.
λ
tm1tm2···
B(p)(t)= ϕλ 1 2 ∈F(0) ,
λ ρm !m !··· n
1 2
ρ∈XPnc(p)
where ϕλ is the irreducible Brauer character corresponding to λ, evaluated at the
ρ
p-regular conjugacy class ρ. These functions form a basis for the space F(p) =
n
8 KAZUYAAOKAGE,HIROSHIMIZUKAWAANDHIRO-FUMIYAMADA
F(p)∩F(0) , where
n
F(p) =C[t ;j ≥1,j 6≡0(modp)}.
j
Given a Schur function S (t), define the p-reduced Schur function S(p)(t) by
λ λ
”killing” all variables t ,t ,···;
p 2p
S(p)(t)=S (t)| .
λ λ tjp=0
These p-reduced Schur functions are no longer linearly independent. All linear
relationsamongthesepolynomialsareknown(cf. [2]). Thep-decompositionmatrix
D(p) =(d ) is defined by
n λµ
S(p)(t)= d B(p)(t)
λ λµ µ
µ∈XPnr(p)
for λ ∈ P , and are known to satisfy the properties; d ∈ Z , d = 0 unless
n λµ ≥0 λµ
µ≥λ and d =1. Here “≥” denotes the dominance order.
λλ
Now let us go back to the case of p=2. By definition, the Stembridge coefficients
γ (λ∈P ,µ∈SP ) appear as
λµ n n
S(2)(t)= γ Q (t).
λ λµ µ
µ∈XSPn
Looking at the matrices D(2) = (d ) and Γ = (γ ), one observes that they
n λµ n λµ
are “very similar”. We consider the Cartan matrix C(2) = tD(2)D(2) and the
n n n
correspondentG =tΓ Γ .Thereisacompactformulafortheelementarydivisors
n n n
of C(2)([12]): 2ℓ(λ˜)−ℓ(λ) for λ∈SP .
n n
Theorem 4.5. The elementary divisors of C(2) and G coincide.
n n
Proof. We put Z˜n = (2ℓ(λ)−2ℓ(ρ)+ǫζρλ)λ∈SPn,ρ∈OPn, Φ(n2) = (ϕλρ)λ∈SPn,ρ∈OPn and
X(2) =(χλ) . The transition matrix T =(t ) is defined by
n ρ λ∈Pn,ρ∈OPn n λµ λ,µ∈SPn
B(2)(t)= t Q (t).
λ λµ µ
µ∈XSP(n)
BydefinitionofΓ andD(2),wehaveX(2) =Γ Z˜ =D(2)Φ(2) andΦ(2) =T Z˜ .
n n n n n n n n n n
Hence, we have Γ =X(2)Z˜−1 =D(2)Φ(2)Z˜−1 =D(2)T .
n n n n n n n n
The matrix Γ has the following properties; γ ∈ Z , γ =0 unless µ≥ λ and
n λµ ≥0 λµ
γ = 1 ([11]). Fix a total order in the set of partitions which is compatible with
λλ
thedominanceorder,andweshallwrite(d ),(γ )and(t )inplaceof(d ),(γ )
ij ij ij λµ λµ
and (t ), respectively. Looking at the first row of D(2)T , we have
λµ n n
δ =γ = d t .
1j 1j 1k kj
k=1
X
This shows that t =δ . As for the second row of D(2)T , we have
1j 1j n n
δ =γ = d t = d t (j ≥2).
2j 2j 2k kj 2k kj
k=1 k=2
X X
This shows that t =δ . Inductively, we can see that T is a lower unitriangular
2j 2j n
integralmatrix. Therforethe matrixD(2) andΓ aretransformedtoeachotherby
n n
COMPOUND BASIS ARISING FROM THE BASIC A(1)-MODULE 9
1
columnoperations. By a standerdargumentwe see thatthe elementarydivisors of
C(2) and G coincide.
n n
(cid:3)
OurtransitionmatrixA =(a ) canberegardedasacommonextension
n λµ λ,µ∈Pn
of the matrix Γ of Stembridge coefficients and the decomposition matrix D(2).
n n
Theorem 4.6.
|detAn|=2kn,
where k = ℓ(λe)= (ℓ(λ˜r)−ℓ(λr)).
