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J.Aust. Math. Soc. 79(2005), 183-212
CONJUGACY IN SINGULAR ARTIN MONOIDS
RUTH CORRAN
(Received
3
January
2003;
revised 9 September 2004)
Communicated by D. Easdown
Abstract
We define a notion of conjugacy in singular Artin monoids, and solve the corresponding conjugacy
problem for finite types. We show that this definition is appropriate to describe type (1) singular Markov
moves on singular braids. Parabolic submonoids of singular Artin monoids are defined and, in finite type,
are shown to be singular Artin monoids. Solutions to conjugacy-type problems of parabolic submonoids
are described. Geometric objects defined
by
Fenn, Rolfsen and
Zhu,
called (J,
fc)-bands,
are algebraically
characterised, and a procedure is given which determines when a word represents a (j, (fc)-band.
2000
Mathematics
subject
classification:
primary 20F36; secondary 20M05, 20F05.
1.
Preliminaries
Singular Artin monoids are introduced in [9] as a generalisation of singular braid
monoids, defined by presentations related to Coxeter matrices. The singular braid
monoids are singular Artin monoids defined by type A Coxeter
matrices.
Artin groups
(see [7] or [10]) are subgroups of singular Artin monoids, and Coxeter groups (see,
for example, [20]) are quotients of Artin groups. The Artin groups of type A are the
Artin braid groups, and the Coxeter groups of
type
A are the symmetric groups. Thus
singular Artin monoids are generalisations of very natural objects.
The main result of this paper is a solution to the conjugacy problem in singular
Artin monoids. In order to solve this problem, we need an appropriate definition of
conjugacy. Perhaps the most natural choices are the following: to say that V and W
are conjugate if
(1) there exists X such that VX = X
W,
or
© 2005 Australian Mathematical Society 1446-7887/05 $A2.00 + 0.00
183
184 Ruth Con-an [2]
(2) there exist Y and Z such that V = YZ and W = Z Y.
The first is reflexive and transitive, but not necessarily symmetric, while the second
is reflexive and symmetric, but not necessarily transitive. We take the first of these to
be the definition of conjugacy in this paper. This first section establishes and collates
results which allow us to show that it is indeed a symmetric relation in the context of
singular Artin monoids, and Sections 2 and 3 present the solution to the conjugacy
problem in singular Artin monoids of finite type. The second notion of conjugacy
defined above we call 'swap conjugacy', and we verify that the original definition is
precisely its transitive closure.
Fenn, Rolfsen and Zhu [12] introduced the notion of (j, A:)-bands, which are
singular braids satisfying a certain geometric condition on the j th and (/ + l)th
strings of the braid, and they find an equivalent algebraic condition. In the fourth
section, we describe how to determine whether a braid is a (/, &)-band given a word
in the generators which represents it.
In Section 5 we introduce parabolic submonoids of singular Artin monoids, which
are generated by particular subsets of the generators. We show that parabolic sub-
monoids of singular Artin monoids of
finite
type are themselves isomorphic to singular
Artin monoids. This result mirrors that in Coxeter and Artin groups; although in both
of these cases the result has been shown to hold for arbitrary type. The notion of
a (/, A>conjugator (a generalisation of a (/, fc)-band), where J and K define sub-
sets of the generators, is discussed, and a method for determining when a word is a
(/, /O-conjugator is given. We show that the set of (7, JQ-conjugators is not empty
if and only if the,parabolic submonoids defined by J and K are conjugate. We give a
method for determining when two parabolic submonoids are conjugate.
The (j, &)-bands were first introduced in order to prove one case of a conjecture
of Birman [5] about singular braid monoids embedding in the group algebra of the
braid group. The last section of this paper discusses the singular braid monoid
exclusively, particularly the problem of determining when two singular braids close
to give equivalent singular links. Gemein [14] obtained an analogy for singular
braids of Markov's theorem for braids, which describes the algebraic connections,
called Markov moves, between braids which give equivalent links. We show how to
determine when two singular braids are connected by one Markov move, and give
some stronger results for positive braids connected by 'positive Markov moves'.
A solution has recently been obtained for Birman's conjecture by Paris [21]. The
conjecture may be generalised
to
arbitrary Artin
types,
and
a
solution
was
subsequently
obtained for the generalisation of this conjecture to
F
C-type (a distinct case to the
finite type case mostly considered in this article) by Godelle and Paris [17]. However,
the finite type case, other than the case originally solved by Paris, remains open. The
result of [17] used results of Godelle's for Artin groups ([15, 16]) similar to some
obtained here for singular Artin monoids. We hope that the results obtained here
[3] Conjugacy in singular Artin monoids 185
will contribute towards the resolution of the generalised Birman's conjecture for finite
type,
which remains an object of current study (see, for example, [1]).
Some of the techniques in this paper have been adapted from [7] (particularly the
notion of
an
a-chain), in which solutions to the word, division and conjugacy problems
for Artin groups were given. That paper, in turn, generalised notions of Garside [13]
who originally solved those problems for the braid group.
We begin by defining positive singular Artin monoids. These turn out to be
submonoids of the singular Artin monoids (defined in Section 3). Let / denote a finite
set. A Coxeter matrix over / is a symmetric / x / matrix M = (my) where mH = 1
for all i e I, and my e {2, 3, 4, ..., oo} for i ^j. The Coxeter graph TM associated
with a Coxeter matrix M is the graph with vertices indexed by /, and where an edge
labelled my joins the vertices i and j precisely when my > 3. The convention is to
explicitly show this label only when my > 4 — thus an unlabelled edge indicates that
my = 3. Figure 1 shows some Coxeter graphs.
Given a Coxeter matrix M, the positive singular Artin monoid of type M is the
monoid generated by S U T where 5 = {cr, | i e /} and T = {r, | i e
/},
subject to
the relations
£#,
listed below
(PiOj)™"
= (cr;a,)m» whenever 2 < my < oo,
(o-,o))m»~1r<: = Xj{aiaJ}m"''i whenever 2 < my < oo,
(i if my is odd;
and where k = I\j if my is even,
TJ-TJ
= i)T,
whenever
my — 2, and
aj,••
=
T,CT,
for all / in /,
where (ab)p denotes the alternating product aba
• • •
with p factors. Let 5?^ denote
the positive singular Artin monoid of type M. If two words W and V represent the
same element of y^, we say that W and V are equivalent, and write W ~ V. The
symbol = is used to indicate when two words are the same letter for letter (in other
words, equal in the free monoid on S U T, which is denoted (S U T)*). Notice that
since every relation is homogeneous — that is, both sides of the equation are words of
the same length — whenever W ~ V, then the length of W and V must be the same.
