Table Of ContentAN EXPLICIT CONSTRUCTION OF CASIMIR OPERATORS
AND EIGENVALUES : II
H. R. Karadayi and M. Gungormez
Dept.Physics, Fac. Science, Tech.Univ.Istanbul
80626,Maslak, Istanbul, Turkey
7
9 Internet: karadayi@sariyer.cc.itu.edu.tr
9
1
n
Abstract
a
J
3
1
] It is given a way of computing Casimir eigenvalues for Weyl orbits as well as for irreducible represen-
h
tations of Lie algebras. A κ(s) number of polinomials of rank N are obtained explicitly for A Casimir
p N
- operators of order s where κ(s) is the number of partitions of s into positive integers except 1. It is also
h emphasized that these eigenvalue polinomials prove useful in obtaining formulas to calculate weight mul-
t
a tiplicities and in explicit calculations of the whole cohomology ring of Classical and also Exceptional Lie
m algebras.
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I. INTRODUCTION
Ina previouspaper [1]whichwe refer (I)throughoutthe work,we establishthemost general explicit
forms of 4th and 5th order Casimir operators of A Lie algebras. By starting from this point, we want
N
to develop a framework which makes possible to calculate, for the irreducible representations of A Lie
N
algebras, the eigenvalues of Casimir operators in any order. Extensions are also possible to any other
classical or exceptional Lie algebra because any Lie algebra has always an appropriate subalgebra of type
A .
N
For a Casimir operator I(s) of degree s, the eigenvalues for a D-dimensional representation are known
to be calculated in the following form:
1
Trace(I(s)) . (I.1)
D
A direct calculation of (I.1) could become problematic in practice as the dimension of representation grows
high. Additionally to the ones given in (I), we give here some further works [2] dealing with this problem.
Asecondessentialproblemarisenhereisduetothefactthatonemustalsocalculateweightmultiplicities
for representations comprising more than one Weyl orbit. This latter problem is known to be solved by
formulaswhich are due to Kostantand Freudenthal[3]and it is at the rootofWeyl-Kac characterformulas
[4]. Although they are formally explicit, these two formulas are of recursive character and hence they
exhibit problems in practical calculations. One could therefore prefers to obtain a functional formula in
calculating weight multiplicities. This will be dealt in a subsequent paper.
It is known, on the other hand, that trace operations can be defined [5] in two equivalent ways one
of which is nothing but the explicit matrix trace. An expression like (I.1) could therefore not means for a
Weyl orbit, in general. We instead want to extend the concept of Casimir eigenvalue to Weyl orbits. As
we have introduced in an earlier work [6], we replace (I.1) with the following formal definition:
chs(Π)≡ X(µ)s (I.2)
µ∈Π
where Π is a Weyl orbit and the sum is overall weights µ included within Π. The powers of weights in (I.2)
are to be thought of as s-times products
s times
(µ)s =µz ×µ×}|...×µ{ .
Noteherethat(I.2)isdefinednotonlyforWeylorbitsorrepresentationsbutitmeansalsoforanycollection
of weights. We will mainly show in what follows how (I.2) gives us a way to obtain eigenvalues of a Casimir
operator. Due to a permutational lemma given in section (II), the procedure works out especially for
A Lie algebras. It will however be seen in a subsequent paper that it is generalized to any Classical or
N
Exceptional Lie algebra. In section (III), we will give a general formula of calculating ch (Π) by the aid
s
of this permutational lemma. An efficient way of using this formula is due to reduction rules which are
explained in section (IV) and the polinomials representing Casimir eigenvalues will be given in section (V)
and also in appendix.2. We will show in section (VI) that the two formula (I.1) and (I.2) are in fact in
coincidence.
II. A PERMUTATIONAL LEMMA FOR A WEYL ORBITS
N
Inthissection,wegive,forA Lie algebras,apermutationallemma whichsaysthat,modulo permu-
N
tations, there is one-to-one correspondence between the Weyl chamber and the Tits cone [7].
