Annales de l’institut Fourier (AIF)

Wu, F. Y.

King, C.

Lu, W. T.

ANNALES DE L’INSTITUT FOURIER
F. Y. WU
C. KING
W. T. LU
On the rooted Tutte polynomial
Annales de l’institut Fourier, tome 49, no 3 (1999), p. 1103-1114
<http://www.numdam.org/item?id=AIF_1999__49_3_1103_0>
© Annales de l’institut Fourier, 1999, tous droits réservés.
L’accès aux archives de la revue « Annales de l’institut Fourier »
(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-
nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-
tion commerciale ou impression systématique est constitutive d’une in-
fraction pénale. Toute copie ou impression de ce fichier doit conte-
nir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Ann. Inst. Fourier, Grenoble
49, 3 (1999), 1103-1114
ON THE ROOTED TUTTE POLYNOMIAL (*)
by F.Y. WU, C. KING and W.T. LU
1. The Tutte polynomial.
Consider a finite graph G with vertex set V and edge set E. A spanning
subgraph G\S) C G' is a subgraph of G containing all members of V and
an edge set S C E. Let C be a set of q distinct colors. A q- coloring of G is
a coloring of the vertices in V such that two vertices connected by an edge
bear different colors. It is well-known that the number of g-colorings of G
is given by the chromatic polynomial (see [1])
(1) P(G-,q)=^q^\-l^,
SCE
where p(S) is the number of components in the spanning subgraph G^S).
Alternately, we can regard (1) as generating colorings of components of
spanning subgraphs of G with q colors with an edge weight —1.
As an extension of the chromatic polynomial, Tutte [2], [3], [4] intro-
duced what is now known as the Tutte polynomial
(2) Q{G'^v)= ̂ ̂ )^|-M+p(^^t,v) ^ , r f''
SCE
Indeed, one has the relation
(3)________ P(G^q) = (-l^'^G; - g, -1).
(*) This work is supported in part by the National Science Foundation grants DMR-
9614170 (FYW and WTL) and DMS-9705779 (CK).
Keywords: Graph colorings — Rooted Tutte polynomial — Planar partition — Duality
relation.
Math. classification: 05C30 - 05C15.
1104 F.Y. WU, C. KING AND W.T. LU
In view of (3), it is useful to write (2) as
(4) Q(G'^v) = v-^ ̂ (^ l̂,
SCE
so that for vt = q = positive integers, the Tutte polynomial (4) generates
colorings of components of spanning subgraphs of G with q colors and edge
weights z?, instead of v == —1.
For planar G with dual graph Op, it is well-known that the Tutte
polynomial possesses the duality relation
(5) vQ{G'^v)=tQ(GD\v.t),
a relation first observed by Whitney [5].
2. The rooted Tutte polynomial.
We extend the definition (4) to a rooted Tutte polynomial.
A vertex is rooted, or is a root, if it is colored with a prescribed (fixed)
color. A graph is rooted if it contains rooted vertices.
Let R denote a set of n roots located at vertices {r i , r2? • • • ^n}-
A color configuration is a map x: R i—^ G, and as a convenient shorthand we
write x(ri) = Xi for i == 1 ,2 , . . . , n. A component of a spanning subgraph
is exterior if it contains one or more roots, and is interior otherwise. An
exterior component is proper if all roots in the component are of the same
color. A spanning subgraph G'{S) is proper if all its exterior components
are proper. An edge set Sx C E is proper if the spanning subgraph G'{Sx)
it generates is proper.
For a prescribed color configuration {a; 1 ,^2 , . . . ,^n} of the n roots,
we introduce in analogy to (4) the rooted Tutte polynomial^ ̂
(6) Qx^...x^G^v) = v-^ ̂ {vtr-^v^^
S^CE
where the summation is taken over all proper edge sets 63;, and pin (63;) is
the number of interior components of G^&c). Thus, as in (3), we have for
(^ Strictly speaking, it is the expression v\v\Qx^x•2...xn(G\t^v) which is a polynomial
in v and t.
ON THE ROOTED TUTTE POLYNOMIAL 1105
positive integral q the relation
(7) (-l)I^^JG;-^-!)
= the number of g-colorings of G
with color configuration {x\^x^,..., Xn}'
Clearly, the expression (6) depends on how the n roots are partitioned into
subsets of different colors, and the actual colors do not enter the picture.
The coloring configuration {a;i, x^ , . . . , Xn} induces a partition X of R
into blocks (subsets) such that all roots in one block are of one color, and
colors of different blocks are different. Namely, two elements r^rj € R
belong to the same block of X if and only if they have the same prescribed
color Xi = Xj.
Consider now the summation in (6). Let G'{S) be any (not necessarily
proper) spanning subgraph of G. The connected components of G'(5)
induce a partition on the set of vertices V of G. We get hence also a
partition 7r(5') on the set of rooted vertices -R by restricting this partition
to R. Clearly, the spanning subgraph G'(Sx) is proper if and only if the
partition 7r(Sx) is a refinement of the partition X. It follow that we can
rewrite (6) as
(8) Qx{G'^v)= ̂ Fx'(G;^),
x'^x
where
(9) Fx'{G^v)=v-^ ̂ {vtY-^v^.
S^CE
Tr(S^=X'
Here, we have abbreviated Qxix2...xrz by Qx^ which is permitted since the
actual colors do not enter the picture at this point. Also it is understood
that G is now a rooted graph, with root set R.
The expression (8) assumes the form of a transformation of a partially
ordered set. Its inverse is given by the Mobius inversion
(10) Fx(G'^v) = ̂ ^X^X)Qx'(G'^v\
x '
where (see [6])
^^ ^-l^'-W n (^(X')-!)!, i fX^X,
(11) /^(X^X) = < blocks GX
10, otherwise,
n^-^) being the number of blocks of X' that are contained in the block b
of X. Note that for n == 1 we have \X\ = \X' = 1, pin(S^) = p(S^) - 1,
1106 F.Y. WU, C. KING AND W.T. LU
and all edge sets S C E are proper. Hence we have
(12) Fx[G^ v) = Qx(G'^ v) = (vt^Q^ v), n = 1.
This completes the definition and general description of the rooted Tutte
polynomial for any graph G.
3. Planar graphs.
From here on we consider G being planar with the n roots residing
around a single face of G. Without the loss of generality, we can choose
the face to be the infinite face and order the roots in the sequence
{ r i , r 2 , . . . ,rn,r-t} as shown in Fig. 1.
r2 rs
Figure 1. A planar graph G with n roots. The graph is denoted
by the shaded region and the n roots by the black circles.
A partition X of the n roots is non-planar if two roots of one block
separate two roots of another block in the cyclic sequence. Otherwise X is
planar. For a given n, there are bn partitions, where (see [9])
00 00 00
(13) bn= ̂ [n!/]^!)"1^!], ^i/m^=n,
rn^=0 i/=l y=l
and of the bn partitions c- - ̂ ,
are planar (see [7], [8]). We shall adopt the convention of writing
X = { < 7 , H . . . , . . . }
for colors {xi == xj, Xk = x^ = • • • , . . . } , with {ij}, {k£ . . . } , . . . each in order
