Planar algebras and the decategorification of bordered Khovanov homology

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by  Lawrence Roberts

PLANAR ALGEBRAS AND THE DECATEGORIFICATION OF
BORDERED KHOVANOV HOMOLOGY
LAWRENCE P. ROBERTS
5
1
0 Abstract. We give a simple, combinatorial construction of a unital, spherical, non-
2 degenerate ∗-planar algebra over the ring Z[q1/2,q−1/2]. This planar algebra is similar
in spirit to the Temperley-Lieb planar algebra, but computations show that they are dif-
v
ferent. The construction comes from the combinatorics of the decategorifications of the
o
N type A and type D structures in the author’s previous work on bordered Khovanov ho-
mology [10]. In particular, the construction illustrates how gluing of tangles occurs in

5 the bordered Khovanov homology ([9]) and its difference from that in Khovanov’s tangle
homology, [5], without being encumbered by any extra homological algebra. It also pro-

] videsasimpleframeworkforshowingthatthesetheoriesarenotrelatedthroughasimple
T
process,therebyconfirmingrecentworkofA.Manion,[7]. Furthermore,usingKhovanov’s
G
conventions and a state sum approach to the Jones polynomial, we obtain new invariant
. for tangles in Σ×[−1,1] where Σ is a compact, planar surface with boundary, and the
h
t tangle intersects each boundary cylinder in an even number of points. This construction
a naturally generalizes Khovanov’s approach to the Jones polynomial.
m
[

2
v
1 1. Introduction
0
5
Let S be an oriented two-sphere.
5
.
1 Definition 1 (Disc configurations). A disc configuration in S is a non-empty ordered set
0 D = (←D− ;←D− ,...,←D− ), m 0, of closed discs embedded in S, and a choice of points
4 0 1 m ≥
1 ←− ∂←D−i for each i = 0,...,m, such that
: ∗Di ∈
v
(1) ←D− int←D− for each i 1, and
i i 0
X ⊂ ≥
(2) ←D− ←D− = for i,j 1
i j
r ∩ ∅ ≥
a
An example of a disc configuration is given in Figure 1.
Each disc ←D− in D inherits an orientation from S, and ∂←D− is oriented as the bound-
i i
ary ←D− . The discs ←D− will be called inside discs. The outside discs for D are the discs
i i
→−D = S int←D− .
i i
\
Given a d(cid:0)isc con(cid:1)figuration D we identify an oriented sub-surface of S
m
ΣD = ←D−0 int←D−i
\
(cid:32) (cid:33)
i=1
(cid:91)
1
2 LAWRENCEP.ROBERTS
4
5
3
←D− ∗ ∗
1 ←D−
3
6
∗
←D−
2 2
←D−
0 ∗
1
Figure 1. A planar diagram with signature (3;2,2,1) subordinate to the
disc configuration D. ΣD is the shaded planar surface. The points in ∂D0
have been labeled according to their ordering.
Definition 2 (Planar diagram). A planar diagram P subordinate to a disc configuration D
is a set of arcs and circles properly and disjointly embedded in ΣD such that each boundary
circle of ΣD intersects the arcs of P in an even number of points.
Given P let ∂←D−i P be ordered according to the orientation of ∂←D−i ←− . the signature
∩ \{∗Di}
of P, denoted Sign(P), is the tuple (n ;n ,...n ) where 2n = ∂←D− P . The diagram
0 1 m i i
| ∩ |
in Figure 1 is a planar diagram.
To an oriented circle with a marked point and 2n labeled points, we will associate a
free Z[q1/2,q−1/2]-module . Our goal is to∗define for any planar diagram P subordinate
2n
to D, with Sign(P) = (n ;In ,...,n ), a linear map
0 1 m
Z :
P I2n1⊗I2n2⊗···⊗I2nm −→ I2n0
where the tensor products are taken over Z[q1/2,q−1/2].
