The Algebra of Graph Invariants - Lower and Upper Bounds for Minimal Generators

0.28 MB·English

by  Tomi Mikkonen

The Algebra of Graph Invariants - Upper and
Lower Bounds for Minimal Generators
8
Tomi Mikkonen
0
0 Tampere University of Technology
2 Department of Signal Processing
n P.O.BOX 553
a FIN-33101 TAMPERE
J FINLAND

0 tomi.mikkonen@tut.fi
3
and

]
O
Xavier Buchwalder
C Universit´e de Lyon
. Institut Camille Jordan UMR 5208
h
t Universit´e Lyon 1
a
43 boulevard du 11 Novembre 1918
m
69622 Villeurbanne cedex
[
FRANCE

2 [email protected]
v
2
4 Version: February 2, 2008
1
0 Inthispaperwe studythealgebra of graph invariants, focusing mainly on theinvari-
. ants of simple graphs.
2
All other invariants, such as sorted eigenvalues, degree sequences and canonical per-
1
mutations, belong to this algebra. In fact, every graph invariant is a linear combination
7
0 of thebasic graph invariants which we study in thispaper.
: To prove that two graphs are isomorphic, a number of basic invariants are required,
v
which are called separator invariants. The minimal set of separator invariants is also the
i
X minimal basic generator set for the algebra of graph invariants.
We find lower and upper bounds for the minimal number of generator/separator in-
r
a variants needed for proving graph isomorphism.
Finally we find a sufficient condition for Ulam’s conjecture to be true based on Red-
field’s enumeration formula.
1. INTRODUCTION
Let G=(V,E) and H =(V,F) be simple graphs (i.e. unoriented, no loops, no
multiple edges) with n vertices, where V is the common set of vertices and E,F
are the sets of edges. We say that G is a subgraph of H if E ⊆F. Two graphs G
and H are isomorphic, denoted by G ∼= H, if there exists a permutation π of the
set of vertices such that πE =F.
In this paper we study basic graph invariants which count the number of sub-
graphs isomorphic to G in H. We denote by I(G)(H) the number of subgraphs
isomorphic to G in the graph H. Graphs are denoted by monomials a .
(i,j)∈E ij
Q
1
Thus for instance I(a )(a a a )=3 and I(a a )(a a a )=3. The defini-
12 23 24 34 12 23 23 24 34
tion of I(G)(H) depends only on the isomorphism classes of G and H.
Let A be the adjacency matrix of the graph H, i.e. a = 1 if there is an edge
ij
between vertices i and j in H and a = 0 otherwise. Because H is an unoriented
ij
graph we have a = a . Then I(G)(H) is a function in the variables a and can
ij ji ij
be written explicitly as
1
I(G)(H)= a a ···a . (1)
|Stab(a a ···a )| ρ(i1)ρ(j1) ρ(i2)ρ(j2) ρ(id)ρ(jd)
i1j1 i1j2 idjd ρX∈Sn
Here the (i j )-pairscorrespondto the edgesinsome labelingofthe graphG. The
k k
stabilizer is
Stab(a ···a )={ρ∈S |a ···a =a ···a } (2)
i1j1 idjd n ρ(i1)ρ(j1) ρ(id)ρ(jd) i1j1 idjd
with respect to the symmetric group S , where two monomials are considered the
n
same if they have the same variables. The use of the stabilizer in equation (1)
guarantees that the coefficient of each monomial in the sum is one. Note that
every monomial is either 1 or 0 depending on whether the monomial is contained
inH. Thetotaldegreedofthepolynomial,denotedalsoas|G|,correspondstothe
number of edges in G. Examples of these so-called orbit sums are shown below.
The sum in (1) clearly permutes the monomial a ···a in every possible
i1j1 idjd
location in the vertex set V. Thus I(G)(H) depends only on the isomorphism
classes of the G and H, i.e. it is invariant with respect to the labeling of the
vertices.
Consequently, in the definition of I(G)(H) we can represent the graph G using
vertex-edgesets,theadjacencymatrix,amonomialinthevariablesa oragraphic
ij
representation. Forexamplethevertex-edgeset({1,2,3,4},{{1,2},{1,3},{2,3}}),
the adjacency matrix
0 1 1 0
1 0 1 0
 ,
1 1 0 0
 0 0 0 0 
 
 
the monomial a a a and Figure 1 represent the same graph.
12 13 23
Figure 1 Graphic representation.