(see [8]). For example, two partitions for n = 5 are
X^= {123,4,5}, |Xi|=3, planar,
( J - O )
X2 = {24,351}, |X2| = 2, non-planar.
Now if G is planar and X' is non-planar then by definition the summand
in (9) is empty and one has Fx'{G\t, v) == 0. Thus we have
ON THE ROOTED TUTTE POLYNOMIAL 1107
PROPOSITION 1. — For planar G
(16) Fx(G\ t, v) =0, if X is non-planar.
This proposition was first established in [9] for the Potts model
correlation function (see section 6) by considering its graphical expansion
similar to the consideration given in the above.
As a consequence of Proposition 1 and the use of (10), we now have:
COROLLARY 1. — Rooted Tutte polynomials associated with non-
planar partitions can be written as linear combinations of the rooted Tutte
polynomials associated with (refined) planar partitions.
Corollary 1 leads to the sum-rule identities reported in [9] for the
Potts correlation function. In the case of n = 4, for example, the identity
^{13,24} (G;^)=0
leads to the sum rule (see [9])
(17) 0{13,24}(G;^) = Q{13,2,4}(G;^) +Q{1,3,24}(G;^)
-Q{l,'2,3^}{G',t,v).
From here on we shall restrict our considerations to rooted Tutte
polynomials associated with the Cn planar partitions only.
4. The graph G*.
The rooted Tutte polynomial (6) possesses a duality relation for
planar graphs, which relates the rooted Tutte polynomial on a graph G to
that of a related graph G*. Here we define G*. Starting from a planar G,
place an extra vertex / in the infinite face and connect it to each root of G
by an edge. This gives a new graph G", which has one more vertex than G
and n additional edges. The dual graph of G" is also planar, and it has
a face F containing the extra vertex /. Now remove the n edges on the
boundary of F^ and the resulting graph is G*.
It is readily seen that the graph G* has
(18) \V^\=\VD\+n-l
1108 F.Y. WU, C. KING AND W.T. LU
vertices where [Volis the number of vertices of GD, the dual of G, and there
is a one-one correspondence between the edges of G and G*. We denote the
set of n vertices {r^, r^...,r^} of G* surrounding the face F by R*, with
T^ residing between the two edges (/, r^-i) and (/, r^) of G", where r-o = r^.
An example of a G and the related G* for n = 4 is shown in Fig. 2.
r2x-
r i H
rl T
--3
t
X N -
I
I---
- - -1
3
---?
C-..
(
?---
» ' ^
'x''
---y"'
1
- - -X-"
^3
'X'
Figure 2. A graph G (solid lines) and the related graph G*
(broken lines) for n = 4. Black circles denote roots of G and
crosses denote vertices of G*.
Clearly, the relation of G to G* is reciprocal, namely, we have
(G*)* = G.
Now each planar partition X of R induces a partition X* of the set R*
(see [10]). In order to define X*, for each block b of X we choose a point
in the infinite face of G, and connect all roots ri in b to this point by
drawing new edges. Because X is planar, the points for the blocks can be
chosen so that the edges of different blocks do not cross, and the resulting
extended graph is still planar. So this process divides the infinite face into
regions. The induced partition X* is then described by the condition that
all roots of R* in one region are regarded as belonging to one block of X*.
Alternately, for another way of defining X*, let P^, 1 < i < j <: n, be a
partition of R into two blocks, the sets { r ^ , . . . ,rj_i} and { r j , . . . ,7*z_i},
where all numbers are modulo n. Then, the partition X* induced by X is
defined by the condition that the two roots r̂ * and rj belong to the same
block in X* if and only if X is a refinement of P^.
If X induces X*, we write
X X*.
ON THE ROOTED TUTTE POLYNOMIAL 1109
Clearly, X* is planar, and we have
(19) |X|+ |X*|=n+l .
For the planar partition Xi in (15), for example, we have
(20) Xi = {123,4,5} -^ Xi* = {2,3,451}, |Xi| = |X^| == 3.
However, as a result of our labelling convention, the color configuration of
the partition (X*)* further induced by X* is a cyclic shift of that of X,
namely,
(21) {^ i ,^2 , . . . , ^n} —^ X* —^ {a;n,a;i,...,^-i}.
In the example above, for instance, we have
(22) {123,4,5} —— {2,3,451} —— {234,5,1}.
Finally, there is a one-one correspondence between the edge sets E
of G and £'* of G*, an edge set S C E defines a "complement" edge set
5* C E* by the condition that an edge is included in S* if and only if its
corresponding edge is not included in S. Clearly, we have
(5*)* == S.
5. The duality relation.
The rooted Tutte polynomial arises in statistical physics as the
correlation function of the Potts model (see next section). In a recent
paper [10] we have established a duality relation for the Potts correlation
function for planar G. However, the proof of the duality relation given
in [10] is cumbersome and not easily deciphered in graphical terms. Here we
re-state the results as two propositions in the context of the rooted Tutte
polynomial, and present direct graph-theoretical proofs of the propositions.
PROPOSITION 2. — For planar G and G* and the associated planar
partitions X —> X*, we have
(23) v^Fx(G^v)=t^\Fx^G^^^.
1110 F.Y. WU, C. KING AND W.T. LU
Proof. — Let Sx be a proper edge set on G. We have the Euler relation
(24) |5,|+|^|=|V|+|l^|-2
and, after eliminating n and \VD\ using (18), (19) and (24), the identity
(25) |^|+|x|-|y|=|^|+|x*|-|y*|,
which holds for any proper edge set Sx. Note that we have also the fact
(26) Tr(^) = X if and only if 7r(S^) = X\
Let c(S^) the number of independent circuits in the spanning sub-
graph G'(5^). Then we have
(27) p(^)=c(^*)+|X|.
Also, starting from the \V*\ isolated vertices on G*, one constructs G'^S^)
by drawing edges of S^ on G* one at a time. Since each edge reduces the
number of components by one except when the adding of an edge completes
an independent circuit, one has also
(28) p(S^=\Vf\-\S^+c^).
Eliminating c(5^) using (27) and (28) and making use of the relations
fP(^)=Pin(^)+|X|,
(29) <
\p(S^=p^(S^+\X^
one obtains
(30) pin(^) =Pin(^) + m - |̂ | - |X*1.
Proposition 2 now follows from the substitution of (30) into the right-hand
side of (23) where, explicitly,
(31) Fx'(Gt•,v,t)=t-\v'\ ̂ (^••^W,
S^CE'
^(S:)=X"
and the use of the identities (25) and (26). This completes the proof of
Proposition 2.
ON THE ROOTED TUTTE POLYNOMIAL 1111
Proposition 2 was first conjectured in [9] and established later in [10]
in the context of Potts correlation functions (see next section) without the
explicit reference to the polynomial form (9).
Remark. — For n == 1, the duality relation (23) for the rooted Tutte