The length of a word W in the generators 5 U T is denoted t (W). Let &T denote the
set {(U, V), (V, U) | U= Visa relation from & }.
A word V is said to divide a word W if there exists a word X such that W ~ VX.
A set of words Q has a common multiple W if every element of
£2
divides W. A least
common multiple is a common multiple which divides all other common multiples.
Notice that by homogeneity, the length of a divisor cannot exceed the length of its
multiple. Thus only finite sets can have common multiples (as infinite sets contain
186 RuthCorran [4]
words of arbitrary length). However, there are finite sets of words without common
multiples (see the comments following Lemma 1.1, below, for an example).
Many properties of
y^
were discussed in
[9],
where the word and division problems
for such a monoid were solved, and a unique normal form was described. Furthermore,
it was shown that S^jj is both left and right cancellative, and that whenever a set Q
has a common multiple in S^^, it has a least common multiple L(£l) (which is unique
in
S^M)-
The crucial result in proving the above properties is the following reduction
property (so named after [7, Lemma 2.1], 'Reduction Lemma').
LEMMA 1.1 ([9, Lemma 15]). For all a,b € S U T, and for all words X and Y,
the equation aX ~ bY implies there exist words U, V and W in (S U T)* such that
X~ UW, F~ VW and either {aU,bV) is in ^x or aU = bV.
Thus if Wx ~ W2, then there are words V, Ri and R2 such that the first letter of Wx
and /?i coincide, the first letter of W2 and R2 coincide, W\ ~ Rt V and W2 ~ R2 V,
and either (Ru R2) e &z or R\ = R2. This result is very useful: we can immediately
apply it to the problem of the existence of common multiples of pairs of generators.
For example, since there is no pair of the form (r,(/, r; V), where m,, > 2, in ^E,
then the reduction property ensures that {r,, i)} has no common multiple.
In this first section various results are obtained which are useful in the sequel.
Notation and operators defined in full in [9] are more briefly described. The last part
of this section deals specifically with the subset of positive singular Artin monoids
of finite type, which turn out to be precisely the singular Artin monoids associated to
finite disjoint unions of the Coxeter graphs in Figure 1. (The results preceding this
are valid for positive singular Artin monoids of arbitrary type.)
LEMMA 1.2. Suppose that my > 2. Then there is no common multiple of
Xi
and
(o-jO-
i
)
mi
>~
3
z
p
,
where p is i if my is even and] otherwise.
PROOF. If m,j = 3, then, as remarked earlier, the reduction property precludes T,
and tj having a common multiple. Suppose that Wy > 3,
and that X provides a minimal length counterexample to the lemma. By the reduction
property, there is a word W\ such that
W ~ (o)a,)m"-1 Wi and
(cr.-cr,
)m«
"4rp U ~ (cr,^ )m>~\ W,.
Cancelling yields xp U ~
o
p
aq
x
p
W\. By further applications of
the
reduction property,
u
w,
w,
~o-pW2
~ (opoq)m'-2i
~ crp W4
and
r,
W3 and
and {aq
aqxp
,o-p)m»-3Xj
W, -
W2~
w3-
w
2
,
qap
w4.
[5] Conjugacy in singular Artin monoids 187
there are words W2, W3 and W4 such that
and
But this last equivalence gives a common multiple of xp and (<T,<Tp)m*~3r,, and of
length less than the length of X, contradicting the assumption of minimality. The
lemma now follows by induction. •
Let a and b be letters from 5 U T. A nonempty word C is a simple chain with
source a and target b if there are words U and V such that (a
U,
Cb
V)
e <^?E; we also
say that C is a simple a-chain for short. A simple a-chain C is said to be preserving
if (aC, Cb) g ^P^. Inspection of the relations shows that any simple a-chain whose
target is an element of T must be preserving, and that if C is a simple preserving
a-chain to b, then a is in 7 precisely when b is in T.
A word C is called a compound a-chain, or just an a-chain, if C = C\
• • •
Q for
simple chains
C\,...,
Q, where C\ is an a-chain, and the source of Ci+\ is the target
of C, for all i > 1. The source and target of C are defined to be the source of C\
and the target of C* respectively. An a-chain is said to be preserving if each of its
component simple chains is preserving.
REMARK 1.3. It was shown in [9] that if C is an a-chain to b and CD is a common
multiple of a and C, then CD is a common multiple of a and Cb; thus the target of C
divides D. In particular, a does not divide C. For each a in 5 U 7, a partial operator
tfa : (5 U 7")* -> (5 U T)* was then defined, with the properties that
(1) Ka( W) is defined whenever a and W have a common multiple;
(2) when it is defined, Ka{ W) ~ W, and
ATfl( W)
begins with a if a divides W, or
otherwise is an a-chain;
(3) Ka
(WO
is calculable; and
(4) if a does not divide W, but a divides Wb for some generator £, then Ka{ W) is
an a-chain with target b.
LEMMA 1.4. A nonempty word W
is
equivalent to a preserving a-chain to b precisely
when a does not divide W and aW~ Wb. Moreover, any a-chain with target in T is
preserving.
PROOF. Suppose that W is equivalent to C = C\ •
• •
Ck where each C, is a simple
preserving a,_i chain to a,. Then for each /, (a,_iC,
C
t
ai)
e ^>s, so aj-xd ~ Cia{.
Hence
OQC
~ Cak, where a$ = a is the source and ak = b is the target of C. Thus
a W ~ Wfe, and by Remark 1.3, a does not divide W.
188 RuthCorran [6]
Now suppose that W is not divisible by a but aW ~ Wb. By the reduction
property, there is some pair (aCi, U\d) in J?E and a word V such that a W ~ aC\ V
and Wb ~
t/,0
1
V. Thus C, Vfc ~ Wb ~ aW ~ aQV ~
UidV.
Suppose d £ £/,.
Inspection of the relations tells us that aC\ = {(TicTj)mi'~1xp and U\d = ry-(ff,-o)>m«~1
for some i andy , where p = i if m.y is odd andy if m,y is even. Substituting into the
above, {Ojai)n">~lxp Vb ~ T,
(CTJCTJ
)"•»"' V. By the reduction property, there is a word
V such that (a,o) )m»~3Tp Vb ~ t) V", which contradicts Lemma 1.2. Hence Q s (/,.