We canexpressthe sum (1) without divisionby using the quotientof groupsas
follows
I(G)(H)= a a ···a . (3)
ρ(i1)ρ(j1) ρ(i2)ρ(j2) ρ(id)ρ(jd)
ρ∈Sn/Stab(aXi1j1ai1j2···aidjd)
This representation remains valid in fields of finite characteristic and we will use
this as the definition of I(G)(H).
We present examples mostly in the algebra C[a ]/ha2 −a ,a −a i Sn but
ij ij ij ij ji
allresults generalizedirectly to generalpermutationgroupsG. Also generalization
(cid:0) (cid:1)
2
to (⊗ Zn)G is quite obvious and only partially presented here as the purpose of
d +
this paper is to provide an invariant theoretical view to classical graph theory.
Example 1. Choose d = 1, (i j ) = (12), n = 4, corresponding to G = a .
1 1 12
The stabilizer for a is Stab(a )={(1234),(2134),(1243),(2143)}and
12 12
S /Stab(a )={(1234),(1324),(1423),(2314),(2413),(3412)}. (4)
n 12
Theorbitsum(3)becomesinthiscaseI(a )(H)=a +a +a +a +a +a
12 12 13 14 23 24 34
and it calculates the number of edges in a graph with 4 vertices. It is clearly
invariant with respect to all permutations of vertices.
Example 2. Choose d=2, (i j )=(12), (i j )=(13), n=4.
1 1 2 2
TheinvariantI(a a )(H)=a a +a a +a a +a a +a a +a a +
12 13 12 13 12 14 12 23 12 24 13 14 13 23
a a +a a +a a +a a +a a +a a calculatesthenumberofsubgraphs
13 34 14 24 14 34 23 24 23 34 24 34
isomorphic to a a in the graph H. Any permutation of vertices affects only the
12 13
order of summation.
We call the polynomials I(G)(H) basic graph invariants of type G. The basic
graph invariants I(G)(H) are polynomials in the variables a and depend on H
ij
only through the values of these variables. Thus we may consider the basic graph
invariantsassymbolicpolynomialsina andweoftendropthesecondgraph(H in
ij
I(G)(H))fromthenotation. WeusethenotationI(A)forthissymbolicpolynomial,
where A is some monomial in the orbit sum.
In[3]Fleischmanndescribesageneralformulafortheproductoftwoorbitsums
in a graded algebra. In this paper we will modify this product formula so that it
calculates the product of two basic graph invariants, i.e.
I(A)·I(B)= ck I(G ), (5)
A,B k
k
X
asalinearcombinationofbasicgraphinvariantsI(G ). Theresultiscloselysimilar
k
to Kocay’slemma [8]which alsogivesthe coefficients ck . Insections 2 and4, we
A,B
introduce two distinct ways of calculating these coefficients. However the second
looks more efficient, we still believe that the first is of independent interest.
Example 3. ConsiderthegraphF inFigure2. Thereadercanverifybycalcu-
latingthe numberofsubgraphsofagiventypethatI(a )(F)=9,I(a a )(F)=
12 12 13
15,I(a a a )(F)=4,I(a a a )(F)=3,I(a a a )(F)=52andI(a a a )
12 13 14 12 13 23 12 34 45 12 23 34
(F)=16. The algebraic dependence given by the product formula will turn out to
be
I(a )I(a a ) = 2I(a a )+2I(a a a )+3I(a a a ) (6)
12 12 13 12 13 12 23 34 12 13 23
+3I(a a a )+I(a a a )
12 13 23 12 34 45
anditshowsthatthereisanalgebraicdependencebetweentheseinvariants. Indeed
9·15=2·15+2·16+3·4+3·3+52.
Thegraphisomorphism(GI)problemaskstodetermine,whetherforanygraphs
A and B there exists a permutation ρ of vertices of the graph A such that ρ(A)=
B. There is no known polynomial-time algorithm for solving GI and some results
indicate that generalGI might notbelong to P [1], [9]. There are,however,several
GI algorithms which perform very well on average [11],[19]. If the vertex degree
3
Figure 2 Graph F.
(i.e. the number of edges adjacent to a vertex) is bounded, then GI belongs to P
[10].
Insection7wecalculatetheupperandlowerboundsfortheminimalnumberof
basic graphinvariants requiredto prove graph isomorphismbetween two arbitrary
graphs.
Many results presented here were provedindependently by both of the authors
andmoreoverhadalreadybeenpublishedpreviouslybyothermathematicians. The
authors tried their best to provide a self-contained, rather complete, treatment of
the subject, however the interested reader might have a look at the works of J.A.