polynomial becomes the duality relation (5) for the Tutte polynomial. This
is a consequence of (12).
PROPOSITION 3.
1) The rooted Tutte polynomials associated with the Cn planar
partitions for G and G* are related by the duality transformation
(32) Ox(G;^)=^r,(X,y)Qy(G*;^),
y
where Tn is a Cn x Cn matrix with elements
(33) T,(X, Y) = r+1 ^ (^-i^i/^ v'). x' -^ y.
x'-^x
2) The matrix Tn satisfies the identity
(34) [Tn}\X, X') = 6(x^x^6(x^ ̂ )... 6{x^ x[).
Proof. — The transformation (32) follows by combining (8) and (10)
with Proposition 2, and its uniqueness is ensured by the uniqueness of the
Mobius inversion. The property (34) is a consequence of (21). D
Proposition 3 was first given in [10] in the context of the Potts
correlation function (see next section). Explicit expression of Tn for
n = 2,3,4 can be found in [10] and [11].
6. The Potts and the random cluster models.
It is well-known in statistical physics that the Tutte polynomial gives
rise to the partition function of the Potts model (see [12]). In view of the
prominent role played by the Potts model in many fields in physics, it is
useful to review this equivalence and the further equivalence of the rooted
Tutte polynomial with the Potts correlation function.
1112 F.Y. WU, C. KING AND W.T. LU
The estate Potts model [13] is a spin model defined on a graph G.
The spin model consists of \V\ spins placed at the vertices of G with each
spin taking on q different states and interacting with spins connected by
edges. Without going into details of the physics [12] which lead to the Potts
model, it suffices for our purposes to define the Potts partition function
(35) Z(G^v)== ̂ (f^l5!,
SCE
the ?7-point partial partition function
(36) Zx(G'^v)= ̂ ̂ in(̂ l̂
S^CE
and the n-point correlation function
(37) Pn(G;^i, x^... ,xn) = Pn(G;X) = Zx(G;g, v)/Z(G;q, v),
where again, in analogy to notation in Sections 1 and 2, we have denoted
the color configuration {x^, x ^ , . . . , Xn} by the associated partition X. More
generally, for any real or complex g, the partition function (35) defines the
random cluster model of Fortuin and Kasteleyn [14], which coincides with
the Potts model for integral q.
Relating this to the Tutte polynomial, we now have
(Z{G;q,v)=v^Q(G;t,v),
(38) ^ Zx(G;q,v) = v^Qx(G;t,v),
( Pn(G;X) = Qx(G;t,v)/Q(G;t,v),
for q = vt. The duality relation (5) for the Tutte polynomial then implies
the following duality relation for the Potts partition function [10], [13], [15]
(39) vl-^Z(G•,q, v) = (^)l-lv°l^(Gp;g, v*),
where
(40) vv* = q.
One further defines the dual correlation function
(41) PW;X^ = q Z^G^,q^)/Z(GD\qX).
and also the functions Ax and Bx' by
(42) Pn(G;X)= ̂ Ax'(G;q,v)
x'^x
ON THE ROOTED TUTTE POLYNOMIAL 1113
and
(43) P,*(G*;X*)= ^ Bx-(G*;^*).
x'-^x-
Then, Proposition 2 leads to the relation
(44) Ax(G;g,^) = g-WBx^G*;^*), X -. X\
which is the main result of [10].
7. Summary and discussions.
We have introduced the rooted Tutte polynomial (6) as a two-variable
polynomial associated with a rooted graph and deduced a number of
pertinent results.
Our first result is that the rooted Tutte polynomial assumes the
form (8) of a partially order set for which the inverse can be uniquely
determined. For planar graphs and all roots residing surrounding a single
face, we showed that (Proposition 1) the inverse function vanishes for non-
planar partitions of the roots. We further showed that the inverse function
satisfies the duality relation (23) (Proposition 2) which, in turn, leads to
the duality (32) for the rooted Tutte polynomial (Proposition 3). We also
reviewed the connection of the Tutte and rooted Tutte polynomials with
the Potts model in statistical physics.
Finally, we remark that results reported here have previously been
obtained in [9] and [10] in the context of the Potts correlation function. Here,
the results are reformulated as properties of the rooted Tutte polynomial
and thereby permitting graph-theoretical proofs.
Noted added. — The polynomial Q(G;x, y) is now commonly referred
to as the dichromatic polynomial and the dichromate
x(G;:r, y) = (x - ̂ Q^x -1^-1)
is now commonly known as the Tutte polynomial. We are indebted to
Professor W.T. Tutte for this remark.
Acknowledgments. — We are grateful to the referee for providing an
independent proof of Proposition 1 and numerous suggested improvements
on an earlier version of this paper.
1114 F.Y. WU, C. KING AND W.T. LU
BIBLIOGRAPHY
[I] G.D. BIRKHOFF, A determinant formula for the number of ways of coloring of a
map, Ann. Math., 14 (1912), 42-46.
[2] W.T. TUTTE, A contribution to the theory of chromatic polynomials, Can. J.
Math., 6 (1954), 80-91.
[3] W.T. TUTTE, On dichromatic polynomials, J. Comb. Theory, 2 (1967), 301-320.
[4] W.T. TUTTE, Graph Theory, in Encyclopedia of Mathematics and Its Appli-
cations, Vol. 21, Addison-Wesley, Reading, Massachusetts, 1984, Chap. 9.
[5] H. WHITNEY, The coloring of graphs, Ann. Math., 33 (1932), 688-718.
[6] See, for example, L.J. VAN LINT and R.M. WILSON, A course in combinatorics,
Cambridge University Press, Cambridge, 1992, p. 301.
[7] H.N.V. TEMPERLEY and E.H. LlEB, Relations between the percolation and
colouring problem and other graph—theoretical problems associated with regular
planar lattice: some exact results for the percolation problem, Proc. Royal Soc.
London A, 322 (1971), 251-280.
[8] W.T. TUTTE, The matrix of chromatic joins, J. Comb. Theory B, 57 (1993),
269-288.
[9] F.Y. WU and H.Y. HUANG, Sum rule identities and the duality relation for
the Potts n— point boundary correlation function, Phys. Rev. Lett., 79 (1997),
4954-4957.
[10] W.T. LU and F.Y. WU, On the duality relation for correlation functions of the
Potts model, J. Phys. A: Math. Gen., 31 (1998), 2823-2836.
[II] F.Y. WU, Duality relations for Potts correlation functions, Phys. Letters A, 228
(1997), 43-47.
[12] See, for example, F.Y. WU, The Potts Model, Rev. Mod. Phys., 54 (1982),
235-268.
[13] R.B. POTTS, Some generalized order-disorder transformations, Proc. Camb.
Philos. Soc., 48 (1954), 106-109.
[14] C.M. FORTUIN and P.W. KASTELEYN, On the random-cluster model I. Intro-
duction and relation to other models, Physica, 57 (1972), 536-564.
[15] F.Y. WU and Y.K. WANG, Duality transformation in a many-component spin
model, J. Math. Phys., 17 (1976), 439-440.
F.Y. WU & W.T. LU,
Northeastern University
Department of Physics
Boston, Massachusetts 02115 (USA).
fywu@neu.edu
wtlu@albert.physics.neu.edu
&
C. KING,
Northeastern University
Department of Mathematics
Boston, Massachusetts 02115 (USA).
king@neu.edu

