The Higher Transvectants are Redundant

0.34 MB·English

by  Abdelmalek Abdesselam

THE HIGHER TRANSVECTANTS ARE REDUNDANT
ABDELMALEKABDESSELAMANDJAYDEEPCHIPALKATTI
8
0
0
2
n ABSTRACT. Let A,B denote generic binary forms, and letur = (A,B)r denote their
a r-thtransvectant inthesenseofclassicalinvariant theory. Inthispaperweclassifyall
J
thequadraticsyzygiesbetweenthe ur .Asaconsequence,weshowthateachofthe
0 { }
higher transvectants ur : r 2 isredundant inthe sense thatitcan becompletely
1 { ≥ }
recovered fromu0 andu1. Thisresultcanbegeometrically interpreted intermsofthe

] incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence
G
of SL2-representations, and the notion of a 9-j symbol from the quantum theory of
A
angularmomentum.
.
h We give explicit computational examples for SL3,g2 and S5 to show that this result
at haspossibleanaloguesforothercategoriesofrepresentations.
m
[ MathematicsSubjectClassification(2000): 13A50,22E70.

Keywords: angular momentum in quantum mechanics, binary forms, Cauchy exact
1
v sequence,9-jsymbols,representations ofSL2,transvectants.
3
3
5 CONTENTS
1
1. Introduction 1
.
1
2. TheCauchyexactsequence 6
0
8 3. TheincompleteSegreimbedding 15
0
4. SL -representations 18
: 3
v 5. Thestandardrepresentationofg 21
2
i
X 6. ThestandardrepresentationofS 24
d
r 7. Wignersymbols 27
a
References 36
1. INTRODUCTION
TransvectantswereintroducedintoalgebramoreorlessindependentlybyCay-
ley and Aronhold (see [11, 13]). The German school of classical invariant theo-
rists used them dexterously in the symbolical treatment of algebraic forms (for
instances,see [23, 41]). In theirmodernformulation,they encodethe decompo-
sition of the tensor productof two finite-dimensionalSL -representationsover a
2
fieldofcharacteristiczero.
1
2 ABDESSELAMANDCHIPALKATTI
We begin by giving an elementary definition of transvectants. In §1.3-1.5 we
describetheirreformulationinthelanguageofSL -representations.Anoutlineof
2
themainresultsisgivenin§1.9(onpage5)aftertherequirednotationisavailable.
We will use [21, 24] as standard references for classical invariant theory, and
inparticularthesymboliccalculus. Modernaccountsofthissubjectmaybefound
in[15,31,35]. Thereaderisreferredto[19,Lecture6],[40,Ch.3]and[42,Ch.4]
forthebasictheoryofSL -representations.
2
1.1. Let
m n
m n
A = a xm ixi, B = b xn ixi;
i i 1 − 2 i i 1− 2
i=0 (cid:18) (cid:19) i=0 (cid:18) (cid:19)
X X
denote binary forms of orders m,n in the variables x = x ,x . (The coeffi-
1 2
{ }
cientsareassumedtobeinafieldofcharacteristiczero.) Letr denoteaninteger
suchthat0 r min(m,n). Ther-thtransvectantofAandB isdefinedtobe
≤ ≤
thebinaryform
(m r)!(n r)! r r ∂rA ∂rB
(A,B) = − − ( 1)i (1)
r m!n! − i ∂xr i∂xi ∂xi ∂xr i
i=0 (cid:18) (cid:19) 1− 2 1 2−
X
oforderm+n 2r. Inparticular(A,B) istheproductofA,B,and(A,B) is
0 1
−
(uptoamultiplicativefactor)theirJacobian. Byconstruction,
(B,A) = ( 1)r(A,B) . (2)
r r
−
Theprocessoftransvectioncommuteswithachangeofvariablesinthefollow-
α β
ingsense. Letg = denoteamatrixofindeterminates.Define
γ δ
(cid:18) (cid:19)
m
m
A = a (αx +βx )m i(γx +δx )i,
′ i 1 2 − 1 2
i
i=0 (cid:18) (cid:19)
X
andsimilarlyforB etc. Thenwehaveanidentity
′
(A,B ) = (detg)r[(A,B) ].