As will be explained below, such a correspondence appears only when one reformulates everything in terms
of the so-called fundamental weights.
Foranexcellentstudy ofLiealgebratechnologywereferthebookofHumphreys [8]. We give,however,
some frequently used concepts here. In describing the whole weight lattice of a Lie algebra of rank N,
the known picture will be provided by simple roots α and fundamental dominant weights λ where
i i
3
indiceslikei ,i ,..takevaluesfromthesetI ≡{1,2,..N}. AnydominantweightΛ+ canthenbeexpressed
1 2 ◦
by
N
Λ+ =Xri λi , ri ∈Z+ (II.1)
i=1
where Z+ is the set of positive integers including zero. We know that a Weyl orbit Π is stable under the
actions of Weyl group of Lie algebra. This means that all weights within a Weyl orbit are equivalent under
the actions ofWeyl groupandthey canbe obtainedfrom anyone ofthem by performingWeyl conjugations
one-by-one. We thus obtain a description of the whole weight lattice of which any weight is given by
µ=m λ +m λ +...+m λ , ±m ∈Z+ . (II.2)
1 1 2 2 N N i
Our way of thinking of a Weyl orbit is, on the other hand, based on the fact that Weyl reflections
can be replaced by permutations for A Lie algebras. It is seen in the following that essential figures
N
for this are fundamental weights µI which we introduced [9] some fifteen years ago:
µ ≡λ
1 1
(II.3)
µ ≡µ −α , i=2,3,..N +1.
i i−1 i−1
Indices like I ,I ,... take values from the set S ≡ {1,2,..N,N +1}. Recall here that the weights defined
1 2 ◦
in (II.3) are nothing but the weights of (N+1)-dimensionalfundamental representationsof A Lie algebras.
N
To prevent confusion, note here that some authors prefer to call λ ’s fundamental weights.Thoughthere are
i
N+1 number of fundamental weights µ , they are not completely linear independent due to the fact that
I
their sum is zero. The main observation is, however, that (II.2) replaces with
µ=q µ +q µ +..+q µ (II.4)
1 I1 2 I2 N+1 IN+1
when one reformulates in terms of N+1 fundamental weights. The conditions
I 6=I 6=..6=I (II.5)
1 2 N+1
must be taken into account for each particular weight (II.4) and one can always assume that
q ≥q ≥...≥q ≥0 . (II.6)
1 2 N+1
(II.6) receives here further importance in the light of following lemma:
Let P(N) be the weight lattice of A Lie algebra. A dominant weight Λ+ ∈P(N) has always the form
N
of
Λ+ =q µ +q µ +..+q µ (II.7)
1 1 2 2 N+1 N+1
andhencethewholeWeylorbitΠ(Λ+)isobtainedbypermutationsof(II.7)overN+1fundamentalweights.
In the basis of fundamental weights all weights of the Weyl orbit Π(Λ+) are thus seen in the common form
(II.4) where all indices I take values from the set S together with the conditions (II.5).
k ◦
Although it is not in the scope of this work,demonstrationof lemma is a direct resultof the definitions
(II.3). It will be useful to realize the lemma further in terms of (N+1)-tuples which re-define (II.7) in the
form
Λ+ ≡(q ,q ,..q ) . (II.8)
1 2 N+1
Then every elements µ∈Π(Λ+) corresponds to a permutation of q′s:
i
µ=(q ,q ,..,q )
I1 I2 IN+1
.
To this end, let us choose a weight
−λ +2 λ −λ +λ +λ −λ +λ (II.9)
1 2 3 4 5 6 7
4
which is expressed in the conventional form (II.2). By taking inverses
λ ≡µ +µ +...+µ , i∈I (II.10)
i 1 2 i ◦
of (II.2), we can re-express (II.9) as
2µ +3 µ +µ +2 µ +µ +µ (II.11)
1 2 3 4 5 7
which says us that
−λ +2 λ −λ +λ +λ −λ +λ ∈Π(λ +λ +λ ) .
