Table Of ContentFusion procedure for wreath products of finite groups by the
symmetric group
L. Poulain d’Andecy
3 Mathematics Laboratory of Versailles (CNRS UMR 8100), Versailles-Saint-Quentin University,
1 45 Avenue des Etats-Unis 78000 Versailles, France
0
2
n
a Abstract
J
8 Let G be a finite group. A complete system of pairwise orthogonalidempotents is constructed
2 for the wreath product of G by the symmetric group by means of a fusion procedure, that is by
consecutive evaluations of a rational function with values in the group ring. This complete system
]
T of idempotents is indexed by standardYoung multi-tableaux. Associatedto the wreath productof
R G by the symmetric group, a Baxterized form for the Artin generators of the symmetric group is
defined and appears in the rational function used in the fusion procedure.
.
h
t
a
m 1. Introduction
[
The fusion procedure for the symmetric group Sn originates in [9] and has then been developed in
2
the more general situation of Hecke algebras [1]. The procedure allows to express a complete set
v
9 of primitive idempotents, indexed by standard Young tableaux, of the group ring CS via a certain
n
9 limiting process on a rational function in several variables with values in CS .
n
3
4 In[12],analternativeapproachofthefusionprocedureforthesymmetricgrouphasbeenproposed,
.
1 and is based on the existence of a maximal commutative set in CS formed by the Jucys–Murphy
n
0
elements. More precisely, this approach relies on an expression for the primitive idempotents in terms
3
1 of the Jucys–Murphy elements [10, 13] and transforms this expression into a fusion formula. The
: evaluations of the variables of the rational function are now consecutive. This version of the fusion
v
i procedure has then been generalized to the Hecke algebras of type A [6], to the Brauer algebras [3, 4],
X
to the Birman–Wenzl–Murakami algebras [5], to the complex reflection groups of type G(m,1,n) [14]
r
a and to the Ariki–Koike algebras [15].
Let Gbea finitegroup. Theaim of this article is to generalize thefusion procedure,inthe spiritof
[12], for the wreath productG of the group G by the symmetric group S . Therepresentation theory
n n
of the group G is well-known, see e.g. [8] or [11], and the irreducible representations are labelled
n
by multi-partitions of size n.eAnalogues of the Jucys–Murphy elements for the group G have been
n
introduced in e[16, 17] and used for an inductive approach to the representation theory of the chain of
groups G [16]. e
n
e
1
TheirreduciblerepresentationV ofG correspondingtothemulti-partitionλadmitsthefollowing
λ n
decomposition, as a vector space,
e V = W ,
λ T
where the direct sum is over the set of standardMYoung multi-tableaux of shape λ. The subspaces
W are common eigenspaces for the Jucys–Murphy elements j ,...,j of the group G . Let E be
T 1 n n T
the idempotent of CG associated to the subspace W . The idempotents E , where T runs through
n T T
the set of standard multi-tableaux of size n, form together a ceompletee system of pairweise orthogonal
idempotents of CG .e
n
We state here the main result of the article.
e
Theorem. The idempotent E of CG corresponding to the standard multi-tableau T can be obtained
T n
by the following consecutive evaluations
e
1
E = Φ(u ,...,u ,v ,...,v ) ... ... . (1)
T FGλ Fλ 1 n 1 n (cid:12)vi(α)=ξp(αi), αi==11,,......,,nm (cid:12)u1=cG1 (cid:12)un=cGn
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
The function Φ appearing in (1) is a rational function in several variables with values in the group
ring CG and FG, F are complex numbers.
n λ λ
TheeigenvaluesoftheJucys–Murphyelementsarenotsufficienttodistinguishbetweenthedifferent
e
subspaces W . Therefore the commutative family used here for the fusion procedure is formed by the
T
Jucys–Murphy elements j ,...,j , together with an additional set of elements {g(α)} of CG . The
1 n i n
elements g(α) are images in CG of a set of elements {g(α)} which linearly span the center of CG -
i n
the indices i = 1,...,n inedicateein which copies of CG in CG they belong (see Sections 2 aend 3 for
n
precise definitions). e
The set of variables is split into two parts: the variablees v(α) are first evaluated simultaneously
i
(α) (α)
at complex numbers ξ which are eigenvalues of elements g . These variables correspond to the
ν i
positions of the nodes of a multi-partition (their places in the m-tuple). Then the variables u ,...,u
1 n
areconsecutively evaluated attheeigenvalues of theJucys–Murphyelements j ,...,j andarerelated
1 n
to the classical contents of the nodes of a multi-partition.