Thus C\ must be preserving. Since Q Vb ~
C
x
dV,
left cancellativity gives
Vb ~ dV, where V is shorter than W. Furthermore, V is not divisible by d since
if it were, V ~ d V" for some word V", and W ~ C, V ~ C,rf V" ~ ad V", which
contradicts that a does not divide W. So we can continue in this way replacing W
with V, until W ~ d d
• • •
Q where each C, is preserving.
Finally, suppose that C = C\
• • •
C* is an a-chain to
Xj
for somey . Let a,_i be the
source and a, the target of C, for each i. So C* is a simple at_i-chain to
T).
Thus
it must be preserving, and have source ak-\ € T also. Continuing backwards in this
way through C, we have that each simple component is preserving, and that each a,
is in T. D
THEOREM 1.5. For any word W, any generators a and b and r any positive integer,
we have ar W ~ Wbr if and only if a W ~ Wb.
PROOF. The 'if direction is evident; suppose henceforth that ar W ~ Wbr. If a
divides W, then cancellativity and an inductive hypothesis give the result quickly. We
may suppose a does not divide W. We now use various parts of Remark 1.3: firstly,
since Wbr is a common multiple of a and W, then Ka{ W) is defined, and since a does
not divide W, is an a-chain. Let t denote its target. Since Ka(W)br ~ Wbr ~ a' W
is a common multiple of a and W, r divides br, implying t s b. Thus £„(
H
7
)
is an
a-chain to b.
If a e T, since the number of r's is preserved by the relation ~, then a' W ~ W£r
implies that b € 7 as well, so by Lemma 1.4, Ka (W) is preserving, and a W ~ Wfc.
From now on, we suppose that a, and hence b, are in 5. Write a = or,-.
First we show that W is not divisible by Xj for m,j > 2. Let e > 1 and 10 =
r7(CT,a;)m*~l((CTjCr,)m»)e~1; then u> ~ a'v where
((o-;0r,-)m(i""')'"'{o-jai}m''-1xj if m,j even;
u= ((o>orl-)ms"1(or,-a/-)"Ii'"1)('"1)/2(o-;-a,-)'"«-2T/ if m,y and e odd;
((o-jOi)
m
"
-' (ff,-o)
)m*
~')('~2)/2{cr/
or,-)m*
-' (o-.o,
)m*
"2T,
otherwise.
Further, u is a singleton ~-equivalence class in each case. Since T, and a(€ have a
common multiple in w, by [9, Corollary 13] they have a least common multiple, say
[7] Conjugacy in singular Artin monoids 189
L = a'u. Since L divides w, we have u divides v. Since the number of occurrences
of letters from T is preserved under ~, L must contain at least one such occurrence, so
u cannot properly divide v; they must be equivalent. Thus w is the lcm of i) and of.
This lcm has length my
e
> 3e. Now for any n, anr W ~ Wbnr, so if W were divisible
by tj, then anr W would be a common multiple of of and r,
,
and hence would have
length greater than 3nr. Thus for every n, the length of W would have to be at least
2nr, which is absurd. Thus W must not be divisible by
T,
for my > 2.
Thus we have that W begins with x where either x =
OJ
e S, or x = Xj where
i = j or m(j = 2. From inspection of the relations, we see that the lcm of a and x
must be aC ~ Cd where C begins with x and d e S; furthermore, arC ~ Cdr is
the lcm of ar and x. Thus arC divides ar W ~ Wi/, so W ~ CV for some word V.
Thus ar
W
~ Of
V
~ CVbr ~ W, so dr V ~ W. By an inductive hypothesis,
dV~ Vb, so aW ~ aCV ~ CdV ~ CVb~ Wb, completing the
proof.
D
For
any
words Vi,...,
Vk
over
5U
T with
a
common multiple, a word L
(Vj,...,
H)
can be calculated which is a least common multiple of Vi,...,
Vk
(see
[9,
Lemma 12]).
Let A(
U)
denote the set of letters from SU T which divide the word U. Since U is a
common multiple, L(A(f/)) always exists, and divides U. If r, and
T,
are in A(f/),
then {/ ~
T,
U\
~
T,
U
2
for some f/i and f/2. so applying the reduction property,
my = 2. Thus the product of the elements of A(U) n T is a common multiple of
A(f/) n T; and is, in fact, equivalent to L(A(U)
D
T).
LEMMA 1.6. Let U be a nonempty word such that A(f/) C T. Suppose that
VU ~ UW for some words V and W and A(U) D A(V) = 0. Then L(A((/))
commutes with
V and
there
is a word U' shorter
than U such
that VU' ~
U'
W.
PROOF. Take any
Xj
inA(U). Since
T,
and
V
have a common multiple, Remark 1.3
says KtJ(V) is defined, and since
Xj
is not in A(V), KTj(V) is a Ty-chain. Let b be
the target of ATr.(V). Again by Remark 1.3, since r, divides ^rj(V)(/ ~ VU, b
divides U. So b = xk for some k,
K
Zj
(V) is preserving, and
Xj
V ~ Vxk.
Thus for every rp in A(U), there is a corresponding xq in A(f/) such that
T
P
V
~
Vr9. Moreover, if rr
V
~
VT,
then xp V ~ rr
V;
right cancellation then gives
T
P
= xr.
Thus V defines a permutation 7rv on A((/) by xp V ~ V
/
7TV(T
/
,). By the comment
preceding the statement of the lemma, L(A(f/)) is the product of the elements of
A(U) in some order, and the letters all commute. Thus
V =
T,-
• • •
T
4
V
~ ^v(r,,)
• • •
nv(xik)
Now U ~ L(A(U))
U'
for some word (/' which is shorter than U. Thus
VL(A(U))U'
190 Ruth Con-an
[8]
Cancelling L(A(U)) from
the
left,
VU' ~ U'W, and U is
shorter than
U, as
desired.
•
The operator
rev
:
(S
U
T)* i—>
(S
U
T)* maps
a
word to its reverse:
LEMMA
1.7. Lef C
fee
an a-chain to b. Then
b e S
if a
e
S, and b
does not divide
rev(C)
whenever
a
€ S
or b
e T.
PROOF.