On the rooted Tutte polynomial

Gnedbaye, Allahtan Victor

A non-abelian tensor product of Leibniz algebra

Krötz, Bernhard

Norm estimates for unitarizable highest weight modules

Sabbah, Claude

Harmonic metrics and connections with irregular singularities

Abánades, Miguel

Kucharz, Wojciech

ANNALES DE L’INSTITUT FOURIER
MIGUEL ABÁNADES
WOJCIECH KUCHARZ
Algebraic equivalence of real algebraic cycles
Annales de l’institut Fourier, tome 49, no 6 (1999), p. 1797-1804
<http://www.numdam.org/item?id=AIF_1999__49_6_1797_0>
© Annales de l’institut Fourier, 1999, tous droits réservés.
L’accès aux archives de la revue « Annales de l’institut Fourier »
(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-
nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-
tion commerciale ou impression systématique est constitutive d’une in-
fraction pénale. Toute copie ou impression de ce fichier doit conte-
nir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Ann. Inst. Fourier, Grenoble
49, 6 (1999), 1797-1804
ALGEBRAIC EQUIVALENCE
OF REAL ALGEBRAIC CYCLES
by M. ABANADES & W. KUCHARZ
1. Introduction.
Let X be a nonsingular, n-dimensional, quasiprojective variety over
R (that is, an irreducible, yi-dimensional, quasiprojective scheme over M,
smooth over R). We endow the set X(M) of R-rational points of X with
the topology induced by the usual metric topology on R, and assume that
X(M) is nonempty and compact. Thus X(R) is a C7°°, closed, n-dimensional
manifold. Given a nonnegative integer k, we let Zk{X} denote the group of
algebraic {n — ^)-cycles on X (that is, the free Abelian group on the set of
closed, (n — A^-dimensional subvarieties of X). There exists a unique group
homomorphism
cb:Zk(X)-^Hk(X(R)^/<2)
such that for every closed, (n — A^-dimensional subvariety V of X,
the cohomology class cl^V) is Poincare dual to the homology class in
^-A;(X(R),Z/2) determined by V(R) (cf. [5] for the definition of this ho-
mology class). In the present paper we study the cohomology classes of the
form cliR(^), where z is a cycle in Z^X) algebraically equivalent to 0 (we
refer to [7] for the theory of algebraic equivalence of cycles). Such cohomol-
ogy classes need not be trivial, but as we shall see below they must satisfy
quite restrictive conditions.
Both authors were partially supported by NSF Grant DMS-9503138.
-Keywords: Real algebraic variety — Algebraic cycles — Algebraic cohomology.
Math. classification: 14P25 - 55N22.
1798 M. ABANADES & W. KUCHARZ
The extreme cases, k = 0 and k = n, are easy to analyze. Obviously, a
cycle z in Z°(X) is algebraically equivalent to 0 if and only if z = 0. On the
other hand, every cycle in Zn(X) of the form XQ - a;i, where XQ and x^ are
points in X(R), is algebraically equivalent to 0. We have cip^o - x^) ̂ 0
whenever a-o and a;i belong to distinct connected components of X(R).
It follows that a cohomology class u in ^(^(R), Z/2) can be written
as u = c\]§i(z) for some cycle z in Zn{X) algebraically equivalent to 0 if
and only if the homology class in Ho(X(R),Z/2) Poincare dual to u is
represented by an even number of points of X(R). In view of these facts,
we concentrate our attention on the intermediate cases, 1 ̂ k <^ n — 1.
Given a continuous map f : M —> N between topological spaces, we
denote by H^f) : H^N, Z/2) -. H^M, Z/2) the homomorphism induced
by /. Recall that a cohomology class u in Hk(M,Z/c2) with k ^ 1 is said
to be spherical if u = Hk(f)(c), where / : M -^ 6^ is a continuous map
into the unit ^--sphere 6^, and c is the generator of H k ( S k , Z / c 2 ) ^ Z/2.
We denote, as usual, by U and < -, - > the cup product of cohomology
classes and the Kronecker index (pairing) of cohomology and homology
classes, cf. [11]. If M is a (7°°, closed manifold of dimension n, we denote
by Wk(M) the A;th Stiefel-Whitney class of M and by IJLM the fundamental
homology class of M in Hn(M, Z/2).
THEOREM 1.1. — Let X be a nonsingular, n-dimensional, quasipro-
jective variety over R with X(R) nonempty and compact. Let z be a cycle
in Z^X) that is algebraically equivalent to 0. Then the cohomology class
d^(z) in ^(X(]R),Z/2) satisfies clp(z) U cl^) = 0 in H2k{X(R)^/2)
and
< cb(z) U Wi,(X(R)) U ... U w,,(X(R)), ^(K) > = 0
for all nonnegative integers Z i , . . . , ir with ̂ -{-'••-^-ir=n-k. Furthermore,
i f k = l o r i f k = n - l > l with X(R) connected, then the cohomology
class c\^(z) is spherical.
Let us note that, in general, the cohomology class d^(z) of Theo-
rem 1.1 need not be spherical. Indeed, suppose X = X' x X" (product over
SpecR), where X' and X" are nonsingular, projective varieties over R such
that X'(R) is nonempty and X'^R) is disconnected. Let z ' be any algebraic
cycle on X ' . Choose two points po and pi in X"(R) that belong to distinct
connected components. Since the 0-cycle z" = po-Pi on X" is algebraically
equivalent to 0, the cycle z ' x z " on X is algebraically equivalent to 0 as
well. Furthermore, the cohomology class cip^' x z " ) = cip^') x ciR^") is
ALGEBRAIC EQUIVALENCE OF REAL ALGEBRAIC CYCLES 1799
spherical if and only if the cohomology class dpO') is spherical (for po and
pi belong to distinct connected components of X^R)). Taking X' = Pj^,
we have cl^^X')) = Hk(Xf(R)^/2), and the unique nontrivial co-
homology class in Hk(Xf(R),Z/2) ^ Z/2 is not spherical, provided that
1 ̂ k < m- 1 and m is even. In particular, "connected" cannot be omitted
in the last part of Theorem 1.1.
If dp (z) is spherical, then dR(»Ud]R(» = 0 is automatically satisfied,
and Theorem 1.1 is in some sense the best possible result. More precisely,