e e
The rational function Φ can be written as the product of two functions, the first one containing
(α)
the variables u ,...,u and the second one containing the variables v . As in the fusion procedure
1 n i
for the symmetric groups [12], the function related to the contents (that is, containing the variables
u ,...,u ) is build up from a “Baxterized” form of the generators of S inside CG . However, the
1 n n n
Baxterized form used here is a non-trivial generalization, associated to the finite group G, of the usual
Baxterization for the symmetric groups. e
The coefficient (FGF )−1 appearing in (1) only depends on the multi-partition λ and not on the
λ λ
standard multi-tableau T of shape λ. The element F is the product of the hook lengths of the nodes
λ
in the multi-partition λ and is independent of the group G. The additional factor FG depends on the
λ
numbers ξ(α), and in turn on the choice of the set {g(α)} of central elements of CG.
ν
2
In general the subspaces W are not one-dimensional, and therefore the idempotent E of CG
T T n
is not in general primitive. In fact, the subspaces W are all one-dimensional if and only if G is an
T
Abelian finite group. In this situation the idempotents E , where T runs through the set of standaerd
T
multi-tableauxofsizen,formtogetheracompletesystemofprimitivepairwiseorthogonalidempotents
of CG .
n
For an Abelian finite group G, we provide a simplified fusion procedure for the wreath product
e
G . This simplified version is obtained by replacing, in the function Φ appearing in (1), the set {g(α)}
n
of central elements of CG by a smaller subset.
e If G is the cyclic group of order m then the group G is isomorphic to the complex reflection group
n
of type G(m,1,n). A fusion procedure for the complex reflection group of type G(m,1,n) has been
given in [14]. The fusion procedure presented here, ineits simplified version, is slightly different than
the one in [14], namely the Baxterized forms are different. The Baxterized form used here, when G is
the cyclic group of order 2, has been used in [2] for a fusion procedure for the Coxeter groups of type
B (that is, the complex reflection groups of type G(2,1,n)).
The paper is organized as follows. Sections 2 and 3 contain definitions and notations about the
finite group G and the wreath product G . The Jucys–Murphy elements for the group G are defined
n n
in Section 4. In Section 5 we introduce the Baxterized form, associated to the wreath product G ,
n
for the Artin generators of the symmeetric group and prove that these Baxterized eleements satisfy
the Yang–Baxter equation with spectral parameters. In Section 6, we recall standard results on tehe
representation theory of the groups G and give the formula for the idempotents E in terms of the
n T
(α)
elements j ,...,j and g . The main result of the article, which gives the fusion procedure for the
1 n i
group G , is proved in Section 7. In eSection 8, we consider the case of an Abelian finite group G and
n
provide aesimplifieed fusion formula in this situation.
e
Notation.
We denote by CH the group ring over the complex numbersof a finite group H. For a vector space V,
we denote by Id the identity operator on V. Symbols v and v , for an integer i, stand for m-tuples
V i
of variables, namely v := (v(1),...,v(m)) and v := (v(1),...,v(m));
i i i
2. Definitions
Let G be a finite group and let {C ,...,C } be the set of all its conjugacy classes. We denote by
1 m
g(1),...,g(m) the following central elements of the group ring CG:
1
g(α) := g , α = 1,...,m. (2)
|C |
α
gX∈Cα
Let ρ ,...,ρ bethe pairwise non-isomorphicirreduciblerepresentations of G and W ,...,W be
1 m 1 m
thecorrespondingrepresentation spacesofdimensions,respectively, d ,...,d . Denotebyχ ,...,χ
1 m 1 m
(α)
the associated irreducible characters. We define complex numbers ξ , α,ν = 1,...,m, by:
ν
1
ξ(α) := χ (g(α)) , α,ν = 1,...,m. (3)
ν d ν
ν
3
The central elements g(α) act in the irreducible representations of G as multiples of the identity
operators, namely:
ρ (g(α)) = ξ(α)·Id for α,ν = 1,...,m.
ν ν Wν
It is a standard fact that the elements g(α), α = 1,...,m, span the center of CG and therefore, if
ν 6= ν′, there exists some α ∈ {1,...,m} such that ξ(α) 6= ξ(α); that is, the eigenvalues of the elements
ν ν′
g(α), α = 1,...,m, distinguish between the irreducible representations of G.