By
inspection
of
the relations,
b e S if a € 5, and
always,
b
does
not
divide rev(C)
if C is
simple. Suppose henceforth that
a e S or b e T, so
either
{a, b)
c
5,
or
{a, b)
C 7\
We will show
(*)
if
C
is simple and W any word
where
b
divides
rev(
WC),
then
a divides rev( W).
The result then follows by induction on the number
of
simple components
of C.
The proof
of (*)
falls into two cases: either (i) rev(C)
is a
simple
fe-chain
to a,
or (ii)
C =
Xj {PiOj
)r
for
some
i
and
j
with
Wy
> 2,
and some
0 < r <
m,,
—
1. In
case (i), we observe that rev( WC)
is a
common multiple
of
rev(C) and b, and so,
by the comments beginning Remark 1.3,
a
(the target of
the
fc-chain
rev(C)) divides
rev(W).
In case (ii),
C
is
a
simple
a =
a,-chain
to
b, where
b =
Oj
if r is
odd and
b = a,
otherwise. However, rev(C) is not simple, but a compound
fo-chain
to a) consisting
of simple chains rev((cr,<7j)r)
and i;.
Again using Remark
1.3, a,
must divide
rev(
W),
say
W ~
XCT,
;
hence
Now rev((cr/cr,)r+1)
is a
i-chain
to a,, so a,
must divide rev(Xi))
= r;
rev(X).
By the reduction property, there exists
a
word
V
such that rev(X)
~
(CTJO))"1'"1
V.
Thus
rev(VV)
~
Oj rev(X)
~
o> (ff.-o))"*"1 V
s
(ay<T,>m* V
~
(or.-o))1"*
V,
so we have
<r,
s a divides rev(
W).
D
We thank the referee
for
the stronger version
of
this result
as
given above. He also
pointed out that
it
is now as strong as
it
can be—there are a-chains
C
with target
b e S
for which
b
divides rev(C).
An
example
is in
type
A2,
where
C =
a1<r2cr1CT2
is a
compound r2-chain
to a\, but C ~
O\O\OiO\,
so
rev(C)
is
divisible
by
its target,
crt.
For the remainder of this section, we suppose that A
=
L(CTI,CT2,
..., an)
is defined,
that is, the elements
of 5
have
a
common multiple,
and A is a
least common multiple
[9]Conjugacy in singular Artin monoids191
Type
An
(n>l)
Bn
(n>2)
D
n
(n>4)
En (n = 6,7,8)
him) (m>5)
Coxeter graph
FIGURE
1. The irreducible Coxeter graphs of finite type. Unlabelled edges have value 3.
of S. Then we say that M is of finite type. It is known (see for example, [20,
Chapter 2]) that the graphs for these types are precisely finite disjoint unions of those
shown in Figure 1. The reader will notice that this list is closed under taking complete
subgraphs. The element A is called the fundamental element, and has a number of
properties which will be referred to shortly. First we need the definition of a square
free word. A word W has a quadratic factor if there are words U and V over 5 U T
and a letter a such that W = Uaa V. A word is square free if all words equivalent to
it have no quadratic factor. The following results can all be found in [7, Sections 5
and 8] and [9, Section 4].
REMARK 1.8. Properties of the fundamental element A. ([7, Sections 5, 8]; [9,
Section 4].)
(1) A is an element of the positive Artin monoid (that is, a word over S).
(2) A is both a left and right least common multiple of S.
(3) rev A ~ A.
(4) A word over S is square free precisely when it is a divisor of A.
(5) A generates the centre of S^^ for all finite Artin types except types
A „
for n > 2,
Dyc+i,
^6 and hQ.q-\- 1), in which cases A2 generates the centre.
We will denote by £ either A or A2, the generator of the centre of yjj- Define
192 RuthCorran [10]
to be the set of
words
over 5 for which there is a word A such that A A ~ A—
in other words, J2^(S) is all words (including the empty word) which divide the
fundamental element. According to the previous result, we could define £}&{S)
equivalently to be the square free words over 5 (the letters 21 & stand for quadrat frei,
or square free). There are only finitely many words whose length is at most l(A), so
£1&{S) is finite. In contrast, there are infinitely many square free words over S
U
T,
as there are square free words of arbitrary length over 5
U
T, for example (r,r;)* is
square free for any k, provided mtj > 2.
LEMMA 1.9. (1) Suppose that BxC ~ B'yC where B, B', C, C are
words
over
S andx, y € T. Then BC~ B'C.
(2)
Suppose
that
A
is in Q&iJS) and b
is
any
letter in
SUT. Ifb
does not left divide
A,
then bA is square free. Ifb does not
right divide
A, then Ab is square free.
PROOF. Part (1) follows by observing that the only relations involving an element t
of T which may be applied here are of
the
form (tw, wu) or (wt, uw) for some u e T
and word w over S. The result then follows by induction on the number of such
applications required to transform BxC into B'yC.
If b € 5, then the statement of part (2) follows from [7, Lemma 3.4] and an
application of rev. If b € T, then by part
(1),
bA and Ab are square free whenever A
is in
£&(S).
•
LEMMA 1.10. Let B and V be words over SUT. Suppose that A is the longest
square free
word
over S
which
divides
A
B,
and
moreover,
that A divides
VA
B.
Then
A divides VA.
PROOF. By (3) and (4) of Remark 1.8, there is a word D over 5 such that DA ~ A.
So A divides DVAB. Hence every letter of S divides D VAB. In particular, DVAB
is a common multiple of a, and D VA for all
er,
in 5, so Kai(D VA) is always defined.
Suppose
CT,
does not divide DVA. Then KOj(DVA) is a a,-chain to b, for some
b e S. Further, b must divide B, since at divides DVAB (Remark 1.3). Moreover,
Lemma 1.7 says b does not right divide DVA, so in particular b does not right
divide A. Hence Ab is squarefree (by Lemma 1.9) and divides AB, contradicting the
maximality of A.
Thus each <x, divides DVA, so their least common multiple A ~ DA divides
D VA. Cancelling D from the left, A must divide VA. •
2.
Conjugacy in
5?%,
when M is of finite type
Suppose that V and W are words over SUT. We say that V is conjugate to
W
(relative
to &), denoted V x W, if there exists a word X over SUT such that
[11] Conjugacy in singular Artin monoids 193
VX ~ X W. In this case we say V x W by X. It is not immediately obvious whether
conjugacy is an equivalence relation. It is certainly reflexive, and if V x W by X and
IV x Z by K then V x Z by X Y, so conjugacy is transitive.