′ ′ r r ′
Inclassicalterminology,(A,B) isajointcovariantofA,B.
r
1.2. NowletA,B denotegenericformsofordersm,n,thatistosay,theircoef-
ficients are assumed to be independentindeterminates. Write u = (A,B) for
r r
the r-th transvectant.1 Broadlyspeaking,the main resultof this paperis thatthe
highertransvectants u : r 2 are redundantin the sense that each of them
r
{ ≥ }
canberecoveredfromtheknowledgeofu andu . Webeginwithanillustration.
0 1
1
‘Uberschiebung’inGerman.
HIGHERTRANSVECTANTS 3
Example1.1. Assumem = 5,n = 3. Thenwehaveanidentity
21 21 315
(u ,u ) + (u ,u ) + u2 = u u , (3)
8 0 0 2 16 0 1 1 256 1 0 2
which gives a formula for u in terms of u ,u . (This is an instance of general
2 0 1
formulaetobeprovedbelow.) Similarly,
20 20 25
(u ,u ) + (u ,u ) + u u = u u , (4)
0 1 2 0 2 1 1 2 0 3
3 9 14
which indirectly expresses u in terms of u ,u . Our result shows the existence
3 0 1
ofsuchformulaeingeneral.
Theorem1.2. Assumem,n,r 2. Withnotationasabove,thereexistconstants
≥
c Qsuchthatwehaveanidentity
i,j
∈
1
u = c (u ,u ) .
r i,j i j r i j
u0 −−
0 i j<r
≤X≤
Since the right hand side depends only on u ,...,u , it follows by in-
0 r 1
{ − }
duction that u ,u determine the rest of the higher transvectants. In fact, more
0 1
generally we will exhibit explicit formulae for all the quadratic syzygies between
the u ,ofwhich(3)and(4)arespecialcases.
i
{ }
The titleof thepapershouldnotbeunderstoodto meanthat‘highertransvec-
tion’ is redundant. Notice, for instance, that the formula for u itself involves
2
(u ,u ) .
0 0 2
1.3. SL -representations. Throughoutthispaperweworkoveranarbitraryfield
2
k of characteristic zero. Let V denote a two-dimensional k-vector space with
basis x = x ,x . For m 0, the symmetric power S = SymmV is the
1 2 m
{ } ≥
space of binary m-ics, with an action of the linearly reductive group SL(V) =
ϕ End(V) : det ϕ = 1 . The S : m 0 areacompletesetofirreducible
m
{ ∈ } { ≥ }
SL(V)-representations, and any finite-dimensional representation decomposes
asa directsumofirreducibles. By Schur’slemma,ifalinearmapS S is
m m
−→
SL(V)-equivariant,thenitisnecessarilyascalarmultiplication.
Henceforth, V will not be explicitly mentioned if no confusion is likely; for in-
stance,S (S )willstandforSymm(SymnV)etc.
m n
1.4. Itwillbeconvenienttointroduceseveralpairsofvariables
y = (y ,y ), z = (z ,z ),...
1 2 1 2
all on equal footing with x. Then, for instance, an elementof the tensor product
S S can be representedas a bihomogeneousform F(x,y) of orders m,n
m n
⊗
inx,yrespectively.DefineCayley’sOmegaoperator
∂2 ∂2
Ω = ,
xy
∂x ∂y − ∂x ∂y
1 2 2 1
4 ABDESSELAMANDCHIPALKATTI
andthepolarisationoperator
∂ ∂
y∂ = y +y .
x 1 2
∂x ∂x
1 2
Ifc standsforthesymboliclinearformc x +c x ,then
x 1 1 2 2
m!
(y∂ )ℓcm = cm ℓcℓ.
x x (m ℓ)! x− y
−
The operators Ω , y∂ etc. are similarly defined. The symbolic bracket (xy)
xz z
standsforx y x y ,andlikewisefor(xz)etc.
1 2 2 1
−
1.5. Wehaveadirectsumdecompositionofthetensorproduct
min(m,n)
S S S , (5)
m n m+n 2r
⊗ ≃ −
r=0
M
usuallycalledtheClebsch-Gordandecomposition.Let
π :S S S
r m n m+n 2r
⊗ −→ −
denotetheprojectionmap,whichactsbytherecipe
F(x,y) πr f(m,n;r)[Ωr F(x,y)] ; (6)
xy y x
−→ →
where
(m r)!(n r)!
f(m,n;r)= − − .
m!n!