The module comes from the idempotents of the differential bigraded algebra Γ
2n 2n
I B
used in bordered Khovanov homology [9], [8], and which supports the decategorification of
differential bigraded modules over Γ , [10]. The maps Z are a combinatorial generaliza-
2n P
B
tion to planar surfaces of the combinatorics of decategorification in [10]. Our motivation,
besides the simplicity of the construction, is 1) to use these maps to compare the construc-
trions of bordered Khovanov homology to Khovanov’s tangle homology, and 2) to give a
simple generalization of the Jones polynomial to tangles with good compositional proper-
ties, that is different from those the author could find in the literature.
PLANAR ALGEBRAS AND THE DECATEGORIFICATION OF BORDERED KHOVANOV HOMOLOGY 3
1.1. The modules . To an oriented circle C, embedded in an oriented sphere S, with
2n
I
a marked point C, and a subset Q C of 2n 0 points, we assign the free
∗ ∈ ⊂ \{∗} ≥
Z[q1/2,q−1/2]-module generated by the elements of the following set:
Definition 3 (Cleaved links). is the set of equivalence classes of pairs (L,σ) where
n
CL
(1) L is a link embedded in S so that each component of L transversely intersects C,
(2) L C = Q
(3) σ :∩Circles(L) +,
−→ { −}
where two such pairs are equivalent if there is an orientation preserving diffeomorphism of
S which maps the equator to the equator, carries L to L , and preserves the marked points
1 2
and decorations on each component.
The elements of ∞ will be called cleaved links and the maps σ will be called deco-
n=0CLn
rations. We note that is the empty link in a sphere with an marked equator.
0
(cid:83) CL
Definition 4. For each n 0, is the free Z[q1/2,q−1/2]-module generated by the ele-
2n
≥ I
ments of . The generator corresponding to (L,σ) will be denoted I .
n n (L,σ)
CL ∈ CL
Examples: When n = 0, there is only one equivalence class I of cleaved links, so =
0 0
Z[q1/2,q−1/2]I . The generators of are cleaved links with one component K which iInter-
0 2
I
sectstheequatorC exactlytwice. K canbedecoratedwitheithera+ora ,corresponding
to the two generators I+ and I− for 2. Thus, 2 ∼= Z[q1/2,q−1/2]I+ Z[q1−/2,q−1/2]I−. For
I I ⊕
, there are twelve generators corresponding to the elements in , depicted and labeled
4 2
I CL
in Figure 2.
1.2. The maps Z . TodefinethemapZ wedescribeitonthegeneratorsof
P P I2n1⊗I2n2⊗
and then extend linearly. We start with some preliminaries.
···⊗I2nm
Definition 5 (Multiply Cleaved Links). A decorated, multiply cleaved link (M,σ) subor-
dinate to a disc configuration D in S is a (possibly empty) embedded submanifold M S,
consistingoffinitelymanycircles, Circles(M), andadecorationmapσ : Circles(M)⊂
−→
+, , such that
{ −}
(1) M intersects the boundary ∂←D−i of each disc ←D−i transversely, and away from ←− ,
∗Di
and
(2) M has no component contained in int→−D , or in int←D− for i = 1,...,m.
0 i
Each multiply cleaved link is built on top of a planar diagram:
Definition 6. For a decorated, multiply cleaved link (M,σ) subordinate to D, P(M) is the
associated planar diagram M ΣD.
∩
Then P(M) is subordinate to D and inherits the orderings and marked points of M. For
example, given the multiply cleaved link (M,σ) in Figure 3, P(M,σ) is the planar diagram
inFigure1. WeuseP(M)toimportallthenotionsdefinedforplanardiagramstomultiply
cleaved links.
4 LAWRENCEP.ROBERTS
2 1 2 1
∗ ∗
3 4 3 4
A B
± ±±
2 1 2 1
∗ ∗
3 4 3 4
C D
±± ±
Figure 2. The twelve generators of grouped based on their inside and
4
I
outside matchings. There a two generators of type A and D, and four of
type B and C, as determined by the choice of decorations.
Definition 7. Let (M,σ) be a decorated, multiply cleaved link. The set of equivalence
classes of decorated, multiply cleaved links (M,σ) with signature (n ;n,...,n ) will be
0 1 m
denoted (n ;n,...,n ).