Functions g(α)(v). For α = 1,...,m, we define S(α) to be the set formed by the pairwise different
(α)
numbers among the ξ , i = 1,...,m, that is:
i
S(α) := {ξ(α),ξ(α),...,ξ(α)} with 1 =i < i < ··· < i ≤ m ,
i1 i2 ikα 1 2 kα
(α) (α) (α)
such that the numbers ξ ,ξ ,...,ξ are pairwise different and, for any i ∈ {1,...,m}, we have
i1 i2 ikα
ξ(α) = ξ(α) for some a ∈ {1,...,k }. By construction, we have in CG that:
i ia α
g(α) −ξ(α) = 0 for α= 1,...,m. (4)
ξ(α)Y∈S(α)(cid:16) (cid:17)
We define the following rational functions in v with values in CG:
(v−ξ(α))
g(α)(v) := ξ(α)∈S(α) for α = 1,...,m. (5)
v−g(α)
Q
The rational functions g(α)(v) can be rewritten as polynomial functions in v as follows. Fix α ∈
(α) (α) (α)
{1,...,m} and let a ,a ,...,a be the complex numbers defined by
0 1 kα
(X −ξ(α))= a(α) +a(α)X +···+a(α)Xkα ,
0 1 kα
ξ(α)Y∈S(α)
where X is an indeterminate. Define the polynomials a(α)(v), i= 0,...,k , in v by
i α
a(α)(v) = a(α)+a(α) v+···+a(α)vkα−i for i= 0,...,k .
i i i+1 kα α
Then we have that:
kα−1
g(α)(v) = a(α)(v) g(α) i = a(α)(v)+a(α)(v)g(α) +···+a(α)(v) g(α) kα−1 . (6)
i+1 1 2 kα
i=0
X (cid:0) (cid:1) (cid:0) (cid:1)
kα−1
Indeedonecandirectlyverifythat(v−g(α)) a(α)(v) g(α) i = (v−ξ(α)) (intheverification,
i+1
Xi=0 (cid:0) (cid:1) ξ(αY)∈S(α)
it is useful to use the recursive relation a(α)(v) = v−1(a(α)(v)−a(α)) for i = 0,...,k −1, together
i+1 i i α
with the initial condition a(α)(v) = a(α) +a(α)v+···+a(α)vkα).
0 0 1 kα
4
Examples.
− If S(α) = {ξ(α)} then we have g(α)(v) = 1;
− if S(α) = {ξ(α),ξ(α)} then we have g(α)(v) = g(α) +v−ξ(α) −ξ(α);
1 2 1 2
− if S(α) = {ξ(α),ξ(α),ξ(α)} then we have
1 2 3
g(α)(v) = g(α) 2+(v−ξ(α)−ξ(α)−ξ(α))g(α)+v2−v(ξ(α)+ξ(α)+ξ(α))+ξ(α)ξ(α)+ξ(α)ξ(α)+ξ(α)ξ(α) .
1 2 3 1 2 3 1 2 1 3 2 3
(cid:0) (cid:1)
3. Wreath product
Definitions. Let Gn := G×···×G, the Cartesian product of n copies of G. The symmetric group
S on n letters acts on Gn by permuting the n copies of G. We denote the action by σ(a), σ ∈ S
n n
and a ∈ Gn. The wreath product G of the group G by the symmetric group S is the semi-direct
n n
product Gn ⋊S defined by this action. The group G consists of the elements (a,σ), a ∈ Gn and
n n
σ ∈ S , with multiplication given bye
n
e
(a,σ)·(a′,σ′) = (aσ(a′),σσ′), a,a′ ∈ Gn and σ,σ′ ∈ S .
n
The groups G form an inductive chain of group:
n
e {1} ⊂ G1 ⊂ G2 ⊂ ...··· ⊂ Gn−1 ⊂ Gn ⊂ ... , (7)
where the subgroup of Gn isomorpheic toeGn−1 is formedeby theeelements of the form (a,σ), a ∈ Gn−1
and σ ∈ S . This allows to consider elements of CG as elements of CG and we will often do
n−1 n−1 n
this without mentioninge. e
e e
We denote by s ,...,s the following elements of G :
1 n−1 n
si = 1Gn, πi , i = 1,e...,n−1, (8)
(cid:16) (cid:17)
where1Gn istheunitelementofGn andπi isthetranspositionofiandi+1. Theelements s1,...,sn−1
satisfy the following relations:
s2 = 1 for i = 1,...,n−1,
i
s s s = s s s for i = 1,...,n−2, (9)
i i+1 i i+1 i i+1
s s = s s for i,j = 1,...,n−1 such that |i−j| > 1,
i j j i
and generate a subgroup of G isomorphic to the symmetric group S .
n n
For j = 1,...,n, let ̟ be the injective morphism from CG to CGn defined by
j
e
̟ (g) := (1 ,...,1 , g,1 ,...,1 ) for g ∈ CG, (10)
j G G G G
where 1 is the unit element of the group G and, in the right hand side of (10), g is in the j-th
G
position. Let also ι be the natural injective morphism from CGn to CG , given by
n
ι(a) := (a,1Sn) for a ∈ CGn, e
5
where 1 is the unit element of the symmetric group S .