Bondy, W. L. Kocay, V. B. Mnukhin, M. Pouz´et, B.D. Thatte and N. Thi´ery ([2],
[7],[12],[14],[16],[17],[18]),eachofthemhavingitsowndistinctpointofview(e.g.
N. Thi´ery took a classicalinvarianttheoretic approachin [18]). We mustnote that
the classical invariants form a graded algebra unlike the invariants in this paper.
This is due to reduction a2 =a since a ∈{0,1} for simple graphs.
ij ij ij
ThispaperisbasedontheproductformulaandwhatwecalltheG-posettheory
whichisroughlythefinitesetsystemtheorywithapermutationgroup. Theproduct
formula and G-poset theory are quite essential in the reconstruction problem. In
section 7 we show a sufficient condition for this conjecture to be true.
In the followingsection we calculate the product of basic graphinvariantsI(A)
and I(B). In section 3 we show a couple of examples and consequences of the
product formula. In section 4 we show how all graph invariants can be written as
a linear combinationofthe basic graphinvariants. Insection 6 we derivea simpler
formulaforthe productoftwographinvariants. Insection7we study the minimal
set of generator/separatorinvariants.
2. PRODUCT FORMULA FOR GRAPH INVARIANTS
Fleischmann’s product formula for two orbit sums is not directly applicable to
graph invariants of simple graphs where a ∈ {0,1}. We use a simple example to
ij
showthis. LetG⊆S beanypermutationgroup. Foranynumberofverticesn,the
n
permutation (23) ∈ Stab(a a ) but (23) ∈/ Stab(a2 a ), in fact Stab(a a ) =
12 13 12 13 12 13
h(23),Stab(a2 a )i. However a2 a =a a if a ∈0,1 i.e. there is a reduction
12 13 12 13 12 13 ij
a2 =a . Thus
ij ij
a2 a =2 a a . (7)
ρ(1)ρ(2) ρ(1)ρ(3) ρ(1)ρ(2) ρ(1)ρ(3)
ρ∈G/StXab(a212a13) ρ∈G/StXab(a12a13)
4
In general by the the orbit-stabilizer theorem
ae1 ae2 ···aed (8)
ρ(i1j1) ρ(i2j2) ρ(idjd)
ρ∈G/Stab(aeiX11j1aei22j2···aeiddjd)
|Stab(a a ···a )|
= i1j1 i2j2 idjd a a ···a ,
|Stab(ae1 ae2 ···aed )| ρ(i1j1) ρ(i2j2) ρ(idjd)
i1j1 i2j2 idjd ρ∈G/Stab(aiX1j1ai2j2···aidjd)
making higher degree invariants redundant. Since n a is the highest degree
i<j ij
monomial and it contains n variables, we observe that n provides an upper
2 Q 2
boundfor the degreeofbasicgraphinvariantswhicharegivenbysums oftype (8).
(cid:0) (cid:1) (cid:0) (cid:1)
Fleischmann’s product formula for the product of orbit sums of monomials A
and B over an arbitrary permutation group G is
|Stab(AgB)|
I(A)I(B)= I(AgB), (9)
|Stab(A)∩gStab(B)|
g∈[Stab(AX):G:Stab(B)]
where [G : G : G ] denotes the cross-section of the group G with subgroups
1 2
G ,G EG s.t. G=⊎ G gG . This product formula applies directly to
1 2 g∈[G1:G:G2] 1 2
invariants of multigraphs where a are in a commutative algebra.
ij
Let A denote ae1 ···aed modha2 −a ,...,a2 −a i, i.e.
i1j1 idjd i1j1 i1j1 idjd idjd
b ae1 \···aed =a ···a . (10)
i1j1 idjd i1j1 idjd
With this notation, we can express the Orbit Lemma as
|Stab(G)|
I(G)= I(G). (11)
|Stab(G)|
b
To get a product formula for graph invariantbs of simple graphs where a ∈
ij
{0,1}, we expand the terms in (9) by the formula (8). This results in
I(A)I(B) (12)
|Stab(AgB)|
= I(AgB) (13)
|Stab(A)∩gStab(B)|
g∈[Stab(AX):G:Stab(B)]
[
|Stab(AgB)| |Stab(AgB)| [
= I(AgB)
|Stab(A)∩gStab(B)||Stab(AgB)|
g∈[Stab(AX):G:Stab(B)]
[
|Stab(AgB)| [
= I(AgB).