0 1 m
MCL
The advantage to using multiply cleaved links is that the have “boundaries”:
Definition 8. For each i = 0,...,m, the ith-boundary map
∂ : (n ;n,...,n )
i MCL 0 1 m −→ CLni
takes (M,σ) to the cleaved link ∂ (M,σ) defined by
i
(1) ∂ (M,σ) is subordinate to (←D− ),
i i
(2) ∗∂i(M,σ) = ∗←D−i
(3) Circles(∂ (M,σ)) is the subset of the components of M which intersect ∂←D− ,
i i
(4) σ is the restriction of σ to Circles(∂ (M,σ))
∂i(M,σ) i
For example, the 2nd boundary of the multiply cleaved link on the left of Figure 3 is the
diagram on the right of that figure.
PLANAR ALGEBRAS AND THE DECATEGORIFICATION OF BORDERED KHOVANOV HOMOLOGY 5
4
−
5
3
+
←D−1∗ ←D−3∗
∂
2
∗←D−2
6
+
∗
←D−2 2 −
+
←D−
0 ∗ −
1
Figure 3. The left diagram is a decorated, multiply cleaved link (M,σ)
with P(M,σ) equal to the planar diagram in Figure 1. The inside and
outside matchings are shown as dashed lines. The 2nd boundary ∂ (M,σ) is
2
shown on the right.
Definition 9. Let P be a planar diagram subordinate to D. Then
(P) = (M,σ) (Sign(P)) P(M) = P
M ∈ MCL
If we are given (Li,σi) ∈ CLni(cid:8)for i = 0,...,m, then (cid:12)(cid:12) (cid:9)
(L ,σ );(L ,σ ),...,(L ,σ ) = (M,σ) (P) ∂ (M,σ) = (L ,σ )
P 0 0 1 1 m m i i i
M ∈ M
We will us(cid:0)ually shorten this to P(L0;L(cid:1)1,..(cid:8).,Lm) with the un(cid:12)derstanding that th(cid:9)ere are
M (cid:12)
decoration maps σ on each of the cleaved links. The signature and the associated planar
i
diagram do not depend on the decorations. The decorations, however, return in the weight
of the link:
Definition 10. The weight W(M,σ) Z[q1/2,q−1/2] of a decorated, multiply cleaved link
∈
(M,σ) is the quantity
W(M,σ) =  q1−21NM(C)  q−1+21NM(C)
·
 C ∈Ci(cid:89)rcles(M)   C ∈Ci(cid:89)rcles(M) 
 σ(C)=+   σ(C)=− 
   
   
where N (C) is the number of circles ∂←D− , i 0, which non-trivially intersect the circle
M i
C Circles(M). If Circles(M) = then W≥(M,σ) = 1.
∈ ∅
6 LAWRENCEP.ROBERTS
We are now in a position to define Z . For a generator
P
ξ = I I I
(L1,σ1)⊗ (L2,σ2)⊗···⊗ (Lm,σm) ∈ I2n1 ⊗I2n2 ⊗···⊗I2nm
with (L ,σ ) let
i i ∈ CLni
Z (ξ) = W(M,σ)I
P ∂0(M,σ)
(M,σ)(cid:88)∈M(P)
∂i(M,σ)=(Li,σi),i≥1
Recall that = Z[q1/2,q−1/2], so this definition applies even if P does not intersect some
0
I
of the boundaries ∂←D− for i 0.
i
≥
Warning: Changing the marked point on ∂←D− can change the identification with the
i
generators of , and thus change the map Z . An example below illustrates this phe-
I2ni P
nomenon.
1.3. Planar algebra structure. Planar diagrams can be composed, and we will de-
scribe how the partition maps behave under this composition. Suppose T and R are
planar diagrams subordinate to disc configurations D and D with signatures Sign(T) =
T R
(n ;n ,...,n )andSign(R) = (n(cid:48);n(cid:48),...,n(cid:48) )suchthatn = n(cid:48) forsomei 1,...,m(cid:48) .