Wehavewritteny xforthesubstitutionofx ,x fory ,y respectively,sothat
1 2 1 2
→
therighthandsideof(6)isoforderm+n 2r inxasrequired.
−
InparticularifA(x) S ,B(x) S ,thenastraightforwardbinomialexpan-
m n
∈ ∈
sionshowsthattheimageπ (A(x)B(y))coincideswiththetransvectant(A,B)
r r
asdefinedin(1). Insymbols,ifA = αm,B = βn,thenwehavetheformula
x x
(A,B) = (αβ)rαm rβn r. (7)
r x− x−
Theinitialscalingfactorin(6)issochosenthat(7)hasthesimplestpossibleform.
1.6. The map π is a splitsurjection,letı : S S S denoteits
r r m+n 2r m n
− −→ ⊗
section. Forcm+n 2r S ,itisgivenby
x − m+n 2r
∈ −
cm+n 2r ır g(m,n;r)(xy)rcm rcn r,
x − x− y−
−→
where
m n
g(m,n;r) = r r . (8)
m+n r+1
(cid:0) (cid:1)r−(cid:0) (cid:1)
Define
(cid:0) (cid:1)
(m+n 2r+1)!
h(m,n;r) = f(m,n;r)g(m,n;r) = − . (9)
(m+n r+1)!r!
−
HIGHERTRANSVECTANTS 5
Nowobservethatbytheformulaon[24,p.54],
1
Ωr [(xy)rcm rcn r] = cm+n 2r,
{ xy x− y− }y→x h(m,n;r) x −
whichverifiesthatπ ı is theidentitymaponS (alsosee[17]and[30,
r r m+n 2r
◦ −
§18.2]).
1.7. Angular momenta. There is a process analogous to transvection in the
quantumtheoryof angularmomentum. In brief, the eigenvectors(of the Casimir
element for the Lie algebra su ) can exist in any of the states j labelled by the
2
nonnegativehalf-integers 0,1/2,1,3/2,... . The coupling of two states j ,j
1 2
{ }
producesafinitesetofangularmomentumstates
j j , j j +1, j j +2,...,j +j .
1 2 1 2 1 2 1 2
| − | | − | | − |
If we letm = 2j ,n = 2j , then thisreducesto the Clebsch-Gordandecompo-
1 2
sition. (Thestandardaccountofthis theorymaybefoundin [6, 16].) Ata crucial
placeinourstudyoftransvectantsyzygieswewillneedtheconceptof9-jsymbol
whicharises from the possiblecouplingsoffour angularmomentumstates. This
is further explained in §7, where an introductionto the relevant notions from the
quantumtheoryofangularmomentumwillbegiven.
1.8. Self-duality. ThemapS S kestablishesacanonicalisomorphism
m m
⊗ −→
of S with its dual representationS = Hom(S ,k). It identifiesA S with
m m∨ m ∈ m
thefunctional
S k, B (A,B) .
m m
−→ −→
Consequently,everyfinite-dimensionalSL -representationiscanonicallyisomor-
2
phic to its own dual.2 We have a canonical trace element in S S which
m m
⊗
correspondstotheform(xy)m.
1.9. Results. We can now state the main results of this paper. Let the u be
i
{ }
as in §1.2. For an integer r such that 2 r min(m,n), define a (quadratic)
≤ ≤
syzygyofweightrtobeanidentity
ϑ (u ,u ) = 0, ϑ Q (10)
i,j i j r i j i,j
−− ∈
X
wherethesummationisquantifiedoverallpairs(i,j)suchthat
0 i j, i+j r.
≤ ≤ ≤
Noticethatonlyonesummandin(10)involvesu ,namelyϑ u u .
r 0,r 0 r
2ThisisnolongertrueofSLN-representationswhenN >2. Insomecontextsthisself-duality
leadstosimplification,andinsomeotherstoconfusion.
6 ABDESSELAMANDCHIPALKATTI
Let K(m,n;r) denote the vector space of weight r syzygies. In §2.3–2.4 we
will show that there is a natural isomorphism of K(m,n;r) with the space of
equivariantmorphisms
Hom (S , 2S 2S ).
SL(V) 2(m+n r) m n
− ∧ ⊗∧
ThiswillimplythatK(m,n;r)hasabasiswhichisinnaturalbijectionwiththeset
ofintegralpoints
r 2
Π(m,n;r) = (a,b) N2 :a+b − .