Throughout the whole of this section, M is assumed to be of finite type. We will
see that this restriction is enough to ensure that conjugacy is also symmetric (although
in the case when M is not of finite type, this is not known).
Let N = £#(S) U T. Say that V is ^-conjugate to W if V x W by some element
of N; that is, if there is a word A in N such that VA ~ A W. This is denoted V xK W.
If V x« W then V x W. We will show that x is contained in the equivalence relation
generated by xN. (This will turn out to be x.)
LEMMA 2.1. Suppose that V x W. Then there is a positive integer p and words
XO,XU...,XP
with XO=V,XP = W and X,_, x* XJor i = 1, ..., p.
PROOF. Suppose V x W by U, so VU ~ UW. The argument is by induction on
the length of U. If l(U) = 0 or 1, then U e N so p = 1. Suppose now that l(U) > 1,
so A([7) must be nonempty.
First suppose that A(U) n S ^ 0. Let A be the longest square free word over 5
which divides U. So U ~ A B for some word B and A is not empty. Since A divides
VAB,
Lemma 1.10 says there is a word X such that VA ~ AX. But A 6 N so
Vx« X, and moreover AXB ~ AB W, so after cancelling, XB ~ fi W. The result
now follows by induction applied to B.
Now we suppose that A( U) c T. Suppose that A(U) n A (V) ^ 0. Then there is a
r, e 7 such that U ~ r, £/' and V ~ r, V. Let X = V'r,. Then Vr, ~ r, V'r, = r,X,
so V
XN
X. Moreover, xtX U ~ r, V'r, (/' ~ V(/ ~ £/W ~ r, [/' W. By cancellation,
X [/' ~ {/' W, and the result follows by induction applied to U'.
The only case left is when A(f/) c T and A({/) n A(V) = 0. These are
precisely the conditions of Lemma 1.6, so there is a word U' shorter than U such that
VU' ~ t/' W, and so the result follows by induction applied to £/'. •
LEMMA 2.2. // V xK W //ie« W x V.
PROOF. Suppose that A is an element of £&(S) such that VA ~ AW. Then there
is a word D such that A Z) ~ A, and ADA ~ A2 is certainly central, so
AWDA ~ VADA~ VA2 ~ A2V~ADAV,
and after cancelling
A
we have WDA ~ DA V, and so W x V.
Suppose alternatively that Vr, ~ r,
W
for some r, in T. If r, divides V then
V ~ r,X for some word X, so r,W ~ r,Xr,, after which cancelling gives W ~ Xr,,
yielding WX ~
XT,X
~ X V, and so W x V. If r, does not divide V, then
A"
r
,(
V)
194 RuthCorran [12]
is a r,-chain to r,, by (4) of Remark 1.3. In this case, by Lemma 1.4, KTl(V) is
preserving, so r, W ~ Vr, ~ r,
V;
cancelling gives W ~ V,so W x V. •
THEOREM 2.3.
Conjugacy
is an
equivalence
relation.
PROOF.
AS
remarked earlier, conjugacy is reflexive and transitive. Suppose that
V x W. Then by Lemma 2.1 there is an integer p and words Xo, X\, ..., Xp such
that V = Xo, Xp = W and X,_i xK Xt for i = 1,..., p. By the previous lemma, for
each i = 1,..., p, X, ; x X^\. By transitivity, Xp x Xo; so W x V. Thus conjugacy
is also symmetric, and hence an equivalence relation. •
Let E be a set of words over 5
U
T. Then define
= {X | VxHX for some V e £}.
Observe that if V is conjugate to W, then there is an integer p and words
Xo,...,
Xp
such that V = Xo xK X, xK
• • •
xN Xp = W,so W €
(pp({
V}).
Furthermore, homo-
geneity forces £(X) = l(
V)
whenever V xN X, so
¥>*({ V})
c {X € (5
U
7)* | t(X) = l( V)},
which is finite—it has at most
(2n)
tiV)
elements, where |5| = \T\ = n. By reflexivity
of xK-conjugacy,^*(E) c ^+1(£) for
all
numbers handsets E.
then
<pk+r(Z)
-
<pk(X)
for all r > 0. So,
Let
Then
<I>( V)
is precisely the set of all words conjugate to V.
If W = bU for some letter
&,
then we write
b~'W
= £/. If
W
does not begin with b
then *"'
W is
not defined.
We
now define a partial operator (/) : (5U
T)*
x (SU
T)*
-»•
(5
U
T)* which does the job of division. Suppose V s a,a2
• • •
ak. If V divides W,
then
(w/ V) = T1^ (... ^Kai (°;'Kax
(
MO)
• •
•).
and (W/ V) is undefined otherwise. (In fact, if
V
does not divide W then in trying to
perform the calculation described will result in an 'undefined' answer at some stage.)
Moreover, if V divides W then, by [9, Lemma
6],
W ~ V(W/ V).
THEOREM
2.4. The set
4>( V)
of all
words conjugate
to V is
calculable.
[13] Conjugacy in singular Artin monoids
195
PROOF. For any *
>
0,
<pk+l({V})
=
Ux&wMW)- Consider
X e
<pk({V}).
Then 1{X) = l(V), and
tp([X})
= {Y e
(S U
T)IW | XA ~
A
K
for some word A in K}
= {K
€
(5 U
T)
e{V)
|
(AK/X)
f£
oo and ((AY/X)/A)
is
the
empty word for some word A in X}.
So to calculate ^({X}), for each of the elements
A of
K and for each of the
(2n)
l(V>
elements
Y of
(5 U
T)/(V),
the two calculations (A
Y/X)
and ((A
Y/X)/A)
must be
made. There are at most
|
T\
+
(|5|
+
1)'
<A)
=n
+
(n+
1)'
(A)
words in N, so this adds
up to at most 2(2n)e(V) (n + (n
+
1)<(A)) calculations. Since thsre can be no more than
(2n)'
(V)
elements of
<pk({
V}), at most 2(2n)t(V)(n
+
(n
+
l)<(A))(2/x)'(V) calculations
need to be done
to
calculate (pk+1({V}), once <pk({V})
is
calculated. By definition
d>(
V)
=
^(2n)'(10({ V}), so at most
1(1~\l(V)( , / , i\<(A)\/')M\«(V)/o \t(V) o3<(V)+l 3<(V)/' , / , i\«(A)\
L\£.n)
\n
-\-
\n
-\-
i)
/(•£'i)
K^M) =
Z
n \n
-t-
\n
•+•
i) \
calculations are required to determine
<!>(
V).