Sn n
For any j = 1,...,n, the composition ι◦̟ is an injective morphism from CG to CG . We will
j n
use the following notation:
e
g := ι◦̟ (g) for g ∈ CG and j = 1,...,n. (11)
j j
For j = 1,...,n, we define, similarly to (5),
(v−ξ(α))
(α) ξ(α)∈S(α)
g (v) := for α= 1,...,m. (12)
j (α)
Q v−g
j
Now consider the following element of the group ring of G×G:
1
e:= (g,g−1) . (13)
|G|
g∈G
X
As noticed in [16], the element e satisfies
(g,h)e = e(h,g) for any g,h ∈ G, (14)
and this implies that e acts as follows in the irreducible representations of G×G:
ρν ⊗ρν′(e) = δdν,ν′ PWν⊗Wν for ν,ν′ = 1,...,m, (15)
ν
where P is the permutation operator of the space W ⊗W (P(u⊗v)= v⊗u for u,v ∈ W ).
Wν⊗Wν ν ν ν
Elements e . For any i,j = 1,...,n such that i 6= j, the map ̟ ⊗̟ is an injective morphism
i,j i j
from the group ring of G×G to CGn, and thus the composition ι◦(̟ ⊗̟ ) is an injective morphism
i j
from the group ring of G×G to CG . We define, for i= 1,...,n, e := 1 and
n i,i
e := ι ̟ ⊗̟ (e) for i,j = 1,...,n such that i 6=j, (16)
i,j i e j
With the notation (11), elem(cid:0)ents e can(cid:1)be written as
i,j
1
e = g g−1 , i,j = 1,...,n .
i,j |G| i j
g∈G
X
By construction, we have e = e for i,j =1,...,n, and
i,j j,i
s e = e s for i,j = 1,...,n and k = 1,...,n−1, (17)
k i,j πk(i),πk(j) k
where we recall that π is the transposition of k and k+1.
k
For 1 ≤ i< j ≤ n, the element ̟ ⊗̟ (e) acts in the irreducible representations of Gn as follows
i j
(this is implied by (15))
δ
ρ ⊗···⊗ρ ̟ ⊗̟ (e) = νi,νj P , for any ν ,...,ν ∈{1,...,m}, (18)
ν1 νn i j d i,j 1 n
νi
(cid:0) (cid:1)
6
where P is the operator on W ⊗···⊗W which permutes the i-th and j-th spaces and acts as
i,j ν1 νn
the identity operator anywhere else, that is
P (u ⊗...u ...u ... ⊗u )= u ⊗...u ...u ... ⊗u , u ∈ W for a = 1,...,n.
i,j 1 i j n 1 j i n a νa
The following relations are consequences of (16) and (18):
e e = e e for 1 ≤ i< j < k < l ≤ n such that k−j > 1,
i,j k,l k,l i,j
e e = e e for i = 1,...,n−1 and k,l = 1,...,n, (19)
i,i+1 k,l πi(k),πi(l) i,i+1
e e e = e e e for i= 1,...,n−2.
i,i+1 i+1,i+2 i,i+1 i+1,i+2 i,i+1 i+1,i+2
4. Jucys–Murphy elements
The Jucys–Murphy elements j ,...,j for the group G are defined [16, 17] by the following initial
1 n n
condition and recursion:
e e e
j = 0 and j = s j s +e s for i = 1,...,n−1. (20)
1 i+1 i i i i,i+1 i
The Jucys–Murphy eelements j1,...,ejn form aecommutative set of elements,
e jkejl = jljk for k,l = 1,...,n; (21)
moreover they satisfy, for k = 1,..e.,en, e e
s j = j s for i= 1,...,n−1 such that i 6= k−1,k,
i k k i
(22)
g j = j g for l = 1,...,n and for any g ∈ G.
l k k l
e e
Recall that, for α =e1,...e,m, the element g(α) is the central element of CG defined by (2), and
that,forl = 1,...,n,theelementg(α) istheimageinCG ofg(α) bytheinjectivemorphismι◦̟ . The
l n l
(α)
relation (21) and the second relation in (22) imply that the set of elements {g , α = 1,...,m, l =
l
e
1,...,n} together with the Jucys–Murphy elements j ,...,j form a commutative set of elements of
1 n
CG .
n
e e
We state here the following Lemma, which will be used later in the proof of the fusion formula in
e
Section 7.