0 1 m 0 1 m 0 i ∈ { }
Then there is a planar diagram R T subordinate to
i
◦
D D = (←D−(cid:48);←D−(cid:48),...,←D−(cid:48) ,←D− ,...,←D− ,←D− ,...,←D−(cid:48) )
R◦i T 0 1 i−1 1 m i+1 m(cid:48)
obtained by gluing the discs ST\int→−D0 and SR\int←D−(cid:48)i along their boundaries, so that ∗←D−0
is glued to ∗←D−(cid:48)i, and gluing induces and ordered bijection ∂←D−0∩T −→ ∂←D−i(cid:48) ∩R.
The definition of planar algebras in [2], [3] provides a composition Z Z of Z and Z
R i T R T
◦
along the ith entry in the tensor product of the domain of ZR. In particular, if ai 2n(cid:48)
∈ I i
and b then
j ∈ I2nj
(1) ZR◦iT(a1⊗···⊗ai−1⊗b1⊗···⊗bm(cid:48)⊗ai+1⊗···⊗am)
:= ZR(a1 ai−1 ZT(b1 bm(cid:48)) ai+1 am)
⊗···⊗ ⊗ ⊗···⊗ ⊗ ⊗···⊗
In section 2 we prove
Theorem 11 (Composition). Let T and R be planar diagrams so that R T is defined.
i
◦
Then the partition map Z equals the map Z Z
R◦iT R◦i T
In addition,
Theorem 12. The maps Z have the following properties:
P
(1) (Temperley-Lieb property) If P has a circle component embedded in the interior of
ΣD then
ZP = (q+q−1)ZP(cid:48)
where P(cid:48) is the planar diagram found by removing this component,
PLANAR ALGEBRAS AND THE DECATEGORIFICATION OF BORDERED KHOVANOV HOMOLOGY 7
(2) (Conjugation property) For p Z[q1/2,q−1/2] let the conjugate p∗ be the polynomial
∈
p(q−1). There is an involution on which allows us to extend to a morphism
n
CL
ξ ξ∗ on . For this map
2n
−→ I
Z (ξ∗) = Z (ξ)∗
P P
In the language of [3] these properties show that the planar algebra is a unital, spherical,
-planar algebra. It is unital since each planar diagram for a disc configuration (←D− ) gives
0
∗
rise to a map Z which satisfies the composition requirement, [3]. It is spherical since
P
dim = 1 (over Z[q1/2,q−1/2]) and the maps are invariant under isotopies. Finally, the
0
I
conjugation ξ ξ∗ acts correctly for it to be a -planar algebra.
−→ ∗
Furthermore, it is non-degenerate in the sense that a standard pairing considered in [?]
is non-degenerate on each , see section 2.3. However, the pairing is not positive definite,
2n
I
as usually desired in the planar algebra approach to studying subfactors, [?].
Following the terminology of planar algebras, as in [2], we will now call Z the parti-
P
tion map for P.
In the next section 2 we justify the properties cited above. In section 3 we describe how
this can be extended throught the usual Kauffman state summations to give invariants of
tangles which generalize the Jones polynomial and compose well. In section 4 we provide
some example computations. Finally, in section 5, we compare this construction to the
Temperley-Lieb planar algebra and the decategorification of Khovanov’s tangle homology.
2. Properties of the partition maps for planar diagrams
We now verify the Temperley-Lieb, Conjugation, and Composition properties for the par-
tition maps of planar diagrams.