we have the following.
THEOREM 1.2. — Let M be a C°°, closed, n-dimensional manifold
and let u be a spherical cohomology class in H1^ (M, Z/2) with 1 < k ^ n-1.
Then the following conditions are equivalent :
(a) There exist a nonsingular, projective algebraic variety X over R
and a C°° diffeomorphism ̂ : X(R) -^ M such that Hk^)(u) = clp(^) for
some cycle z in Z^X) algebraically equivalent to 0;
(b) < uUw,, (M)U.. .Uw^(M), UM >= 0 for all nonnegative integers
% i , . ..,ir with %i + • • • +^ = n- k,
Let us mention that Theorem 1.2 is an improvement upon inefficient
[10], Theorem 2.4.
2. Proofs.
Let X be a nonsingular, n-dimensional, quasiprojective algebraic
variety over R with X(R) nonempty and compact. Recall that if an
algebraic cycle z in Zk(X) is rationally equivalent to 0, then cl^O) = 0
(cf. [5], 5.13) and hence dp induces a homomorphism, also denoted by dp,
from the Chow group Ak{X) of X into ^(X(R),Z/2). It is known that
dp : A*(X) -^ ^(X^.Z^) is a homomorphism of graded rings [5],
p. 495. Thus
%g(X(IR),Z/2) = dR(Z*(X)) = dM(A*(X))
is a graded subring of ^*(X(R),Z/2). We shall need the following result
[10], Theorem 2.1 :
(1) < dpO) U v, /^C(K) > = 0
for all cycles z in Zk{X') algebraically equivalent to 0 and all v in
^(X(R),Z/2).
1800 M. ABANADES & W. KUCHARZ
Assume now that X is projective. Then the set X(C) of C-rational
points of X is a compact complex manifold of complex dimension n. There
exists a unique group homomorphism
cIcr^PO-^^Q.Z)
such that for every closed, (n - A;)-dimensional subvariety V of X,
the cohomology class clc(V) is Poincare dual to the homology class in
^2n-2fc(X(C),Z) determined by V(C) (cf. [5] for the definition of this ho-
mology class). In other words, if TT : Xc = X xspecR SpecC -^ X is the
canonical projection, then clc(^) is the cohomology class corresponding to
the pullback algebraic cycle 7r*{z) on Xc, cf. [5], 4.2 or [7], Chapter 19. In
particular,
(2) clc(^) = 0
for all cycles z in Zk(X) algebraically equivalent to 0, cf. [5], 4.14 or [7],
Proposition 19.1.1. Furthermore, it follows from the proof of [2], Theorem A
that
(3) CI]R(^) U d^(z) = the reduction modulo 2 ofr(clc(^))
for all z in ^(X), where r : ^(X^.Z) -^ ^(XO^.Z) is the
homomorphism induced by the inclusion map X(R) ^-> X(C).
Proof of Theorem 1.1. — By Hironaka's resolution of singularities
theorem [8], 3, we may assume that X is projective.
We obtain ch(z) U cl^O) = 0 directly from (2) and (3).
It follows from [5], p. 498 that w,(X(R)) is in ^g(X(R),Z/2), and
hence if z i , . . . , ir are nonnegative integers with zi + • • • + Zy, = n — k, then
the cohomology class
v = w,, (X(R)) U . . . U w^(X(R))
belongs to 7y(X(R),Z/2). In view of (1), we have < d^z) U v,
/^X(R) >= 0, which completes the proof of the first part of the theorem.
Given an invertible sheaf C on X, we denote by C^ (resp. £c)
the topological real (resp. complex) line bundle on X(R) (resp. X{C))
determined by C in the usual way. If C corresponds to a Well divisor D on
X, then
wi(/:R) = cb(D) and ci(£c) = clc(^),
where wi (-) and ci (-) stand for the first Stiefel-Whitney class and the first
Chern class, respectively, cf. [5], p. 498, p. 489. Note that the restriction
£c|X(lR) of £,c to X(R) is the complexification of £]R, and hence
ci(/:c|X(R))=/3(wi(/:R)),
ALGEBRAIC EQUIVALENCE OF REAL ALGEBRAIC CYCLES 1801
where (3 : H1{X(]K)^/2) -> ̂ (X^.Z) is the Bockstein homomorphism
that appears in the long exact sequence
... -^ ̂ (XW.Z) ̂ ^(XW.Z) -> ̂ (XW.Z^) ̂ ^(XW.Z)
cf. [II], Problems 15-C and D. The last equality can be written in an
equivalent form
(4) r(clc(D))=/3(clR(D)),
where r : ^(^(C^Z) -^ ^(X^.Z) is the homomorphism induced by
the inclusion map X(R) ̂ X(C).
Suppose now that k = 1, that is, z is a Weil divisor on X. By (2)
and (4), we have /3(c\^(z)) = 0, which means that c\^(z) is the reduction
modulo 2 of a cohomology class in ^(^(R), Z). This last fact implies that
the cohomology class c\^(z) is spherical, cf. [9], p. 49.
Let us now assume that k == n — 1 > 1 and X(R) is connected. We
already know that < c\^(z) U Wi(X(IR)), ^x(R) >= 0; which in view of the
connectedness of X(M) is equivalent to c\^(z) U wi(X(R)) = 0. The last
condition implies that the homology class in H-i(X(R),Z/2) Poincare dual
to cliR^) can be represented by a C7°°, closed curve in X(R), with trivial
normal vector bundle, cf. for example [4], p. 599. This in turn implies that
c\^(z) is spherical, cf. [12], Theoreme 11.1. Thus the proof is complete. D
Proof of Theorem 1.2. — By Theorem 1.1, (a) implies (b), and we
show below that (b) implies (a).
Choose a nonsingular, irreducible algebraic subset W ofM^1, which
has precisely two connected components Wo and VFi, each diffeomorphic
to the unit A;-sphere 5^ (for example, W = { (a '1 , . . . ,rc/c+i) € M^1 | x\ —
^x\ + 1 + x^ + • • • + a'i_^i = 0}). Let c be the unique generator of the
group ^(TVo, Z/2) ^ Z/2, viewed as a subgroup of ^(W, Z/2). Since the
cohomology class u is spherical, there exists a C°° map h: M —> W such
that h(M) C WQ and u = ̂ (/^(c). Choose a regular value yo of h in Wo.
Then u is Poincare dual to the homology class in Hn-k(M, Z/2) represented
by the C°° submanifold h~l(yo) of M, cf. [5], 2.15. Clearly, there exists a
unique C°° map f : M —> W such that for every connected component S
of M and every point x in 5, we have f{x) = h(x) if S D h~l(yo) ̂ 0 and
f[x) = yo if S n h~l(yo) = 0. The map / satisfies
(5) /(M) C Wo and ^ = ̂ (/)(c).
1802 M. ABANADES & W. KUCHARZ