n λ∈Pn λ∈Pn
Proof. We hPave four bases Pof F(0) ; S = (S (x)) , S˜ = (S (x,x)) , V =
n λ λ∈Pn λ λ∈Pn
(P (x)S (x2)) and W = (Q (x)S (x2)) . From Corollary 4.2, W and
λr λd λ∈Pn λr λd λ∈Pn
V aredualtoeachotherwithrespecttothe innerproducth, i . Likewise,S˜and
−1
S are dual to each other. Hence we obtain
tM(S,V) M(S˜,W) =I,
n n
where M(S,V) denotes the transition matrix from the basis S to the basis V for
n
F(0) . Since
n
M(S,V) =M(S,S˜) A M(W,V) ,
n n n n
we see that
1
(detA )2 = .
n detM(S,S˜) detM(W,V)
n n
Let X =(χλ) be the character table of S . We put R =diag(z ;ρ∈P ) and
n ρ λρ n n ρ n
L =diag(2ℓ(ρ);ρ∈P ). Then we see that
n n
detM(S,S˜) =detM(S,p) detM(p,S˜)
n n n
=detX R−1detL−1tX
n n n n
=detL−1,
n
and
detM(W,V) = 2ℓ(λr).
n
λY∈Pn
Hence we have
detA2 = 2ℓ(λ)−ℓ(λr) = 22ℓ(λd) = 22ℓ(λe).
n
λY∈Pn λY∈Pn λY∈Pn
(cid:3)
Here is a small list of k .
n
n 1 2 3 4 5 6 7 8 ···
.
k 0 1 1 4 5 11 15 28 ···
n
Next we consider the “Cartan-like” matrix tA A . The Frobenius formula for
n n
W reads
λ
p p = 2−ℓ(λr)XλrχλdW (x)
σ 2ρ σ ρ λ
λ∈PXn0,n1
10 KAZUYAAOKAGE,HIROSHIMIZUKAWAANDHIRO-FUMIYAMADA
for σ ∈OP and ρ∈P , where the Green function Xλ is defined by
n0 n1 σ
Q (x)= 2ℓ(σ)z−1Xλp
λ σ σ σ
σ
X
for λ ∈ SP . This formula shows that the transition matrix M(p,W) is, after a
n n
suitable sorting of rows and columns, decomposed into diagonalblocks, each block
indexed by the pair (n ,n ) with n +2n =n.
0 1 0 1
We have
tA A =tM(p,W) tM(S˜,p) M(S˜,p) M(p,W)
n n n n n n
=tM(p,W) (tL X−1)(X R−1L )M(p,W)
n n n n n n n
=tM(p,W) tL R−1L M(p,W) .
n n n n n
SinceL2R−1isdiagonalmatrix,tA A isblockdiagonalmatrix,eachblockindexed
n n n n
bythepair(n ,n ).LetdenoteB thecorrespondingblockintA A .Notethat
0 1 n0,n1 n n
the “principal” block B is nothing but the matrix G .
n,0 n
Example 4.7.
(3,∅) (21,∅) (1,1)
(3,∅) 3 1 0
tA A = (21,∅) 1 1 0 ,
3 3
(1,1) 0 0 2
(4,∅) (31,∅) (∅,2) (∅,12) (2,1)
(4,∅) 4 2 0 0 0
(31,∅) 2 3 0 0 0
tA A = (∅,2) 0 0 3 1 0 .
4 4
(∅,12) 0 0 1 3 0
(2,1) 0 0 0 0 4
Theorem 4.8.
P (ℓ(λ˜r)+ℓ(λd)−ℓ(λr))
|detBn0,n1|=2 λ∈Pn0,n1 .
Proof. We have
tA A =tM(p,W) tL R−1L M(p,W)
n n n n n n n
=tM(p,W) tL R−1L M(p,p˜) M(p˜,V) M(V,W)
n n n n n n n
=tM(p,W) tL R−1M(p˜,V) M(V,W) ,
n n n n n
where p˜ (x)=p (x,x). Note that tL , R−1, M(V,W) are diagonal matrices. By
ρ ρ n n n
requiring that hp ,p i = 2−ℓ(ρ)z δ , we obtain hp ,p˜i = 2ℓ(σ)−ℓ(ρ)z δ .
ρ σ −1 ρ ρσ ρ σ −1 ρ ρσ
Hence,
det(M(p˜,V) tM(p,W) Z−1)=detI.
n n n
We recall that tA A is block diagonal. We have
n n
P (ℓ(λ˜r)+ℓ(λd)−ℓ(λr))
|detBn0,n1|=2 λ∈Pn0,n1 .
(cid:3)