|Stab(A)∩gStab(B)|
g∈[Stab(AX):G:Stab(B)]
This proves
Theorem 1. The product formula for graph invariants I(A) and I(B), where
A,B are simple graphs, is
[
|Stab(AgB)| [
I(A)I(B)= I(AgB), (14)
|Stab(A)∩gStab(B)|
g∈[Stab(AX):Sn:Stab(B)]
5
Thisformulaisquitedifficulttousebutwecaninterpretthesetofpermutations
g ∈[Stab(A) :S :Stab(B)] by using colored graphs. We associate a monomial A
n
in the variables a with coloredgraphs by equating the color of the edge (ij) with
ij
the exponent of the variable a in the monomial A.
ij
Example 4. Themonomiala a a2 a2 a3 correspondstothegraphinFigure
13 14 24 25 35
2.
Figure 3 The colored graph corresponding to the monomial a a a2 a2 a3 .
13 14 24 25 35
NowconsiderthepermutationgroupG=S andallgraphsAρB,whereρ∈S ,
n n
suchthattheedgesofAhavecolor1,theedgesofB havecolor2,theverticesofB
are permuted over all permutations and whenever two edges coincide the color of
the edgeis 3. Thenthe setofisomorphismclassesofthese coloredgraphs,denoted
by C(A,B), corresponds to the set of monomials AgB. The
g∈[Stab(A):Sn:Stab(B)]
coloring of graphs corresponds to the modification of monomials AgB such that
S
all the variables in part A are raised to the power 1 and all the variables in B are
raised to the power 2.
Proposition 1. Themapφbetweenthesets{AgB|g ∈[Stab(A):S :Stab(B)]}
n
and C(A,B) coloring AgB as above is bijective.
Proof.
Since φ is clearly onto, all we have to show is that the set of colored graphs
{AgB|g ∈[Stab(A):S :Stab(B)]} (15)
n
doesnotcontaintwoelementsAgBandAhB(whereg,h∈[Stab(A):S :Stab(B)]
n
andh6=g)suchthatπAgB =AhB forsomepermutationπ ∈S . Supposewehad
n
such a pair πAgB = AhB. Because of the coloring we can recover the location of
the edges of A in AgB, namely, every edge in AgB with color 1 or 3 corresponds
to an edge in A. This implies that the permutation π ∈ Stab(A) since it maps
πA = A. Also because of the coloring we have πgB = hB which implies that
∃b∈Stab(B) s.t. πg =hb as we can take b=h−1πg.
We can now solve g =π−1hb and so g ∈Stab(A)hStab(B) which implies g =h
by the choice of g,h∈[Stab(A):S :Stab(B)]. This is a contradictionand thus φ
n
is an injective map.
Remark that the identity (9) gives
|Stab(AgB)|
I(A) ρB = I(AgB). (16)
|Stab(A)∩gStab(B)|
ρ∈SnX/Stab(B) g∈[Stab(AX):Sn:Stab(B)]
6
Kocay’s lemma [8] on the other hand says that the coefficient ck equals the
A,B
numberofpairs(C,D)whereC ∼=A,D ∼=Bandg =C∪D,foranyrepresentative
k
ofG . Wecangroupthesepairsaccordingtothetypeofcoloringtheygiveandsee
k
thatthegroupsarepreciselytheorbitsofthecoloringundertheactionofStab(G ).
k
Thus the two results are the essentially the same, any monomial in the resulting
product being split into its distinct type of coloring in the Fleischmann formula.
Example 5. Consider the product I(a a )2, which can be calculated using
12 13
Theorem 1.
I(a a )2 = 4I(a a a a )+2I(a a a )+2I(a a a a ) (17)
12 13 12 23 34 14 12 23 34 12 23 24 34
+2I(a a a a )+6I(a a a )+6I(a a a )+I(a a ).
12 23 24 34 12 13 14 12 13 23 12 13
ThetermI(a a a )arrivesforinstancefromthemonomiala a2 a ≡a a a
12 13 14 12 13 14 12 13 14
mod a2 −a . The coefficient of I(a a a ) is 6 because the numerator |Stab
13 13 12 13 14
(a a a )|=6andinthedenominatortheintersectionofstabilizersStab(a a )=
12 13 14 12 13
{(1234),(1324)} and Stab(a a )= {(1234),(1243)} is the trivial group and thus
13 14
the denominator is 1.