2.1. Properties allowing simplifications of the diagram:
Proposition 13 (Temperley-Lieb Property). Let P be a planar diagram and C be a circle
component of P. Let P(cid:48) be the planar diagram after deleting C from P, then Z = (q +
P
q−1)ZP(cid:48)
Proof. Thereisa2:1map (P) (P(cid:48))whichtakes(M,σ)to(M(cid:48),σ(cid:48)), where(M(cid:48),σ(cid:48))
M −→ M
hasthesameconfigurationofdiscs,markedpoints,andarcsasM,butthecircleC hasbeen
removed. Under this map σ(cid:48) is the restriction of σ to the remaining circles. The pre-image
of(M(cid:48),σ(cid:48))consistsofatwoelementset M(cid:48) wherewehaveaddedC backwithdecoration
±
{ }
σ(C) = . If we partition (P) into (P) (P), based on the value of σ(C), then
+ −
± M M (cid:116)M
Z (ξ) can similarly be decomposed. From the definition of Z the term corresponding to
P P
+(P) is seen to be qZP(cid:48)(ξ) while the other is q−1ZP(cid:48)(ξ). (cid:3)
M
Corollary 14. Let P be a planar diagram subordinate to D = (←D− ;←D− ) with signaturw
0 1
(0;0). Then P consists solely of some number N of circles, and Z is the map
P
Z[q1/2,q−1/2] Z[q1/2,q−1/2] which multiplies by (q+q−1)N.
−→
8 LAWRENCEP.ROBERTS
Note that (q+q−1)N is also the Jones polynomial of such a tangle, using Khovanov’s con-
ventions from [4].
In addition, any disc ←D− with n = 0 can be “removed:”
i i
Proposition 15. Suppose P is a planar diagram subordinate to D = (←D− ;←D− ,...,←D− )
0 1 m
with m 1. Suppose ←D− has n = 0 for some i 1. Let P(cid:48) be planar diagram determined by
i i
P and s≥ubordinate to D(cid:48) = D Di , with the inh≥erited order. Then ZP(cid:48) equals ZP under the
\{ }
canonicalisomorphism with
I2n1⊗···⊗I2ni⊗···⊗I2nm I2n1⊗···⊗I2ni−1⊗I2ni+1⊗···⊗I2nm
Proof. Since n = 0, = Z[q1/2,q−1/2] in the tensor product , thereby
i I2ni I2n1 ⊗···⊗I2nm
establishing the canonical isomorphism in the statement. The image of each generator is
determined only by the configuration of circles in multiply cleaved links in (P). As these
M
are the same in the two diagrams, the partition maps ZP and ZP(cid:48) are identical, under the
isomorphism. (cid:3)
Corollary 16. If P is the empty tangle, (i.e. it has no components ) then both
I2n1⊗···⊗
and areisomorphictoZ[q1/2,q−1/2], andZ istheidentitymapZ[q1/2,q−1/2]
I2nm I2n0 P −→
Z[q1/2,q−1/2] under the isomorphisms.
2.2. Partition maps and conjugation: Multiply cleaved links admit an involution we
will call conjugation:
Definition 17. Let (M,σ) be a decorated, multiply cleaved link. The conjugate (M,σ)∗ is
the decorated, multiply cleaved link (M,σ(cid:48)) where σ(cid:48)(C) = if and only if σ(C) = .
± ∓
We now consider the effect of the conjugation (M,σ) (M,σ)∗ on the partition map.
−→
Definition 18. For p Z[q1/2,q−1/2], p∗(q) = p(q−1) is the conjugate of p.
∈
The following proposition follows directly from the definitions.
Proposition 19. Let (M,σ) be a decorated multiply cleaved link. Then Sign(M,σ)∗ =
Sign(M,σ), and W (M,σ)∗ = W(M,σ)∗ in Z[q±1/2].
Weextendthesecon(cid:0)jugatem(cid:1)apstothetensorproductsI2n1⊗···I2nm bylinearlyextending
p(q)I I p∗(q)I I
(L1,σ1)⊗···⊗ (Lm,σm) −→ (L1,σ1)∗ ⊗···⊗ (Lm,σm)∗
Proposition20(Conjugationproperty). LetP beaplanardiagramwithsignatureSign(P) =
(n ;n ,...,n ), and ξ , then Z (ξ∗) = Z (ξ) ∗.