Lemma 1. For l = 1,...,n−1, we have
n−1
j s s ...s = s s ...s j − s s ...s ...s e , (23)
l l l+1 n−1 l l+1 n−1 n l l+1 k n−1 k,n
k=l
X
e e b
where s s ...s ...s stands for the product s s ...s with s removed.
l l+1 k n−1 l l+1 n−1 k
b
7
Proof. We prove the formula (23) by induction on n−l. The basis of induction (for l = n−1) is
j s = s j −e ,
n−1 n−1 n−1 n n−1,n
which comes immediately from the recurrence relation in (20).
e e
For l < n−1,we use j s = s j −e to write
l l l l+1 l,l+1
jlslesl+1...sne−1 = sljl+1sl+1...sn−1−el,l+1sl+1...sn−1 .
The formula (23) follows, using the induction hypothesis and e s ...s =s ...s e . (cid:3)
e e l,l+1 l+1 n−1 l+1 n−1 l,n
5. Baxterized elements
Recall the standard Artin presentation of the symmetric group S : the group S is generated by ele-
n n
ments s ,...s with defining relations as in (9), with s replaced by s . The standard Baxterization
1 n−1 i i
for the generators s ,...,s of S is:
1 n−1 n
1
s (c,c′):= s + for i =1,...,n−1;
i i c−c′
the parameters c and c′ are called spectral parameters.
The elements s ,...,s , defined by (8), of G generate a subgroup isomorphic to the symmetric
1 n−1 n
group S . We define a generalization of the Baxterized elements s (c,c′), associated to the group G ,
n i n
as follows: e
e
s (c,c′):= s + i,i+1 for i =1,...,n−1. (2e4)
i i c−c′
Proposition 2. The Baxterized elements s (c,c′) satisfy the Yang–Baxter equation with spectral pa-
i
rameters:
s (c,c′)s (c,c′′)s (c′,c′′)= s (c′,c′′)s (c,c′′)s (c,c′) for i = 1,...,n−2; (25)
i i+1 i i+1 i i+1
they satisfy also
s (c,c′)s (d,d′) = s (d,d′)s (c,c′) for i,j = 1,...,n−1 such that |i−j| > 1,
i j j i
e2 (26)
s (c,c′)s (c′,c) = 1− i,i+1 for i = 1,...,n−1.
i i (c−c′)2
Proof. The first relation in (26) is immediate using that s s = s s and e e = e e
i j j i i,i+1 j,j+1 j,j+1 i,i+1
for i,j = 1,...,n−1 such that |i−j| > 1 (see (9) and (19)).
The second relation in (26) follows from a direct calculation in which one uses that s e =
i i,i+1
e s for i= 1,...,n−1 (see (17)).
i,i+1 i
To finish the proof of the Proposition, we develop successively both sides of (25) and, using (17),
we move in each term all elements e on the left of the elements s and s . We compare the two
k,l i i+1
results:
8
− The cubic terms in the elements s and s coincide since s s s = s s s .
i i+1 i i+1 i i+1 i i+1
e
i+1,i+2
− In both sides, the coefficient in front of s s is .
i i+1 c′−c′′
e
i,i+1
− In both sides, the coefficient in front of s s is .
i+1 i c−c′
− In the left hand side, the coefficient in front of s is
i
e e e e
i,i+1 i+1,i+2 i,i+2 i,i+1
+ .
(c−c′)(c−c′′) (c−c′′)(c′−c′′)
e e
i,i+1 i+1,i+2
Using the second line in (19), we transform this expression in , which coincides with
(c−c′)(c′−c′′)
the coefficient in front of s in the right hand side (we also use (19) here).
i
− A similar calculation shows that the coefficient in front of s in the right hand side is equal
i+1
e e
i+1,i+2 i,i+1
to , which coincides with the coefficient in front of s in the left hand side.