{ ∈ ≤ 2 }
Since (a,b) = (0,0) is such as point, there exist nontrivial syzygies of any
(p)
weight r 2. For an arbitrary p = (a,b) Π(m,n;r), let ϑ denote the
≥ ∈ i,j
correspondingsyzygycoefficients.
(p)
In§2.10wewillgiveanexplicitformulafortherationalnumberϑ . Itwillfollow
i,j
(p)
thatifwespecialisetop = (0,0),thenϑ = 0. Wecanthenrewriteidentity(10)
0,r 6
intheform
(p)
1 ϑ
i,j
u = (u ,u ) ,
r i j r i j
u0 −ϑ(p) −−
0,r
X
and thereby complete the proof of Theorem 1.2. In Theorem 3.1 we prove the
thematicallyrelatedresultthatthemorphism
PS PS P(S S )
m n m+n m+n 2
× −→ ⊕ −
whichsendsapairofforms(A,B)to(AB,(A,B) ),isanimbeddingofalgebraic
1
varieties.
Of courseit wouldbe of interestto findsimilarredundancytheoremsfor other
categoriesof representations. In sections 4,5 and 6, we give one example each
ofthisphenomenonrespectivelyforrepresentationsofSL ,g andS .
3 2 5
2. THE CAUCHY EXACT SEQUENCE
Inthissectionweestablishthebasicset-upwhichleadstothecharacterisation
ofquadraticsyzygiesbetweentransvectants.
2.1. Given any two finite-dimensionalvector spaces W ,W , we have a short
1 2
exactsequence(see[4,§III.1])ofGL(W ) GL(W )-representations
1 2
×
0 2W 2W δ S (W W ) ǫ S (W ) S (W ) 0, (11)
1 2 2 1 2 2 1 2 2
−→ ∧ ⊗∧ −→ ⊗ −→ ⊗ −→
C
which we may call the Cauchy exact sequence. (The correspondingformula on
| {z }
charactersisduetoCauchy–see[19,AppendixA].)
Letthedotstandforsymmetrisedtensorproduct,i.e.,wewriteg hinsteadof
·
1(g h+h g). Withthisnotation,ǫisthe‘regrouping’map
2 ⊗ ⊗
(g g ) (h h ) (g h ) (g h ),
1 2 1 2 1 1 2 2
⊗ · ⊗ −→ · ⊗ ·
HIGHERTRANSVECTANTS 7
andδ isthemap
(g h ) (g h ) (g g ) (h h ) (g h ) (h g ).
1 1 2 2 1 2 1 2 1 2 1 2
∧ ⊗ ∧ −→ ⊗ · ⊗ − ⊗ · ⊗
Theexactnessof(11)isaninstanceofageneralresultaboutSchurfunctors(see
loc. cit.), but it is elementary to check in this case. Indeed, it is immediate that
ǫ δ = 0, implyingimδ kerǫ. Now write w = dimW , and observethat the
i i
◦ ⊆
dimensionsofthefirstandthethirdvectorspaceadduptothesecond:
w w w +1 w +1 w w +1
1 2 1 2 1 2
+ = ,
2 2 2 2 2
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
henceimδ = kerǫ.
2.2. ConsidertheSegreimbedding
PS PS P(S S ), [(A,B)] [A B]
m n m n
× −→ ⊗ −→ ⊗
with image X, and ideal sheaf . Since X is projectively normal, we have an
X
I
exactsequence
0 H0( (2)) g H0( (2)) h H0( (2)) 0. (12)
X P X
−→ I −→ O −→ O −→
Letusintroduceaseriesofgenericforms
m n
m n
A= a zm kzk, B = b zn kzk, (13)
k k 1 − 2 k k 1− 2
k=0 (cid:18) (cid:19) k=0 (cid:18) (cid:19)
X X
ofordersm,n,and
m+n 2ℓ
− m+n 2ℓ
Uℓ = k− qk,ℓ z1m+n−2ℓ−kz2k, (14)
k=0 (cid:18) (cid:19)
X
of orders m + n 2ℓ for 0 ℓ min(m,n). (That is to say, the a,b,q are
− ≤ ≤
assumedtobesetsofdistinctindeterminates.)Considerthepolynomialalgebras
Q =k[ q ], R = k[a ,...,a ;b ,...,b ].