0 1 m ∈ I2n1 ⊗···⊗I2nm P P
Proof. Let ξ = I(L1,σ1)⊗···⊗I(Lm,σm) be a generator of I2n(cid:0)1 ⊗···(cid:1)⊗I2nm and (M,σ) be a
decorated multiply cleaved link in (P) with ∂ (M,σ) = (L ,σ ) for i 0. Then (M,σ)∗
i i i
M ≥
has P(M∗) = P. Thus conjugation is a bijection on (P). In addition, ∂ (M,σ)∗ =
i
M
(L ,σ )∗ for i 0. Since W((M,σ)∗) = W(M,σ)∗ in Z[q1/2,q−1/2]
i i
≥
W(M,σ)I = W(M(cid:48),σ(cid:48))∗I
∂0(M,σ) ∂0(M(cid:48),σ(cid:48))∗
(M,σ)(cid:88)∈M(P) (M(cid:48),σ(cid:48)(cid:88))∈M(P)
∂i(M,σ)=(Li,σi)∗,i≥1 ∂i(M(cid:48),σ(cid:48))=(Li,σi),i≥1
An thus, by definition, Z (ξ∗) = Z (ξ) ∗. (cid:3)
P P
(cid:0) (cid:1)
PLANAR ALGEBRAS AND THE DECATEGORIFICATION OF BORDERED KHOVANOV HOMOLOGY 9
2n 1
2n1 2
−
2n2 3
−
∗ ←D−1 ←D−2 ∗
3 2n−2
1 2 2n−12n
Figure 4.
2.3. Non-degeneracy.
In this section we show that the planar algebra is non-degenerate. In fact, the pairing
depicted defined by the diagram in Figure 4 is a sum of hyperbolic pairs.
Definition 21. Let (L,σ) be subordinate to (←D−. Then the dual (L,σ) of (L,σ) is
n
∈ CL
the cleaved link subordinate to (→−D) obtained by changing the orientation on the sphere, but
fixing the ordering of the points L ∂←D−.
∩
For L,L(cid:48) , let L,L(cid:48) be the image of L L(cid:48) under the map Z :
n P 2n 2n
∈ CL (cid:104) (cid:105) ⊗ I ⊗ I −→
Z[q1/2,q−1/2] for the diagram P in Figure 4.
Proposition 22 (Non-degeneracy). Let L,L(cid:48) , then
n
∈ CL
0 L(cid:48) = L
L,L(cid:48) = (cid:54)
(cid:104) (cid:105) 1 L(cid:48) = L
(cid:26)
Proof. ToseethiswecomputethepartitionmapforthediagraminFigure4. Thediagrams
in (P) are obtained by picking two inside matchings m and m and filling in the discs
←−1 ←−2
M
←D−i withthem. Then∂1(P 1←m−1 2←m−2) = ←m−1#→−m2 where→−m2 isthedualof←m−2,i.e. thesame
◦ ◦
matching considered in an outside disc. On the other hand, ∂ (P m m ) = m #m .
2 1←−1 2←−2 →−1 ←−2
◦ ◦
These boundaries are dual, so only dual cleaved links can have non-zero pairing. To see
that these pair to give 1, note that any circle, with any decoration, in the multiply cleaved
link necessarily intersects both inside discs. Thus the total weight will be multiples of
q1−1/2·2 = 1 and q−1+1/2·2 = 1. (cid:3)
2.4. Composition of the partition maps for planar diagrams. We now verify that
the partition maps compose according to the requirements of a planar algebra.
2.5. Gluing multiply cleaved links: First, wewilldefineagluing formultiplycleaved
i
◦
links. Suppose the multiply cleaved link (M,σ) is subordinate to D = (←D− ;←D− ,...,←D− )
0 1 m
and (N,ν) is subordinate to D(cid:48) = (←D−(cid:48);←D−(cid:48),...,←D−(cid:48) ) with ∂ (M,σ) = ∂ (N,ν). Then
0 1 m(cid:48) 0 i
(N,ν) (M,σ) is the (equivalence class) of decorated, multiply cleaved links subordinate
i
◦
to D(cid:48) D found by removing the disc ←D−(cid:48) from the sphere S and gluing in the disc ←D− so
◦i i N 0
10 LAWRENCEP.ROBERTS
that ∗←D−0 is glued to ∗←D−(cid:48)i, and N ∩∂←D−(cid:48)i is identified with M ∩←D−0, in an order preserving
manner. The decoration map for (N,ν) (M,σ) is obtained by restriction of ν and σ
i
◦
to the circles intersecting ←D− and →−D(cid:48). Since ∂ (M,σ) = ∂ (N,ν) circles which intersect
0 i 0 i
C = ∂←D− = ∂←D−(cid:48) receive the same decoration from ν and σ.