1 2 3 4 5 6 7 1 3 6
It is obvious that this last knowledge is not so transparent in (II.9).
One must further emphasize that the lemma allows us to know the dimensions of Weyl orbits directly
from their dominant representatives. For this and further use, let us re-consider (II.1) in the form
Λ+ ≡u λ +u λ +...+u λ , u ∈Z+−0 (II.12)
1 i1 2 i2 σ iσ σ
with
i ≤i ≤...≤i , σ =1,2,..N. . (II.13)
1 2 σ
Then, it is seen that the number of weights within a Weyl orbit Π(Λ+) is
(N +1)!
dimΠ(Λ+)= (II.14)
ξ(Λ+) (N +1−i )!
σ
where
σ
ξ(Λ+)≡ Y(ij −ij−1)! , i0 ≡0 . (II.15)
j=1
We therefore assume in the following that dimΠ(Λ) is always known to be a polinomial of rank N.
III. EIGENVALUES FOR WEYL ORBITS
Asismentionedabove,eigenvaluesare,infact,knowntobedefinedforrepresentations. Arepresentation
R(Λ+) is, on the other hand, determined from its orbital decomposition:
R(Λ+)=Π(Λ+) + X m(λ+ <Λ+) Π(λ+) (III.1)
λ+∈Sub(Λ+)
where Sub(Λ+) is the set of all sub-dominant weights of Λ+ and m(λ+ <Λ+)’s are multiplicities of weights
λ+ within the representation R(Λ+). Once a convenient definition of eigenvalues is assigned to Π(λ+) for
λ+ ∈ Sub(Λ+), it is clear that this also means for the whole R(Λ+) via (III.1). In the rest of this section,
we then show how definition (I.2) can be used to obtain orbit eigenvalues as N-dependent polinomials.
Letusnowmakesomedefinitionswhichareusedfrequentlyfordescriptionofsymmetric polinomials
encounteredinthe rootexpansionswhichtakeplaceheavilyinthe recentlystudiedelectromagneticallydual
supersymmetrictheories [10]. These will, ofcourse,be givenhere interms offundamentalweights µ . The
I
essential role will be played by generators
N+1
µ(s)≡ X(µI)s , s=1,2,... (III.2)
I=1
and their reductive generalizations
N+1
µ(s1,s2,..,sk)≡ X (µI1)s1(µI2)s2... (µIk)sk . (III.3)
I1,I2,..Ik=1
5
For (III.3), the conditions
s ≥s ≥...≥s (III.4)
1 2 k
are alwaysassumed and no two of indices I ,I ,..I shall take the same value for eachparticular monomial.
1 2 k
Note also that µ(s,0,0,..0)=µ(s).
Asthefirststep,wenowmakethesuggestion,inviewof(I.2),thatorbiteigenvaluescanbeconveniently
calculatedbydecomposingch (Π)intermsofquantitiesdefinedin(III.3)andthisprovidesusthepossibility
s
to calculate orbit eigenvalues with the same ability regardless
( i) the rank N of algebra,
( ii) the dimension dimR(Λ+,N) of irreducible representation,
(iii) the order s of Casimir element.
To give our results below, we will assume that the set
s/k≡{s ,s ,...,s } (III.5)
1 2 k
represents, via (III.4), all partitions
s=s +s +...+s , s≥k
1 2 k
ofpositiveintegerstok-numberofpositiveintegerss ,s ,..s . Itisusefultoremarkherethateachparticular
1 2 k
partition participating within a s/k gives us, modulo (N+1), a dominant weight in P(N) and the whole
subdominant chain Sub(s λ ) is in one-to-one correspondence with the partitions within a s/k. This must
1
always be kept in mind in the following considerations.