k,ℓ 0 m 0 n
{ }
The former is graded by N, and the latter by N N. If we write U = (A,B)z
× ℓ ℓ
(where the transvectant is taken with respect to z variables) and equate coeffi-
cientsinz,theneachq isgivenbyapolynomialexpressionina,b. Thisdefines
k,ℓ
a ring morphismQ R. Now, we have isomorphismsof graded(respectively
−→
bigraded)rings
Q ∼ Se([Sm Sn]∨),
−→ ⊗
e 0
M≥
R −∼→ Se(Sm∨)⊗Se′(Sn∨)
e,e′ 0
M≥
definedasfollows: observethat
( 1)k (U ,zm+n 2ℓ kzk)z = q ,
− × ℓ 2 − − 1 m+n−2ℓ k,ℓ
8 ABDESSELAMANDCHIPALKATTI
hence we identify q with the functional in [S S ] which sends the biform
k,ℓ m n ∨
⊗
αmβn S S to
x y m n
∈ ⊗
( 1)k ((αβ)ℓαm ℓβn ℓ,zm+n 2ℓ kzk)z .
− × z − z− 2 − − 1 m+n−2ℓ
This extends to give an isomorphism of Q with the symmetric algebra on the
space [S S ] . The second isomorphism is defined similarly. The induced
m n ∨
⊗
map Q R on vector spaces may be naturally identified with the map h
2 2,2
−→
from(12).
2.3. Consideraformalexpression
Ψ = ϑ (U ,U )z ,
i,j i j r i j
−−
i,j
X
where ϑ are arbitrary elements in Q. We should like to determine whether Ψ
i,j
correspondstoaweightr syzygy. Now,thedatumΨisequivalenttoamorphism
ofSL(V)-representations
f :S Q , H(z) (H(z),Ψ)z .
Ψ 2(m+n−r) −→ 2 −→ 2(m+n−r)
Thisistobeinterpretedasfollows: Ψ,H(z)arebothformsoforder2(m+n r)
−
inthez-variables.Henceaftertransvectiontherighthandsidehasnoz-variables
remaining,andwegetaquadraticexpressioninthe q .
k,ℓ
{ }
Now Ψ represents a bona fide weight r syzygy iff the following condition is
satisfied: if we substitute (A,B) for U , then Ψ vanishes. This is equivalent to
i i
the requirement that h f = 0, i.e., f factor through kerh. Hence we have
Ψ Ψ
◦
provedthefollowing:
Proposition 2.1. The vector space K(m,n;r) of weight r syzygies is naturally
isomorphictoHom (S ,H0( (2))). (cid:3)
SL(V) 2(m+n r) X
− I
2.4. Now,byspecialising(11)wehavetheexactsequence
0 2S 2S δ S (S S ) ǫ S (S ) S (S ) 0. (15)
m n 2 m n 2 m 2 n
−→ ∧ ⊗∧ −→ ⊗ −→ ⊗ −→
C D E
Byself-du|ality(s{eze§1.8})weca|nide{nztifyH}0(P(S|m Sn),{zP(2))a}ndH0( X(2))
⊗ O O
respectivelywith and ,inducinganisomorphismofH0( (2))with .
X
D E I C
2.5. Wehaveisomorphisms
m−1
⌊ 2 ⌋
2S S (S ) S ,
m 2 m 1 2(m 1) 4a
∧ ≃ − ≃ − −
a=0
M
andsimilarlyfor 2S . Hence,foreachpairp = (a,b)intheset
n
∧
Π(m,n;r) = (a,b) N2 : 2(a+b+1) r , (16)
{ ∈ ≤ }
HIGHERTRANSVECTANTS 9
wehaveamorphismφ definedtobethecomposite
a,b
S θ1 S S θ2 S (S ) S (S )
2(m+n r) 2(m 1) 4a 2(n 1) 4b 2 m 1 2 n 1
− −→ − − ⊗ − − −→ − ⊗ −
θ3 2S 2S .
m n
−→ ∧ ⊗∧
Hereθ is dualto the(r 2a 2b 2)-th transvectantmap,θ is dualtothe
1 2
− − −
tensorproductof2a-thand2b-thtransvectantmaps,andθ isanisomorphism.
3
By construction the φ : (a,b) Π form a basis of the space of SL(V)-
a,b
{ ∈ }
equivariantmorphisms S . Let K(a,b) denote the corresponding
2(m+n r)
− −→ C
weightr syzygy,writtenas
κ (u ,u ) = 0, (17)
i,j i j r i j
−−
X
wherethesumisquantifiedoverpairs(i,j)suchthat0 i,j randi+j r.