0 i
It is straightforward that this provides a well-defined map
: (L(cid:48);L(cid:48),...,L(cid:48) ,L,L(cid:48) ,...,L(cid:48) ) (L;L ,...,L )
◦i MR 0 1 i−1 i+1 m(cid:48) ×MP 1 m
(L(cid:48);L(cid:48),...,L(cid:48) ,L ,...,L ,L(cid:48) ,...,L(cid:48) )
−→ MR◦iP 0 1 i−1 1 m i+1 m(cid:48)
Theorem 23. Let P and R be planar diagram with signatures Sign(P) = (n ;n ,...,n )
0 1 m
and Sign(R) = (n(cid:48);n(cid:48),...,n(cid:48) ), with n = n(cid:48). The partition map Z equals the map
0 1 m(cid:48) 0 i R◦iP
Z Z
R i P
◦
Proof. AsbothmapsarelinearoverZ[q1/2,q−1/2]itsufficestochecktheresultongenerators
ξ = IL(cid:48)1⊗···⊗IL(cid:48)i−1⊗IL1⊗···⊗ILm⊗IL(cid:48)i+1⊗···⊗IL(cid:48)m(cid:48)
of the tensor product that is their common domain:
I2n(cid:48)1⊗···⊗I2n(cid:48)i−1⊗I2n1⊗···⊗I2nm⊗I2n(cid:48)i+1⊗···⊗I2n(cid:48)m(cid:48)
Thus, we will show that Z (ξ) equals Z Z (ξ).
R◦iP R◦i P
(2) ZR◦iZP (ξ) = ZR(IL(cid:48)1⊗···⊗IL(cid:48)i−(cid:0)1⊗ZP(IL1(cid:1)⊗···⊗ILm)⊗IL(cid:48)i+1⊗···⊗IL(cid:48)m(cid:48))
Usingl(cid:0)inearityan(cid:1)dthedefinitionsofZRandZP,weseethat ZR◦iZP (ξ) = L(cid:48)∈CLn(cid:48)0 WL(cid:48)IL(cid:48)
where
(cid:0) (cid:1) (cid:80)
(3) WL(cid:48) = W(N,ν) W(M,σ)
·
L∈(cid:88)CLn0 (cid:0)(N,ν),(M,σ)(cid:1)∈M(cid:88)(L(cid:48);L(cid:48)1,...,L,...,L(cid:48)m)×
M(L;L1,...,Lm)
However, we will show that
Lemma 24. The composition induces a bijection
i
◦
Ψ : (L(cid:48);L(cid:48),...,L(cid:48) ,L,L(cid:48) ,...,L(cid:48) ) (L;L ,...,L )
MR 1 i−1 i+1 m(cid:48) ×MP 1 m
L∈(cid:91)CLn(cid:0) (cid:1)
(L(cid:48);L(cid:48),...,L(cid:48) ,L ,...,L ,L(cid:48) ,...,L(cid:48) )
−→ MR◦iP 1 i−1 1 m i+1 m(cid:48)
such that W N M) = W(N,ν) W(M,σ).
i
◦ ·
Then the dou(cid:0)ble sum in equation (3) is a sum over the elements of the union in lemma 24,
and, assuming the lemma, we conclude that
WL(cid:48) = W(D,η)
(D(cid:88),η)∈
MR◦iP(L(cid:48);L(cid:48)1,...,L(cid:48)i−1,L1,...,Lm,L(cid:48)i+1,...,L(cid:48)m(cid:48))
which is the coefficient of IL(cid:48) in ZR◦iP(ξ) . Thus, ZR◦iZP (ξ) = ZR◦iP(ξ).
(cid:0) (cid:1)
We now prove the lemma.