On the other hand, instead of (II.1), it is crucial here to use (II.7) in the form
σ
Λ+ ≡X qi µi (III.6)
i=1
where σ = 1,2,..N +1. Note here that this is another form of (II.12). Due to permutational lemma given
above,wenowknowthatallweightsofaWeylorbitarespecifiedwiththesameparametersq ,(i=1,2,..σ).
i
It is only of this fact which allows us to obtain the following formula in expressing orbital eigenvalues:
σ
1
Ωs(q1,q2,...,qσ,N)= X(N +1−k)! ξ(s/k) Factors(s/k) (III.7)
(N +1−σ)!
k=1
where we define, for all possible partitions (s/k),
Factors(s/k)≡M(s ,s ,...,s ) q(s ,s ,...,s ) µ(s ,s ,...,s ) (III.8)
1 2 k 1 2 k 1 2 k
and the multinomial
(s +s +...+s )!
1 2 k
M(s ,s ,...,s )≡
1 2 k
s !s !...s !
1 2 k
together with the condition that
M(s ,s ,...,s )≡0 for s<k. (III.9)
1 2 k
ξ(s/k)hereisdefinedasin(II.15)because,asweremarkjustabove,anypermutationwithinas/kdetermines
a dominant weight. As in exactly the same way in (III.3), we also define
σ
q(s1,s2,..,sk)≡ X (qI1)s1(qI2)s2... (qIk)sk . (III.10).
s1,s2,..sk=1
After all, one obtains a direct way to compute (I.2) in the form
1
ch (Λ+,N)= Ω (q ,q ,...,q ,N) (III.11)
s ξ(Λ+) s 1 2 k
forallq ≥q ≥..≥q . Forcaseswhichweconsiderinthiswork,wewillgiveinappendix.1someexemplary
1 2 k
expressions extracted from (III.7).
6
IV. REDUCTION FORMULAS
Althoughithasanexplicitform,thesimplicityofformula(III.7)isnotsotransparenttoanexperienced
eyelooking forits advancedapplications. This point canbe recoveredby recursivelyreducing the quantities
(III.9) up to generators µ(s) defined in (III.2). We call these reduction rules. We will only give the ones
which we need in the sequel. It would however be useful to mention about some of their general features.
As is known, elementary Schur functions Sk(x) are defined by expansions
∞
X Sk(x) zk ≡expX xk zk (IV.1)
k∈Z+ k=1
with the following explicit expressions:
xk1 xk2
Sk(x)= X 1 2 ... , k>0 . (IV.2)
k ! k !
1 2
k1+2 k2+3 k3..=k
The complete symmetric functions h (µ ,µ ,..µ ) are defined, on the other hand, by
k 1 2 N
N
1
Y ≡Xhk(µ1,µ2,..µN) zk . (IV.3)
(1−z µ )
i
i=1 k≥0
It can be easily shown that the known equivalence
h (µ ,µ ,..µ )≡S (x) (IV.4)
k 1 2 N k
is now conserved by the reduction rules with the aid of a simple replacement
µ(s)→s x .
s
A simple but instructive example concerning (IV.4) for k=4 is
h (µ ,µ ,µ ,µ )=µ(4)+µ(3,1)+µ(2,2)+µ(2,1,1)+µ(1,1,1,1) (IV.5)
4 1 2 3 4
with the corresponding reduction rules
1 1 1 1 1
q(1,1,1,1)= q(1)4− q(1)2 q(2)+ q(2)2+ q(1) q(3)− q(4) ,
24 4 8 3 4
1 1
q(2,1,1)= q(1)2q(2)− q(2)2−q(1) q(3)+q(4) ,
2 2 (IV.6)
q(3,1)=q(1) q(3)−q(4) ,
1 1
q(2,2)= q(2)2− q(4) .
2 2
For other cases of interest, the reduction rules will be given in appendix.1 respectively for the partitions of
5,6 and 7.
V. EXISTENCE OF EIGENVALUE POLINOMIALS
After all these preparations,we are now in the position to bring out the mostunexpected partof work.