Furthermore, each connected component of f~l(yo) is a C°° submanifold
of M of dimension either n - k or n. Also, each connected component of
M contains a connected component of /^(^A))- Since n — k ^ 1, we can
find a C°° closed curve C in M such that
(6) f(C) = {^L
the normal vector bundle of C in M is trivial, and each connected compo-
nent of M contains a connected component of C. Choose an integer d with
2n+1 < d and let D be a compact, nonsingular, irreducible, 1-dimensional
algebraic subset of R^ that has the same number of connected components
as C. Replacing M by its image under a suitable C°° embedding into R^,
we may assume that
(7) D = C C M C R<
By Tognoli's theorem [13] or [I], Corollary 2.8.6, there exists a
nonsingular real algebraic subset A of R^, for some p, diffeomorphic to
M. Consider the disjoint union N = M II A and the C°° map F : N —> W
defined by F(x) = f(x) for x in M and F(x) = yo for x in A. We assert that
if w is a cohomology class in H^^W, Z/2) and if j i , . . . ,js are nonnegative
integers with ji + • • • + js = n - £, then
< H\F){w) U Wj, (TV) U ... U w^ (N), ̂ >= 0.
Indeed, first note that w = 0, unless £ = 0 or £ = k. If £ = 0, then either
H°(F)(w) = 0 or H°(F)(w) = 1. In the latter case the assertion holds
since M and A are diffeomorphic. If £ = k, then either Hk(F)(w) = 0 or
^(F^w) = u (we view Hk(M, Z/2) as a subgroup of Hk{N, Z/2)). In the
latter case the assertion follows from condition (b). Thus the assertion is
proved. It implies that there exist a C°° compact manifold N with boundary
9N = N and a C°° map F : N -^ W satisfying F\N = F, cf. [6], 17.3.
In other words, the map / : M —> W and the constant map A —» W,
which sends A to ?/o, represent the same class in the unoriented bordism
group of W. By construction, the normal bundle of D in M is trivial, so
the restriction v\D to D of the normal bundle v of M in R^ admits an
algebraic structure. Therefore, by [I], Theorem 2.8.4 and in view of (6) and
(7), one can find a nonnegative integer e, a nonsingular algebraic subset
V of R^ x R6, a C°° diffeomorphism ^ : V —^ M, and a regular map
g : V —^ W (the latter designates the restriction to V of a rational map
from R^ x R6 into R^4"1 which has no poles on V and maps V into W)
such that g is homotopic to / o (p and D x {0} C V. Since D is irreducible
and each connected component of V contains a connected component of
ALGEBRAIC EQUIVALENCE OF REAL ALGEBRAIC CYCLES 1803
D x {0}, it follows that V is irreducible as well (this is the only place where
D is needed).
Irreducibility of V and W allows us to choose nonsingular, quasipro-
jective varieties T and Y over R with T(R) = V and Y(R) = W. By
Hironaka's resolution of singularities theorem [8], 3, we may assume that T
and Y are projective (and still nonsingular). Let g : U -^ Y be an algebraic
morphism over R, denned on a Zariski open neighborhood of T(R) = V
in T, such that g\T(R) = g . By applying Hironaka's theorem on removing
points of indeterminacy [8], 3, we can find a nonsingular, projective alge-
braic variety X over R and an algebraic morphism G : X -^ Y over R
satisfying X(R) = T(R) and G\X(R) = g.
Let 2/1 be a point in W^ and let (3 be the class in A^V) of the 0-cycle
yo - 2/1 on V. By (5), u = H^f)^^)) (although, of course, clR(/3) ^ c).
Since G\X(R) = g is homotopic to / o ̂ we obtain
H\^(u) = ̂ )(^(/)(cl^))) = H\f o ̂ )(cl^))
= ^(G|X(M))(clK(/3)) = clM(^*(^)),
where the last equality is a consequence of the functorial property of
clip : A* -^ H\ cf. [5], 5.12. Let z be a cycle in Z^X) that represents in the
Chow group Ak{X) the pullback class C7*(/3). Then cl^) = clM(G*(/?)) =
^ ((/:?)(n)• T^ proof is now complete since the cycle 2/0-2/1 is algebraically
equivalent to 0 on Y and hence the cycle z is algebraically equivalent to 0
on X. Q
BIBLIOGRAPHY
[1] S. AKBULUT and H. KING, Topology of Real Algebraic Sets, Mathematical Sciences
Research Institute Publications, Springer, 1992.
[2] S. AKBULUT and H. KING, Transcendental submanifolds of R71 Comm Math
Helv., 68 (1993), 308-318.
[3] E. BIERSTONE and P. MILMAN, Canonical desingularization in characteristic zero
by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997)
207-302.
[4] J. BOCHNAK and W. KUCHARZ, Algebraic models of smooth manifolds Invent
Math., 97 (1989), 585-611.
[5] A. BOREL et A. HAEFLIGER, La classe d'homologie fondamentale d'un espace
analytique, Bull. Soc. Math. France, 89 (1961), 461-513.
[6] P.E. CONNER, Differentiable Periodic Maps, Lecture Notes in Math., Vol. 738
Berlin-Heidelberg-New York, Springer, 1979.
1804 M. ABANADES & W. KUCHARZ
[7] W. FULTON, Intersection Theory, Ergebnisse der Math., Vol. 2, Berlin-Heidelberg-
New York, Springer, 1984.
[8] H. HIRONAKA, Resolution of singularities of an algebraic variety over a field of
characteristic zero, Ann. of Math., 79 (1964), 109-326.
[9] S.T. Hu, Homotopy Theory, New York, Academic Press, 1959.
[10] W. KUCHARZ, Algebraic equivalence and homology classes of real algebraic cycles,
Math. Nachr., 180 (1996), 135-140.
[11] J. MILNOR and J. STASHEFF, Characteristic Classes, Ann. of Math. Studies,
Vol. 76, Princeton Univ. Press, 1974.
[12] R. THOM, Quelques proprietes globales de varietes differentiables, Comm. Math.
Helv., 28 (1954), 17-86.
[13] A. TOGNOLI, Su una congettura di Nash, Ann. Scuola Norm. Sup. Pisa, 27 (1973),
167-185..
Manuscrit recu Ie 16 avril 1999,
accepte Ie 26 mai 1999.
M.ABANADES & W.KUCHARZ,
University of New Mexico
Department of Mathematics and Statistics
Albuquerque, NM 87131-1141 (USA).
abanades@math.unm.edu
kucharz@math.unm.edu