≤ ≤ ≤
(We havenotyetimposedtheconditioni j.) In orderto extractthe coefficient
≤
κ ,wewillconstructasequenceofmorphisms
i,j
S (S S ) η1 (S S ) 2 η2 ( S ) ( S )
2 m n m n ⊗ m+n 2i m+n 2j
⊗ −→ ⊗ −→ − ⊗ −
i j
M M
η3 S S η4 S ,
m+n 2i m+n 2j 2(m+n r)
−→ − ⊗ − −→ −
whereη isthenaturalinclusion
1
1
v v (v v +v v ),
1 2 1 2 2 1
· −→ 2 ⊗ ⊗
η isanisomorphism,η isthetensorproductofnaturalprojections,andη isthe
2 3 4
(r i j)-thtransvectionmap.
− −
In §2.6 – 2.7 below, we will give precise symbolic formulae for these maps.
Oncethisisdone,thefollowingpropositionisimmediate.
Proposition2.2. Foranyp = (a,b) Π(m,n;r),theendomorphism
∈
η η η η δ θ θ θ :S S
4 3 2 1 3 2 1 2(m+n r) 2(m+n r)
◦ ◦ ◦ ◦ ◦ ◦ ◦ − −→ −
ξ
| (a{,zb) }
isthemultiplicationbyκ .
i,j
2.6. In order to describe θ we will realise S as the space of order
1 2(m+n r)
−
2(m+n r)formsinz,andS S asthespaceofbihomogeneous
2m 2 4a 2n 2 4b
− − − ⊗ − −
formsoforders(2m 2 4a,2n 2 4b)inx,y respectively.Then
− − − −
θ :f(z)
1
−→
(xy)r 2a 2b 2
− − − [(x∂ )2m 2a+2b r(y∂ )2n+2a 2b rf(z)].
z − − z − −
(2m+2n 2r)!
−
10 ABDESSELAMANDCHIPALKATTI
We realiseS (S ) S (S )as the spaceof quadrihomogeneousforms of
2 m 1 2 n 1
− ⊗ −
orders(m 1,m 1,n 1,n 1)respectivelyinp,q,u,v,whicharesymmetric
− − − −
inthevariablepairsp,qandu,v. Then
(pq)2a(uv)2b
θ :g(x,y)
2
−→ (2m 4a 2)!(2n 4b 2)! ×
− − − −
[(p∂ )m 2a 1(q∂ )m 2a 1(u∂ )n 2b 1(v∂ )n 2b 1g(x,y)].
x − − x − − y − − y − −
2.7. Now realise S (S S ) as the space of forms of orders (m,n,m,n)
2 m n
⊗
respectively in p,u,q,v which are symmetric with respect to the simultaneous
exchange of variable pairs p q,u v. Inside this space, the image of δ
↔ ↔
consists of those forms which are antisymmetric in each of the pairs p,q and
u,v. Then
δ θ : h(p,q,u,v) (pq)(uv)h(p,q,u,v).
3
◦ −→
RealisingS S as biforms in x,y, the compositemorphismη
m+n 2i m+n 2j 3
− ⊗ − ◦
η η sendsQ(p,u,q,v) to
2 1
◦
h(m,n;i)h(m,n;j)[Ωi Ωj Q(p,u,q,v)],
pu qv
followed by the substitutions p,u x and q,v y. The multiplier h is as
→ →
in§1.6. Finally,
η :R(x,y)
4
−→
h(m+n 2i,m+n 2j;r i j)[Ωr i jR(x,y)] .
x−y− x,y z
− − − − →
2.8. The h factors are introduced to ensure that if Ψ = (u ,u ) , then the
i j r i j
−−
map(see§2.3)
η η f : S S
4 1 Ψ 2(m+n r) 2(m+n r)
◦···◦ ◦ − −→ −
is the identity. By contrast, the normalising factors appearing in θ are not so
i
crucial;their purposeis merelyto simplifysome intermediateexpressions. Their
omissionwouldhavetheharmlesseffectofmultiplyingeachsyzygycoefficientby
thesamefactor.
2.9. Torecapitulate,foreach(a,b) Π(m,n;r),theendomorphismofS
2(m+n r)
∈ −
definedbythecomposite