ThereisaprobleminthetermI(a a a a ). Insteadofhavingthisinvariant
12 23 24 34
withcoefficient4wehaveitsplitintotwoparts. Thisisbecausetherearetwonon-
isomorphiccoloringsforthisgraphintheproductI(a a )2. SeeFigure4forthese
12 13
colorings. In section 6 we will solve this problem.
Figure 4 Two non-isomorphic colorings.
3. EXAMPLES
The product formula (14) describes connections between the numbers of differ-
ent subgraph isomorphism classes of graphs. We use two examples to show these
connections in explicit form.
Example 6. Let g = I(a ), g = I(a a ), g = I(a a a ). The mul-
1 12 2 12 13 3 12 13 23
tiplication table of these invariants calculated using Theorem 1 is given in Table
1.
Table 1
Multiplication table for graph invariants with n=3.
g1 g2 g3
g1 g1+2g2 2g2+3g3 3g3
g2 g2+6g3 3g3
g3 g3
7
We can see from Table 1 that the minimal generator set is {g } and the other
1
invariants are given by the relations g = (g2 −g )/2, g = 1/6g3 −1/2g2+g .
2 1 1 3 1 1 1
Thus the transcendence degree of (k[a12,a13,a23]/ha2ij −aiji)S3 is one. The values
of g are limited by 0≤g ≤3.
1 1
Example 7. Letg =I(a ),g =I(a a ),g =I(a a ),g =I(a a a ),
1 12 2 12 34 3 12 13 4 12 23 34
g = I(a a a ), g = I(a a a ), g = I(a a a a ), g = I(a a a a ),
5 12 13 14 6 12 13 23 7 12 23 34 14 8 12 23 24 34
g = I(a a a a a ), g = I(a a a a a a ). The first column of the
9 12 23 34 14 13 10 12 13 14 23 24 34
multiplication table is in Table 2. We can solve for g ,g ,g ,g ,g ,g and g in
2 4 6 7 8 9 10
terms of g ,g and g from Table 2, the solution is below.
1 3 5
Table 2
The first column of the multiplication table for graph invariants with n=4.
g1
g1 g1+2g2+2g3
g2 2g2+g4
g3 2g3+2g4+3g6+3g5
g4 3g4+4g7+2g8
g5 3g5+g8
g6 3g6+g8
g7 4g7+g9
g8 4g8+4g9
g9 5g9+6g10
g10 6g10
g2 = 1/2g12−1/2g1−g3
g4 = 1/2g13−3/2g12−g11g3+g1+2g3,
g6 = g1g3−2g3−2/3g1−g5−1/3g13+g12,
g8 = g1g5−3g5,
g7 = −3/4g13−1/2g1g5+3/2g5+1/8g14−3/4g1
−1/4g12g3+11/8g12+5/4g1g3−3/2g3,
g10 = −47/12g1g5−5/2g1−5g3+5g5+137/24g12
+77/12g1g3−75/16g13−1/12g13g5+g12g5
−1/24g14g3+7/12g13g3−71/24g12g3
+85/48g14−5/16g15+1/48g16,
g9 = −5/4g14−1/2g12g5+7/2g1g5+1/8g15−25/4g12
−1/4g13g3+35/8g13+9/4g12g3−13/2g1g3−6g5+3g1+6g3.
For the reader familiar with general invariant theory (see [15]) we remark that
by calculatingthe Gr¨obnerbasis ofthe relationsinthe multiplicationtable, weget
syzygiesdescribingcompletelythe possiblevaluesofthe graphinvariantsingraphs
withn=4. Forinstanceg satisfiesthesyzygyg7−21g6+175g5−735g4+1624g3−
1 1 1 1 1 1
1764g2+720g = 0 which has roots 0,1,2,...,6, determining the possible values
1 1
for g .
1
The algebraic dependencies in the example above hold only if the number of
vertices is 4. It is easy, however,to construct general products.
8
Lemma 1. General products, i.e. products independent of the number of ver-
tices, can be calculated by selecting n ≥ cv(A)+cv(B), where cv(A) denotes the
number of vertices in connection with the edges of the graph.
Proof. Noticethatcv(G)isthenumberofvertex-indicesinthemonomialsofthe
graph invariant I(G). The maximum number of distinct indices in any monomial
AgB is thus cv(A)+cv(B).