Algebraic equivalence of real algebraic cycles

Cameron, Kathie

Edmonds, Jack

Some graphic uses of an even number of odd nodes

Matsumoto, Makoto

ANNALES DE L’INSTITUT FOURIER
MAKOTO MATSUMOTO
A generalization of Jaeger-Nomura’s Bose Mesner
algebra associated to type II matrices
Annales de l’institut Fourier, tome 49, no 3 (1999), p. 1027-1035
<http://www.numdam.org/item?id=AIF_1999__49_3_1027_0>
© Annales de l’institut Fourier, 1999, tous droits réservés.
L’accès aux archives de la revue « Annales de l’institut Fourier »
(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-
nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-
tion commerciale ou impression systématique est constitutive d’une in-
fraction pénale. Toute copie ou impression de ce fichier doit conte-
nir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Ann. Inst. Fourier, Grenoble
49, 3 (1999), 1027-1035
A GENERALIZATION OF
JAEGER-NOMURA^S BOSE MESNER
ALGEBRA ASSOCIATED TO TYPE II MATRICES
by Makoto MATSUMOTO
1. Spin model and its Bose-Mesner algebra.
Let X be a finite set with cardinality n, and C[X} be the vector
space with basis X over the complex number field C. We consider the set of
matrices MX ''= Hom(C(X), C(X)), which is identified with the algebra on
n x n matrices, by identifying X = { 1 , 2 , . . . , n}. An n x n matrix W € MX
is said to be a (two-weight1) spin model, if it satisfies three algebraic
conditions called type I, type II, and type III conditions. These names
came from the corresponding Reidemeister moves in the Knot theory, see
Jones [6] and Bannai-Bannai [1]. Spin models were introduced to describe
a link invariant. For a matrix M C MX, we denote by M(a,&) its (a, 6)-
component for every a,& € X. The usual product of matrices, denoted by
M'M or M' • M, is defined by M'M{a, &) = E Mf^ x)M(x, 6), while the
xex
Hadamard product M'oM is defined by (M'oM)(a, b) := M'(a, b)M(a, b),
which is commutative. We denote by J a matrix whose components are all
1, which is the unit for Hadamard product. If M is invertible with respect
to the Hadamard product, its inverse is denoted by M°. Then, the type II
condition is described by
M- ̂ M0 = nl,
Keywords: Bose-Mesner algebra - Spin model - Jaeger-Nomura algebra -Type II matrix
- Association scheme - Link invariant.
Math. classification: 05Exx - 05E30 - 13Axx - 18Axx - 57M25.
1 Here a spin model means two weight, unless otherwise stated.
1028 MAKOTO MATSUMOTO
where I denote the unit matrix of size n. The type I condition is auto-
matically satisfied if M has both type II and type III properties. Type III
condition will be explained later.
1.1. A Bose-Mesner algebra associated with a type II matrix.
A matrix B which satisfies the type II condition above is called a
type II matrix. For a type II matrix, we introduced [4] after Jaeger [5] and
Nomura [7] an algebra N(W) as follows. For each 6, c 6 X, define a vector
(W^x)\
yb1c \ W(c x } ) ecW• VV { C , X ) / ̂ ^
Then we define an algebra (possibly called Jaeger-Nomura algebra of W)
N(W) := {M € MX I Yb,c is M's eigen vector for all &, c € X}.
It can be proved that N(W) is a Bose-Mesner (BM) algebra, i.e. a
commutative subalgebra of MX closed under Hadamard product and
transpose with identity I G MX for usual product and identity J € MX
for Hadamard product. (Thus, it corresponds to an association scheme.)
2. Spin models of index two.
Jaeger-Nomura [5] introduced the notion of index of a spin model. I
don't recall the definition, but according to their result we may define a
spin model W of index two to be a spin model with the following form:
A A B - B '
A A -B B
-B B C C
B -B C C
(1) W=
which was fully used in [5] to construct a spin model with chiral link
invariant, i.e. one which distinguishes some link from its mirror image.
PROPOSITION 1 ([5], Proposition 8). — Let A, C be a symmetric
matrix of the same size. Then, the matrix W in (1) is a spin model if and
only if the following conditions are satisfied:
(A) A, C are spin models and B is a type II matrix.
A GENERALIZATION OF JAEGER-NOMURA ALGEBRA 1029
(B) Let n be the size ofW, and D be one of±^/n. Then the following
four equalities hold:
w ix-) l̂ '̂ M-ĵ
(3) ^^-,^r^
(4) j^A^y^B^B^c) = -DG^cr^a^-^a.c)-1
y
(5) ^C^y)-lB^y)B^y) = -DA^c^B^a^B^a)-1.
y
The concrete result of this article is just the following.
THEOREM 2. — Let A,B,C,D be type II matrices. Then the
conditions (2) and (3) are equivalent, and the conditions (4) and (5) are
equivalent.
The first equivalence is a direct consequence of [4], Theorem 1, as
shown below, but the second equivalence is more delicate and I prove it by
introducing something like a duality on something like BM algebra.
Proof of (2) ̂ (3). — Let us define y=6,cas in§l . l . Then (2) for
all a G X is equivalent to
Ay^c=DC(b^)yb^
and thus A G N(B). Also this shows that <^(A) = D ' C°, where ^>w is
defined as follows.
THEOREM 3 ([4], Theorem 1). — Let W be a type II matrix of
size n. Let M C N(W). Write ^iv(M)(a,6) for the eigenvalue ofM with
respect to ya,&. Thus, ^y^(M) € MX' Then, ^w induces a dual pair
<S>w : N{W) -^ N^W),
i.e. a linear isomorphism satisfying <l>w(M'M) = w(M') o ^(M),
<^w{M' oM)= ^iv(M')^(M), and ^(<S>^(M)) = n' tM.n
By this, the condition (2) is equivalently transformed to ^w(A~1) =
D~1C (since inverse is mapped to Hadamard inverse), then by taking <1><^
we have ^^(^(A-1)) = jD-^t^C'). The left hand side is n^A-1),
1030 MAKOTO MATSUMOTO
which is A°, since by type II condition on A we have *A° = n ' A~1. Thus
it is equivalent to t^((7) = DA°, i.e. (3).
For the rest of Theorem 2, we use a similar generalization according
to a categorical concept.
3. A category N .
We shall define a category N (possibly called Jaeger-Nomura cate-
gory), which is a generalization of Jaeger-Nomura algebra N(W). N has a
beautiful duality ̂ which is not a functor unfortunately.
3.1. An abstract nonsense.
Although it is a bit confusing, by y we denote a function X x X —>•
C[X] — {0}, or equivalently n x n nonzero vectors in C[X] indexed by
X x X, i.e.
y=(ya ,beC[X] ,^o |a ,6eX) .
(I.e. y is not a single vector but an ordered set of vectors, from now on.)
Let y' := (y^ ^ \ a, b € X) be another set of vectors. We define the set of
morphisms from y to y' by
Hom7v(y,y') := {M e MX \ My^b = Ca^y^b for some ^,6 ^ C
for all a,6 € X},
which is a subvector space of MX = Hom(C[X],C[X]). It is clear that
a morphism from y to y' can be composed with one from y ' to y " to
obtain one from y to y " (i.e. we defined a cateory C with object y^s and
morphisms Hom^y.y')). We define a linear homomorphism
^ :Hom^(y,y') -^ MX
y , y '
by <^(M) := (c(a,b))a,b- It follows that
^(M'M) = ̂ (M7) o ̂ (M)
for M' 6 Hom^y^y"). Thus, is a functor from C to the category
consisting of one object *, with homomorphisms MX, with composition by
Hadamard product and identity J . We denote this category by MX'