(c−c′)(c′ −c′′) i+1
− The remaining terms in the left hand side are
e e e e
i,i+2 i,i+1 i+1,i+2 i,i+1
+ ,
c−c′′ (c−c′)(c−c′′)(c′−c′′)
which coincide with the remaining terms
e e e e
i,i+2 i+1,i+2 i,i+1 i+1,i+2
+
c−c′′ (c′−c′′)(c−c′′)(c−c′)
of the right hand side, using the third line in (19). (cid:3)
Remark. For i = 1,...,n−1 and α = 1,...,m, the elements s (c,c′) and g(α)(v) satisfy a certain
i i
limit of the reflection equation with spectral parameters (see for example [7]), namely
s (c,c′)g(α)(c)s g(α)(c′) = g(α)(c′)s g(α)(c)s (c,c′) . (27)
i i i i i i i i
(α)
Indeed, due to (12) and (26) and the fact that e commutes with s and g , the equality (27) is
i,i+1 i i
equivalent to
s (c′,c)(c−g(α))s (c′−g(α)) = (c′−g(α))s (c−g(α))s (c′,c) ,
i i i i i i i i
which is proved by a straightforward calculation. We skip the details and just indicate that one uses
(α) (α) (α)
that g e = g e , which follows from (14) together with the fact that g commutes with
i i,i+1 i+1 i,i+1 i
e .
i,i+1
6. Standard m-tableaux and idempotents of G
n
6.1. m-partitions and m-tableaux
e
1. m-partitions. Letλ ⊢ nbeapartition of n(we shallalso say of size n), thatis, λ =(λ ,...,λ ),
1 k
whereλ ,j = 1,...,k,arepositiveintegers,λ ≥ λ ≥ ··· ≥ λ andthesizeofλis|λ| := λ +···+λ =
j 1 2 k 1 k
n.
9
We identify partitions with their Young diagrams: the Young diagram of λ is a left-justified array
of rows of nodes containing λ nodes in the j-th row, j = 1,...,k; the rows are numbered from top to
j
bottom. Note that the number of nodes in the diagram of λ is equal to n, the size of λ.
Recall that, for a partition λ, a node θ of λ is called removable if the set of nodes obtained from
λ by removing θ is still a partition. A node θ′ not in λ is called addable if the set of nodes obtained
from λ by adding θ′ is still a partition.
An m-partition λ, or a Young m-diagram λ, of size n is an m-tuple of partitions,
λ:= (λ(1),...,λ(m)),
such that the total number of nodes in the associated Young diagrams is equal to n, that is |λ(1)|+
···+|λ(m)|= n.
A pair (θ,k) consisting of a node θ and an integer k ∈ {1,...,m} is called an m-node. The integer
k is called the position of the m-node.
Let λ = (λ(1),...,λ(m)) be an m-partition. An m-node θ = (θ,k) ∈ λ is called removable from λ
if the node θ is removable from λ(k). An m-node θ′ = (θ′,k′) ∈/ λ is called addable to λ if the node θ′
is addable to λ(k′). The set of m-nodes removable from λ is denoted by E (λ) and the set of m-nodes
−
addable to λ is denoted by E (λ). For example, the removable/addable 3-nodes (marked with -/+)
+
for the 3-partition ((cid:3)(cid:3),∅,(cid:3)) are
− + + − +
, ,
+ +
(cid:18) (cid:19)
For an m-node θ lying in the line x and the column y of the k-th diagram, we define p(θ) := k and
c(θ):= y−x. The number p(θ) is the position of θ and the number c(θ) is called the classical content
of the m-node θ.
For an m-partition λ, we define
m
FG := ξ(α) −ξ(α) , (28)
λ p(θ)
!
θY∈λ αY=1(cid:16)ξ(α)Y∈S(α)(cid:0) (cid:1) (cid:17)
ξ(α)6=ξ(α)
p(θ)
where the sets S(α) and the numbers ξ(α), for α,ν = 1,...,m, are defined in Section 2.
ν
2. Hook length. Let λ be a partition and θ be a node of λ. The hook of θ in λ is the set of nodes
of λ consisting of the node θ together with the nodes which lie either under θ in the same column or
to the right of θ in the same row; the hook length h (θ) is the cardinality of the hook of θ in λ.
λ
The definition of the hook length is extended to m-partitions and m-nodes as follows. Let λ =
(λ(1),...,λ(m)) be an m-partition and θ = (θ,k) an m-node of λ. The hook length h (θ) of θ in λ is
λ
the hook length of the node θ in the k-th partition of λ, that is
h (θ) := h (θ) .
λ λ(k)
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