The coefficientofI(A[gB)inthe productofI(A)I(B) is |Stab(A[gB)| . This
|Stab(A)∩gStab(B)|
remainsthesamewhennexceedscv(A)+cv(B). Thiscanbeseenbynoticingthat
[ [
Stab(AgB) = Sn−cv(A[gB) ×StabScv(A[gB)(AgB), where Sn denotes the symmetric
[ [
groupandStab (AgB)isthestabilizerofAgB withrespecttopermutations
Scv(A[gB)
[
of the connected vertices in AgB. Thus
[ [ [
|Stab(AgB)|=(n−cv(AgB))!|Stab (AgB)|. (18)
Scv(A[gB)
Next notice that
Stab(A)∩gStab(B)=Sn−cv(A[gB)× Stabcv(A)(A)∩gStabcv(B)(B) (19)
(cid:0) (cid:1)
since no permutation in Stab(A)∩gStab(B) can map a vertex in connection with
the edges in A or B outside the set vertices in connection with the edges. Thus
in the coefficient |Stab(A[gB)| the terms (n−cv(A[gB))! appear both in the
|Stab(A)∩gStab(B)|
denominator and the numerator and cancel each other out.
Example 8. The algebraic dependence
I(a )I(a a )=2I(a a )+2I(a a a )+3I(a a a ) (20)
12 12 13 12 13 12 23 34 12 13 23
+3I(a a a )+I(a a a )
12 13 14 12 34 45
isgeneralholdingforallgraphs,notjustforgraphswith5verticessincecv(a )=2
12
and cv(a a )=3.
12 13
4. G-POSETS AND MNUKHIN-TRANSFORMS
In this section we study orbit sums of monomials in a general context. We
generalize first the notion of I(g )(g ). Let G⊆S be a permutation group acting
i j n
on the variables x ,...,x . Let m+ denote the orbit sum of the monomial m over
1 n
G, i.e. xmρ(1)···xmρ(n).
ρ∈G/Stab(m) 1 n
Todefinebasicinvariantsformultigraphsandmoregeneralobjectsweintroduce
P
thefollowingdifferentialoperatorwhichplaysthecentralroleinCayley’sΩ-process
in classical invariant theory [15].
The differential operator corresponding to m+ is defined as
1 ∂|m|
D := . (21)
m+ m1!m2!···mn!ρ∈G/XStab(m)∂x1mρ(1)···∂xnmρ(n)
TheonlydifferencewiththeoriginalCayley’soperatoristhecoefficient 1
m1!m2!···mn!
which turns this operator into a Hasse derivative.
9
Thevalueofthiscombinatorial invariant,denotedbyI(m)(w)atthemonomial
w is
I(m)(w):={Dm(w)}x=1 (22)
wherem=xm1xm2···xmn andw =xw1xw2···xwn. ThereasonforusingtheHasse
1 2 n 1 2 n
derivativeisthatI(m)(m)shouldbeonemaintainingtheinterpretationofcounting
subgraphsanddefininganunimodularMnukhin-transformwhichwedefineshortly.
Example 9. Take m=x x2 and G=S . Then we calculate
1 2 2
1 ∂3 1 ∂3
D = + (23)
m+ 2∂x2∂x 2∂x ∂x2
1 2 1 2
and
1 ∂3 1 ∂3
x x2+ x x2 (24)
2∂x2∂x 1 2 2∂x ∂x2 1 2
1 2 1 2
= 1.
Thus I(m)(m)={x1}x=1 =1.
Example 10. Take m=x x2,w =x2x2 and G=S . Then
1 2 1 2 2
1 ∂3 1 ∂3
x2x2+ x2x2 (25)
2∂x2∂x 1 2 2∂x ∂x2 1 2
1 2 1 2
= 2x +2x .
2 1
Thus I(m)(w)={2x2+2x1}x=1 =4.
Lemma 2. The invariants I(a) coincide with the orbit sums a+ if a ∈{0,1}.
i
Proof. It is sufficient to consider one monomial a ∈ Orb (a′) and the corre-
G
sponding differential operator D . The monomial a at b equals
a
ba1···baN = b . (26)
1 N i
i:Yai=1
The differential operator at b equals
D xb1···xbN = b xbi−1 xbi (27)
a 1 N i
!
i:Yai=1 i:Yai=0
which equals b at x=1.
i:ai=1 i
Let Q
1 ∂ai
Dai = (28)
i a !∂ai
i i
and notice that Daixbi = bi xbi−ai.
i i ai
Thisgivesusanimportantcluehowtofindthelinearcombinationofdifferential
operators ∞ caiDkixbi(cid:0)s.t(cid:1). the value at x=1 is bai.
k=0 k i i i
The linear equation for the coefficients is
P
Bcai =[0ai,1ai,...,∞ai]T, (29)
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