D
Thus the conjugacy problem
is
solvable
in
positive singular Artin monoids.
To
determine
if
W is conjugate to V, calculate
<t>(
V) (which is finite) and see whether W
is a member.
3.
Conjugacy in singular Artin monoids of finite type
Suppose that
M
is an
/ x /
Coxeter matrix as defined in the first section. Let S"1
denote the set {a~l
\ i e
/} of formal inverses of 5. Then the singular Artin monoid of
type M, denoted
yM,
is the monoid generated by
S
U S~x U
T
subject to the relations
3? described in Section 1, and the free group relations on
5:
OjCr"1
=
crf'cr,
=
1 for all
i e I.
If two words V and W over 5
U
5"1 U T represent the same element in
yM
then write
V %
W.
Throughout the rest
of
this section,
M
will
be of
finite type.
In
this case,
the
singular Artin monoid SM is also said to be of finite type. The reader is reminded that
this means the Coxeter graph
of M is a
finite disjoint union
of
graphs
in
the list
in
Figure 1. The singular Artin monoid of type
An
is commonly known as the singular
braid monoid on n
+
1 strings, as defined in [4] and [5].
Theorem 20 of [9], known as the Embedding Theorem in the sequel, says that
yjj
embeds in
yM.
The proof of this theorem made much use of the fundamental element
196 RuthCorran
[14]
A, and the central element
£
(which
is
either
A or
A2, depending on the type, as per
(5)
of
Remark 1.8). Suppose
V s
U^o'1 U\0~x U2
• • •
t/^icr"1 £4 where each Ui
is a
word over
SUT.
Then, as
in
[9, Section 5], define
6i(V)
=
£/„£,, Ufalh---
t4-i&U
k
,
where £,
=
(£/CT,)> which is defined because every letter
of 5
divides
f.
Thus
v.
Two words V and W over
S
U
S~l U T
are said to be conjugate in
yM if
there exists
a word
X
over
5
U S"1
U T
such that
VX % X
W. The following result shows that
conjugacy in
y^
is just the restriction
of
conjugacy in
yM.
Thus
it
will make sense
to use the notation
V
>c
W for conjugacy in
yM
also.
LEMMA 3.1.
If V and W are
conjugate
in
yM, then there
is a
word
X
over
SUT
such that
VX ^XW.
PROOF. Suppose that
V,
W, and Y are words over SUS~lUT such that
VY
% Y
W.
Since ^^(K)
«s
Y, multiplying through by
^iY)
gives VGi(Y) &
6i(Y)W,
so in
fact
V
and W are conjugate by
a
word X
=
0,
(K) over
5
U
T. D
THEOREM 3.2, Lef V and W be words over SUS~]UT with 62(V)
>
62(W). Then V
is conjugate to W in
yM
precisely when
Gx
(V) is conjugate to f
*<
^-^Wfl, (jy)
,-„
y+m
In particular, words over
SUT
are conjugate in
JS'V
precisely when they are conjugate
in
y^,.
Thus the conjugacy problem is solvable in
yM.
PROOF. Suppose 6>,(V)
is
conjugate to ^-(^-^^e^W)
in yjj.
Then there
is a
word
X
over
SUT
such that 0,(V)X
~
X£*(V)-*(1|r)0i(WO. Multiplying by £-WV),
using the fact that
f
~^(
"^
(V)
^
V, and the centrality of £, we have KX^XW.
On the other hand, suppose that
V
and
W
are conjugate
in yM.
By Lemma 3.1,
there
is a
word
X
over
SUT
such that
VX
«Xiy. Multiplying through
by
f
wv
°
gives 0,(V)X
«
X^^-^^CWO, but, since
62(V) >
62{W), all the words
in
the
equation are over the alphabet
SUT.
By the Embedding Theorem
of
[9], this gives
0i(V)X ~ X^-W-^WOiiW),
so
9i(V)
is
conjugate
to
^w-^^OiiW)
in
yM. •
The remainder
of
this section deals with some results which we hope may explain
our choice
of
the definition
of
conjugacy. There does
not
seem
to be a
general
semigroup theoretic definition
of
conjugacy. Howie [18] introduces
the
following
notion, which
we
call here 'swap conjugacy',
in the
context
of a
certain class
of
semigroups called equidivisibible semigroups. Say that
V
and W are swap conjugate
[15]
Conjugacy
in
singular
Artin
monoids
197
if there exist X and Y such that V ^ XY and W « YX. This notion is a natural one
in the context of singular braids; these may be considered as geometric realisations
of elements of singular Artin monoids of type A. This is discussed in more detail in
Section 6.
Swap conjugacy is clearly reflexive and symmetric; though not necessarily transi-
tive.
In Howie's context, transitivity is easily shown to hold. However, even in the
positive singular Artin monoid of type A2, swap conjugacy fails to be transitive. For
example,
O\O\O
2
is swap conjugate to a1a2<JI ~ o2O\a2, which is swap conjugate to
O\O2o2. Using the fact that
O\O\a
2
is in a singleton equivalence class of words in ^+2,
it is easy to see that the only words which are swap conjugate to it are
<J\O\O
2
,
O\O
2
O\
and
CT
2
CT
1
OT
1
.
Thus o\O\O2 is not swap conjugate to
O\a
2
a
2
in S?^.
We mention that, on the other hand,
O\a\O
2
% (a1cr2)(crlCT2cr1"1) is swap conjugate
to
O\O
2
o
2
~ {O\(J2O\1){G\O2) in S^Al. Thus this relation is coarser in 5?^ than in 5^M\
preventing us from 'bootstrapping' our way from 5^^ up to 5^M as has been the
technique previously.
It turns out that provided we restrict our interest to the transitive closure of swap
conjugacy in S^M, we obtain the same relation as that of conjugacy as defined in this
paper. (At the time of writing, the author does not know if swap conjugacy in 5fM is
transitive.) If V and W are swap conjugate then we write V ;=± W. We remind the
reader that M is assumed to be of finite type.
LEMMA 3.3.
Let V and W be
words over
SUT
such that
V xN W.
Then
V ^ W.
PROOF. There exists A e K = 2&{S) U T such that VA ~ AW. If A e
then A is invertible, and so V « (A)(A~l V) and W « (A"1 V)(A), whence V ^ W.