This is the possibility to extend (III.11) directly for irreducible representations as well as Weyl orbits. We
willshowinasubsequentworkthatthisgivesusthepossibilitytoobtaininfinitelymanyfunctionalformulas
to calculate weight multiplicities and also to make explicit calculations of nonlinear cohomology relations
which are known to be exist [11] for classical and exceptional Lie algebras.
7
In view of the fact that µ(1)≡0, one can formally decompose (III.11) in the form
chs(Λ+,N)≡Xcofs1s2..sk(Λ+,N) µ(s1)µ(s2)..µ(sk) (V.1)
s/k
and this allows us to define a number of polinomials
cof (Λ+,N) dimR(λ ,N)
P (Λ+,N)≡ s1s2..sk k P (λ ,N) . (V.2)
s1s2..sk cof (λ ,N) dimR(Λ+,N) s1s2..sk k
s1s2..sk k
Note here that
cof (λ ,N)≡0 , i<k (V.3)
s1s2..sk i
and also
dimR(λ ,N)=M(N +1,i) , i=1,2,..N. . (V.4)
i
To proceed further, we will work on the explicit example of 4th order for which (V.1) and (V.2) give
ch (Λ+,N)≡cof (Λ+,N) µ(4)+cof (Λ+,N) µ(2)2 , (V.5)
4 4 22
cof (Λ+,N) dimR(λ ,N)
P (Λ+,N)≡ 4 1 P (λ ,N) , (V.6)
4 cof (λ ,N) dimR(Λ+,N) 4 1
4 1
and
cof (Λ+,N) dimR(λ ,N)
P (Λ+,N)≡ 22 2 P (λ ,N) (V.7)
22 cof (λ ,N) dimR(Λ+,N) 22 2
22 2
Ndependencesareexplicitlywrittenabove. Themainobservationhereistochangethevariablesr of(II.1):
i
1 + r ≡θ − θ (V.8)
i i i+1
and to suggest the decompositions
P (Λ+,N)=k (1,N) Θ(4,Λ+,N) +
4 4
k (2,N) Θ(2,Λ+,N)2 +
4
k (3,N) Θ(3,Λ+,N) + (V.9)
4
k (4,N) Θ(2,Λ+,N) +
4
k (5,N)
4
and
P (Λ+,N)=k (1,N) Θ(4,Λ+,N) +
22 22
k (2,N) Θ(2,Λ+,N)2 +
22
k (3,N) Θ(3,Λ+,N) + . (V.10)
22
k (4,N) Θ(2,Λ+,N) +
22
k (5,N)
22
As in (III.2) or (III.10), we also define here the generators
N+1
Θ(s,Λ+,N)≡ X (θi)s . (V.11)
i=1
It is seen then that (V.9) and (V.10) are the most general forms compatible with Θ(1,Λ+,N)≡0. What is
significanthere is the possibility to solveequations(V.6) and(V.7)in view ofassumptions(V.9) and(V.10)
8
but with coefficients k (α,N),k (α,N) which are independent of Λ+ for α=1,..,5. By examining for
4 22
a few simple representations, one can easily obtain the following non-zero solutions for these coefficients:
720
k (1,N)= (N2+2N +2) k (5,N)
4 4
g (N)
4
(V.14)
720
k (2,N)=− (2 N2+4 N −1) k (5,N)
4 4
g (N) (N +1)
4
and
1440
k (1,N)=− (2 N2+4 N −1) k (5,N)
22 22
g (N)
22
720
k (2,N)= (N4+4 N3−8 N +13) k (5,N) (V.15)
22 22
g (N) (N +1)
22
120
k (4,N)=− (N −2) (N −1) (N +1)2 (N +3) (N +4) k (5,N)
22 22
g (N)
22
where
4
g4(N)≡ Y (N +i)
(V.16)
i=−2
g (N)≡g (N) (5 N2+10 N +11) .
22 4
The calculationsgoesjust inthe samewayfor orders5,6and7 andhence we directlygive oursolutions
in appendix.2.