If there is no fear of confusion, we omit the suffix of .
A GENERALIZATION OF JAEGER-NOMURA ALGEBRA 1031
3.2. An object associated with a pair of type II matrices.
Let K^ L be two type-II matrices on the same set X. Define a column
vector by
<^:= W^W^ex.
This gives a set of ordered n x n vectors indexed by X x X, i.e. an object
of C. We denoted it by y^.
Remark. — In this terminology, y^c defined in §1.1 is y^ . More-
over,
N(W) = Hom^y^y^0) = End^y^0),
which becomes an C-algebra by the general fact on additive category. Non
trivial part is that it is closed also under Hadamard product.
3.3. A quasi duality.
The following is the main result of this article.
THEOREM 4. — Let K,L,K' , L ' be arbitrary type II matrices of
same size. We write ^(K L - K ' m for ^
' y^'^ y K f ' L /
(1) ^ [ K , L ; K ' , L ' ) defines a linear map
Hom^y^y^'^') -. Horn^^y^0^).
(2) <E>(tL /, tLO; tft: /o, tJ<) ls a mapping of converse direction, and
^(<L/,<LO;<X/o,^)(^(J<,L;^/,L/)(^r)) = nM
holds. Thus, is a linear bijection.
Proof. — All we need is to prove
tr I tj 0 tr^-fO ty
^(^y^) =nM(g,h)y^K,
or equivalently
(6) ^ ̂ (M)(a, b)L\b^ g)L{^ h)-1 = nM{g, h)K'~\a^ g)K(^ h).
b
The definition of (M) is given by
^ M(c, z)K(a, z}L(b, z) = ̂ (M)(a, b)K\a, c)L'(b, c)
1032 MAKOTO MATSUMOTO
holds for all a, &, c € X. We specialize c to p, then we have
<D(M)(a, b)L'(b^ g) = K\a^ g)-1 ̂ M(c, z)^(a, ̂ )L(6, z).
z
Plug this into the left hand side of (6) to have
^^(M)(a,6)L'(6^)L(&^)-1
b
= K\a, g)-1 ̂ ̂ M(c, ̂ (a, ̂ )L(6, ̂ )L(6, ̂ )-1
6 z
= K1^ g)-1 ̂ M(c, z)^(a, ̂ ̂ L(&, z)L(^ h)-1
z b
= nM(g, h)K' (a,g)K(a, h) ( since L is type II),
as desired. D
From now on, we shall denote
(^L)^^
N(K^ L; K^ L') := Hom^y^, y^^7),
and consider only the objects of the form (Jf, L) in the category C. Let TV
be the full subcategory of C induced by these objects. In other words, we
consider pairs of type II matrices (-ftT, L), and a morphism from (Jf, L) to
(JT,Z/), denoted by N ( K , L \ K ' , L ' ) := Hom^y^y^'^).
Remark. — ^ is a generalization of duality between N(W) and
N^W). We have N(W) = 7v(iy, TV0; W, W°) by definition. If we specialize
to K = K ' and L = L', we have
N ( K , L ' , K , L ) -^N^L^^K0^).
This shows that is not a functor. By the way, the left hand side is
a subalgebra of MX with respect to the ordinary product. From 2 of
Theorem 4, it follows that the right hand side is closed under the Hadamard
product, and these two algebras are isomorphic as an abstract algebra.
Imposing the right hand side to be an algebra with usual product, we may
require (^/L0) = (^°',^), i.e. L = K°. Then the right hand side is
N(^°/K;^°,^),
which is closed also by the usual product, thus we recover the Jaeger-
Nomura algebra. Note that the transpose induces a linear bijection
(7) N(K, L; K ' , L ' ) ̂ 7v(L, K , L\ K ' ) .
A GENERALIZATION OF JAEGER-NOMURA ALGEBRA 1033
By composing with this transpose we have
N { K ^ K ° ^ K ^ K ° ) ^N^^K0^) ̂ 7V(^,^°;^,^0),
i.e. the duality
N(K) -^N^K)
denned in [4], §3.1 (note that N^K) is N^K) there).
4. Application of the category N .
4.1. Proof of (4) ^ (5).
(4) is equivalent to
A° e N^B^B^B0^0) and $(A°) = -DC.
We calculate
^(A°-A)=^(A°)o$(A).
The left hand side is <3>(nJ) by type II property of A, then by definition
of , it is nJ. The right hand side is, by condition (4), -DC o <1>(A). By
moving DC to the left hand side, we have
-DC0 = $(A).
Applying , we have
<S>(C°) = -^-^(^(A)) = DA,
which is nothing but the condition (5).
4.2. Four-weight spin model.
Four-weight spin model was introduced in [1] as a generalizaion of two-
weight model, which was fully used in determination of the link invariant
in [5]. We can paraphrase their definition in terms on TV. For the original
definition, see [1].
PROPOSITION 5. — A pair ofn x n matrix (W^, W^) is a four weight
spin model if and only if the following conditions hold:
1. TVi, W^ are type II matrices.
1034 MAKOTO MATSUMOTO
2. W^NW^W^W^W and^w^^w^wAWi)=D^W^
(which is automatically in N{W^,W^°', W^ W^°)).
3. W, e N{W^W,°^W,°) and ^w^,w^{W^) = DW^°
(which is automatically in N^W-^0, ̂ IVi; ̂ 2°, ̂ 2).
Proof. — In the terminology in [I], Y^ = y^^-. Their condition
IIIi is equivalent to 2, and their condition IIIg is equivalent to 3. D
Remark. — It is shown that IIIi to Ills are equivalent, and IIL) to
IIIi6 are equivalent in [I], where these 16 conditions naturally arize from
Reidemeister move of type III with various orientations. These equivalences
are easily proved by N and ^>. For example, note that IIIi is equivalent to
W^ = ̂ 2° = ^(T^i), and by taking ̂ again, we have the condition III2.
Then, ̂ 3 = W^ and $(A°) = n. (^(A)°) implies Ills, etc.
Remark. — Huang-Guo [2] proved that
1̂° • w^ e N^W^).
This can be interpreted in N by
Wi° G N^W^W^W^W?} (IIIg)
and
and thus
'W^0 eN^W^W^W^W^) (III2)
1̂° • î0 € N^ ̂ 2°; ̂ 2, ̂ 2°) = N^W^).
Acknowledgement. — This study was done while the author visited
Max-Planck-Institute for Mathematics in Bonn. The author discussed on
this subject with Francois Jaeger in his room in the hospital. It became
the last time to see him. I admire his mathematics, and, honesty. He is the
most honest person I have ever met.
BIBLIOGRAPHY
[1] Ei. BANNAI and Et. BANNAI, Generalized generalized spin models (four-weight
spin models), Pac. J. Math., 170 (1995), 1-16.
[2] T. HUANG and H. GUO, Four-weight spin models and related Bose-Mener algebras
preprint 1997. '
A GENERALIZATION OF JAEGER-NOMURA ALGEBRA 1035
[3] F. JAEGER, A talk in RIMS, 1994.
[4] F. JAEGER, M. MATSUMOTO and K. NOMURA, Bose-Mesner algebras related with
type II matrices and spin models, Journal of Algebraic Combinatorics, 8 (1998),
39-72.
[5] F. JAEGER and K. NOMURA, Symmetric versus non-symmetric spin models for
link invariants, submitted (1997).
[6] V.F.R. JONES, On knot invariants related to some statistical mechanical models,
Pac. J. Math., 137 (1989), 311-336.
[7] K. NOMURA, An algebra associated with a spin model, Journal of Algebraic
Combinatorics, 6 (1997), 53-58.
Makoto MATSUMOTO,
Keio University
Faculty of Science and Technolgy
Hiyoshi, Kohoku-ku
Yokohama, 223 (Japan).
matumoto@math.keio.ac.jp

A generalization of Jaeger-Nomura's Bose Mesner algebra associated to type II matrices

Not a member yet

On the rooted Tutte polynomial

A non-abelian tensor product of Leibniz algebra

Norm estimates for unitarizable highest weight modules

Harmonic metrics and connections with irregular singularities

Algebraic equivalence of real algebraic cycles

Some graphic uses of an even number of odd nodes

A generalization of Jaeger-Nomura's Bose Mesner algebra associated to type II matrices

Unitaires multiplicatifs en dimension finie et leurs sous-objets

In search of the invisible spectrum

Conformally equivariant quantization: existence and uniqueness

3,503

3,732