Otherwise, A = r, for some i. Suppose that r, divides V. Then V ~ r, V for some
word
V"
over SU T, and we have r, Vr, ~ r, W. After cancelling, we have W ~ V'r,,
and immediately V ^ W. Finally, suppose that r, does not divide V. However, r,
does divide Vrh so by (4) of Remark 1.3, KT.( V) is a r,-chain to r,. Lemma 1.4 says
this T,-chain must be preserving, and so KTi (V)r, ~ r,
KT.
(V). Since V ~ Kr. (V), we
have r, V ~ Kr, ~
T,
W. After cancelling, we have V ~ W, so V and W are trivially
swap conjugate. •
LEMMA 3.4. // V and W are words over S U 5"1 U T such that V ^ W, then
^my^^m
W
fQf. fl//
PROOF. This is immediate by the centrality off. •
THEOREM 3.5. Let Vand Wbe words over SUT'UT.
(1) lfV^± W, then V
>z
W.
198 RuthCorran
[16]
(2)
If V x
W, then there exists an
integer
p > 0
and
a
sequence Zo,
Z\,...,
Zp
such that
V = Zo ^ Zx ^
• • •
;=±
Zp = W.
PROOF. (1)
If V ^
W, then there exist X and
K
such that
V ^ XY
and
W
^ YX.
Thus VX
«
X YX
%
X
W,
and so
V x W.
(2) Suppose V
x W.
By Lemma 3.1 we may suppose that
VU
«s
f/W for
some
U
€ SL)T.
There exists an integer m
=
max{02(VO, ^(WO) such that V
«
£~m V+
and W
ss
£~m W+
for
some words
V+
and JV+ over SUT. Multiplying through by
£m, and using the fact that
£ is
central, we have V+
U
« t/W+.
By the Embedding
Theorem
of
[9],
V+
[/ ~ UW+.
By Lemma
2.1,
there is an integer
p > 0
and words
Xo, X,,...,XP with
V+^XOXKX^.-.XKXP^
W+.
By Lemma 3.3, we have
Multiplying through by £~m, and invoking Lemma 3.4,
VHZ0^Z,^..-^Zf = W,
where
Z, s
i;-
m
X
t
for
each
i = 0,
1,...,
p. •
4.
Centralisers in singular Artin monoids
of
finite
type
For generators a and
b,
denote by the set Z(a,b) the set of words
W
over
5U
5"'
U T
such that
aW ^ Wb.
The centraliser
of
a generator a, which will be denoted
Z(a),
is then the
set
Z(a, a). Whenever Z(a,b)
is
not empty,
it is
infinite
in
size—since,
for example, whenever
P e Z(a,
b) then
akP is in Z(a,
b) also
for
arbitrary
k. We
assume throughout this section that M is
of
finite
type.
We
will show (Proposition 4.6)
that Z(xi,Xj)
is
not empty precisely when the vertices
i
and./
of
VM are connected
by
a
sequence of edges labelled by odd mab only.
LEMMA
4.1.
The
set
Z(a,
b) is empty if a and b
are
not
both
in S, not
both
in
S~\
and not
both
in
T.
PROOF.
It
can be seen from the relations that the number
of
letters
of T
appearing
in a word is invariant under applications of
the
relations, since each relation either has
no occurrences
of r or
one
on
each side. Thus
an
equation
of
the form a
W
%
Wb
means that either both
a
and b come from
T,
or neither a nor b comes from
T.
[17]Conjugacy in singular Artin monoids199
Now suppose that a{ W «s
WcTj
'. Since W « f-^^fl,
(WO,
multiplying both sides
of the original equation by the central element
f-**"')
gives ot0\(W) « 9i(W)a~\
Thus
ofix(W)<Tj
«
6»,
(HO,
and so by [9, Embedding Theorem], ofix (W)o> ~ 0, (W).
But this cannot be the case, as equivalent words in «$*£ must have the same length. •
It is clear from the definition that Z(pt ',Oj ') = Z(ah Oj), since atW «
precisely when Wo~l « a"1
W7.
Thus we will momentarily restrict our consideration
of Z(a, b) to the case where a and b are both from 5 U T.
It turns out that preserving a-chains are important elements of these sets. Preserving
chains are characterised by the property of Lemma 1.4 that a nonempty word W is
equivalent to a preserving a-chain to b precisely when a does not divide W and
aW ~ Wb. Thus preserving a-chains to b and all words equivalent to them are
in Z(a, b).
LEMMA 4.2. Let a and bbeinSUT and a £ b. A word W over SUTis in Z(a,b)
precisely when it is equivalent to a word of the form am P where m is a non negative
integer and P is a preserving a-chain to b.
A word W over S U T is in Z(a) precisely when it is equivalent to a word of the
form am P where m is a non negative integer and P is either a preserving a-chain to
a or the empty
word.
PROOF. In light of previous comments, we only need to prove the 'only if direction.
Suppose W e Z(a, b). There is some integer m > 0 such that am divides W but
a
m+i
does not divide W. Then W ~ am V for some V which is not divisible by a. Since
a\y ~ Wb, then am+1 V ~ am Vb, and soaV~ Vb by left cancellativity. By
Lemma 1.4, V is equivalent to a preserving a-chain P with target b, and W ~ amP.
The argument for when a = b is almost the same, except that P may be empty. •
Now take any word
Wover
SUS~lU T. Then W s» t;-%(W) where k = 02(W),
a non-negative integer. Since £ is central, then W is in Z(a, b) precisely when 0\ (W)
is in Z(a, b). We know the form of such elements from the previous Lemma. Thus
PROPOSITION
4.3.
Z(a,b)
= W
W «
l;-
k
a
m
P
where k,m>0and P is
a preserving a-chain to b or may be the
empty word ifa = b
LEMMA 4.4. For any i and), Z(at
PROOF. That
Z(CT,
',
o-j
') = Z(<r,, a,) was noted earlier. A quick inspection of
the relations
S&
shows that the source and target of a simple preserving chain must
be both in S or both in T. Further, if P is a simple preserving a,-chain to as,
then r,P ~ Pxh and if P is a simple preserving r,-chain to Xj, then a,P ~ Par
200
RuthCorran [18]
Since
compound preserving chains are just concatenations of suitably matching simple
preserving
chains,
the same results hold
there:
if P is a preserving a,-chain to a,, then
tiP ~ PXj, and if P is a preserving
T,-chain
to r,, then otP ~ Poj.