VI. CONCLUSIONS
In (I), we have obtained the most generalformal operatorsrepresenting 4th and also 5th order Casimir
invariantsofA Liealgebras. Bycomparingwiththeonesappearinginlitterature,theyarethemostgeneral
N
in the sense that both are to be expressedin terms of two free parameters. As is shownin (I), all coefficient
polinomials of 4th order Casimir operators are expressed in terms of u(1) and u(2) while those of 5th order
Casimirs are v(1) and v(2). As is also emphasized there, the existence of two free parameters for both cases
can be thought of as related with the partitions 4=2+2 and 5=3+2. Recall here the polinomials P and
4
P . This gives us the possibility to calculate the trace forms (I.1) directly in any matrix representation of
22
A Liealgebras. Thesetracecalculationsarestraigtforwardandshowthateigenvaluesof4thorderCasimir
N
operatorshavetheformofanexplicitpolinomialwhichdependsontherankNandtwofreeparametersu(1)
and u(2). It is thus seen that there are always appropriate choices of parameters u(1) and u(2) in such a
waythat this same polinomialreproducesP (Λ+,N) orP (Λ+,N)asgivenin(V.9) and(V.10). The same
4 22
is also true for 5th order Casimirs. With the appropriate choice
1
k (5,N)≡ (N +1)2 (N +2) (N +3) (N +4) (VI.1)
4
6!
in (V.9) it is sufficient to take
3 N −8
u(1)=1 , u(2)= (VI.2)
3 N
in order to reproduce
1
Trace(I(4))≡P (Λ+,N)
4
D
with dimR(Λ+,N)=D. The data for other cases of interest are
1
k (5,N)≡ (5 N2+10 N +11) (N +1) (N +2) (N +3) (N +4)
22
6!
(VI.3)
2 2 N2+N +2
u(1)=1 , u(2)=
3 N (N +1)
9
for
1
Trace(I(4))≡P (Λ+,N) ,
22
D
and
(N +1) (N2+2 N −1)
k (2,N)≡−5
5
N (N −1) (N −2) (N −3)
(VI.4)
2 N −5
v(1)=1 , v(2)=
2 N
for
1
Trace(I(5))≡P (Λ+,N) ,
5
D
and
1 (N +1)3 (N +4) (N +5)
k (5,N)≡−
32
12 N (N −1)
(VI.5)
(11 N +5) (N −1)
v(1)=1 , v(2)=
10 N (N +1)
for
1
Trace(I(5))≡P (Λ+,N) .
32
D
Now it is clear that, this would be a direct evidence for equivalence between the formal expressions
(I.1)and(I.2). Inresult,itisseenthatonecanobtainκ(s)numberofdifferentpolinomialsP (Λ+,N)
s1,s2,..sk
representing eigenvalues of A Casimir operators I(s) of order s, with κ(s) is the number of partitions of s
N
to allpositive integers except1. As is knownfrom (I), this is just the number of free parametersto describe
the most general form of I(s).
REFERENCES
[1] Karadayi H.R and Gungormez M: Explicit Construction of Casimir Operators and Eigenvalues:I , sub-
mitted to J.Math.Phys.
[2] Braden H.W ; Jour.Math.Phys. 29 (1988) 727-741and 2494-2498
Green H.S and Jarvis P.D ; Jour.Math.Phys. 24 (1983) 1681-1687
[3] Kostant B. ; Trans.Am.Math.Soc. 93 (1959) 53-73
Freudenthal H. ; Indag.Math. 16 (1954) 369-376 and 487-491
Freudenthal H. ; Indag.Math. 18 (1956) 511-514
[4] Kac.V.G ; Infinite Dimensional Lie Algebras, 3rd edition, Cambridge University Press
[5] the paragraph(3.29) in Carter, R.W: Simple Groups of Lie Type, J.Wiley and sons (1972) N.Y
[6] KaradayiH.R, Jour.Math.Phys. 25 (1984) 141-144
[7] the section 3.12 in ref.4
[8] Humphreys J.E: Introduction to Lie Algebras and Representation Theory , Springer-Verlag (1972) N.Y.
[9] KaradayiH.R: Anatomy of Grand Unifying Groups , ICTP preprints (unpublished), IC/81/213and 224
[10]KutasovD,SchwimmerAandSeibergN:Chiralrings,SingularityTheoryandElectric-MagneticDuality
, hep-th/9510222
[11] Borel A and Chevalley C: Mem.Am.Math.Soc. 14 (1955) 1
Chih-TaYen: SurLesPolynomesdePoincaredesGroupesdeLieExceptionnels,ComptesRendueAcad.Sci.
Paris (1949) 628-630
Chevalley C: The Betti Numbers of the Exceptional Simple Lie Groups, Proceedings of the International
Congress of Mathematicians, 2 (1952) 21-24
Borel A: Ann.Math. 57 (1953) 115-207
Coleman A.J: Can.J.Math 10 (1958) 349-356
10
APPENDIX. 1
In this work, we consider the calculation of eigenvalues for A Casimir operators of orders s=4,5,6,7.
N
It is however apparent that all our results are to be accomplished as in exactly the same way and with the
same ability for all orders. The following applications of the formula (III.7) will be instructive for all other
cases of interest:
1
Ω (q ,N)= (
4 1
(N +1−1)! (A1.1)
1! (N +1−1)! M(4) q(4) µ(4) ) ,
1
Ω (q ,q ,N)= (
4 1 2
(N +1−2)!
1! (N +1−1)! M(4) q(4) µ(4) + (A1.2)
1! (N +1−2)! M(3,1) q(3,1) µ(3,1) +
2! (N +1−2)! M(2,2) q(2,2) µ(2,2) ) ,
1
Ω (q ,q ,q ,N)= (
4 1 2 3
(N +1−3)!
1! (N +1−1)! M(4) q(4) µ(4) +
(A1.3)
1! (N +1−2)! M(3,1) q(3,1) µ(3,1) +
2! (N +1−2)! M(2,2) q(2,2) µ(2,2) +
2! (N +1−3)! M(2,1,1) q(2,1,1) µ(2,1,1) ) ,
and for k≥4
1
Ω (q ,q ,..,q ,N)= (
4 1 2 σ
(N +1−σ)!
1! (N +1−1)! M(4) q(4) µ(4) +
1! (N +1−2)! M(3,1) q(3,1) µ(3,1) + (A1.4)
2! (N +1−2)! M(2,2) q(2,2) µ(2,2) +
2! (N +1−3)! M(2,1,1) q(2,1,1) µ(2,1,1) +
4! (N +1−4)! M(1,1,1,1) q(1,1,1,1) µ(1,1,1,1)) .
Ontheotherhand,foraneffectiveapplicationof(III.7),itisclearthatoneneedstoreducethegenerators
q(s ,s ,..s ) in terms of q(s)’s. Following ones are sufficient within the scope of this work. Together with
1 2 k
the condition that µ(1)≡0, the similar ones are valid also for µ(s ,s ,..s )’s:
1 2 k
q(4,1)=q(1) q(4)−q(5)
q(3,2)=q(2) q(3)−q(5)
1
q(3,1,1)= (q(1)2 q(3)−q(2) q(3)−2q(1) q(4)+2q(5))
2
1
q(2,2,1)= (q(1) q(2)2−2 q(2) q(3)−q(1) q(4)+2 q(5))
2
(A1.5)
1
q(2,1,1,1)= (q(1)3 q(2)−3 q(1) q(2)2−3 q(1)2 q(3)+
6
5 q(2) q(3)+6 q(1) q(4)−6 q(5))
1
q(1,1,1,1,1)= (q(1)5−10 q(1)3 q(2)+15 q(1) q(2)2+20 q(1)2 q(3)−
120
20 q(2) q(3)−30 q(1) q(4)+24 q(5))