Now suppose W e Z(oit
Oj).
Then Proposition 4.3 says W «
%~
k
o™P
where P
is a preserving a,-chain to o) and k and w are non-negative integers. So
r,W « r,fV;P « f-*T,or,mP « f-*cr™T,-P « f-y Pi) «
WT,,
so W 6 Z(r,, r,). Thus we have that Z(cr(, o)) c Z(r,, r,). The reverse inclusion is
analogous. •
Thus it makes sense to denote the set Z(of\ of1) — Z(ah a,) = Z(r,, r,-) by
PROPOSITION 4.5. Whether or not a word W over S U S~l U T is in Z(i,j) is
calculable.
PROOF. The word W is equivalent to
^~
62
^
W)
9
i
(W),
and since £ is central, W is in
Z(i,j) precisely when 0\{W) is. Calculate (6>,(W0/o-*) for k = 0, 1,... ,p where
p is the integer such that U = (0i(W)/o^) is defined, but (0i(WO/af+1) is not. (At
most
£(#i
(W)) calculations need to be done.) Then we need to determine whether or
not U is a preserving cr,-chain to ar According to Lemma 1.4, this is equivalent to
determining whether
cr,
U ~
UOJ
.
This is the case precisely when (Ucfj
/CT,)
is defined
and equivalent to
U,
or in other words when ((
UOJ
/Oi)/
U)
is defined and empty. •
The next results of this section describe precisely when Z(i,j) is not empty. It is
clear that Z(i, i) is never empty as £", a", and r,m are in Z(i, i) for all integers n and
non-negative integers m. If/n,y is odd then (cr/cr,)m»"1 is a preserving a,-chain to <r7,
so
Z(/,
j) is not empty. Moreover if Z(i, j) and Z(j, k) are nonempty then Z(i, k) is
nonempty as Z(i,j)Z(j,
k)={VW\
V e Z(i,y) and W e Z(/, *)} c Z(i, it).
PROPOSITION 4.6. 77ie 5er Z(i,j) is
nonempty
precisely when there is an integer
r > 0
and
a
sequence
i =
i
0
,...,
ir = j
such
that
m
ik
_
iik
is
odd for
each
k = 1, ..., r.
PROOF. If there is such a sequence, then from the preceding comments it is clear
that Z(i't_i, it) is nonempty for each k = 1,..., r; and thus
Z(iJ) 2 Z(i,i,)Z(i,,i2)---Z(ir_i,j)
is nonempty also.
On the other hand, suppose Z(i,j) is nonempty. If / = j then the result holds
with r = 0. Suppose then that i ^ j, and take W e
Z(i,j).
From Proposition 4.3,
[19] Conjugacy in singular Artin
monoids
201
there are non-negative integers k and n and P, a preserving a,-chain to Oj, such that
W « %~ko?P. Thus preserving cr,-chains to cr, exist.
Suppose P = P\P2- •• Pp is such a cr,-chain to o), where each P, is a simple
preserving chain. By excluding the simple components whose source and target
are the same, we obtain a subsequence
j
o
,j\,
... ,jr of
1,2,...,p
such that P' =
Pj0Pj,
• • •
Pjr is a preserving
CT,-chain
to Oj and each Pj, is a simple preserving a,(-
chain to
a,
J+l
where U ^ //+). (So i0 = i and ir = 7)- The only such words are
(ffi,
+
,cr;
J
)
m
''''
+1
~\
with m,,,,+1 odd. (If mP9 is even, then
(a
q
a
p
)
m
>">~
1
is a preserving op-
chain to a,,). Thus the sequence i = i0, i\, ..., ir_i, ir = j satisfies the requirements
of the theorem entirely. •
A word is said to be r-free if it has no occurrences of letters from T.
SCHOLIUM
4.7.
The set Z
(i,
j) is nonempty if and only if there is a x -free preserving
archain to CT;.
The group whose presentation is given by the generators S
U
S~l and relations all
those of yM with no occurrences of letters from T is called the Artin group of
type
M. It is a subgroup of the singular Artin monoid of type M. The Artin group of type
An
is often called the
braid group
onn +
1
strings. Let
ZsdJ) = {We Z(i,j) | Wis r-free} = Z(iJ) n (5U 5"')*.
From Scholium 4.7, we know this is nonempty precisely when Z(i,j) is nonempty.
PROPOSITION
4.8.
The
setZs(i,j) consists
precisely
of
those words W over
SUS"1
such that
W(Tj
and
Oj
W
represent
the same element of
the
Artin group of
type
M.
Membership
ofZs(i, j) is
calculable.
The results of the next corollary follow immediately from Proposition 4.6 and an
examination of the list of Coxeter graphs of finite types given in Figure 1.
COROLLARY 4.9. (1) Ifiandj are not in the same
connected component
of TM,
then Z(i, j) is
empty.
(2) If i and j are in the same connected component of VM and the connected
component is
not of type Bn, F4, G2 or
I
2
(2m),
then Z(i, j) is always
nonempty.
(3) Ifi andj
are in a connected
component of type Bn,
then
Z(i, j)
is
empty precisely
when
1
e U,j] and i ^ j.
(4) ///' andj
are in a connected
component of type
F
4
,
then
Z(i, j)
is
empty precisely
when [ij} is either
(1,
3}, {1, 4},
{2,
3},
or (2, 4}.
(5) If
i and
j are in a connected component of
type
G2 or I2(2m), then Z(i,j) is
nonempty precisely
when i = j.
202 Ruth Corran [20]
COROLLARY 4.10. The results of Corollary 4.9 hold with Zs(i, j) replacing Z(i, j).
Fenn, Rolfsen and Zhu [12] introduced the notions of (j, k)-bands and singular
(j, k)-bands in the braid group and singular braid monoid—which in this article are
called the Artin group of type A and the singular Artin monoid of type A respectively.
These bands turn out to be geometric analogues to elements of Zs(j, k) and Z(j,k)
respectively. The relevant result, [12, Theorem 7.1], is the following, which shows
that a singular braid with a (possibly singular) 0'. &)-band has precisely the required
property for belonging to Z(/, k) in 5^An.
THEOREM 4.11 (Fenn, Rolfsen and Zhu). For a singular braid x in the singular
braid
monoid,
the following are equivalent:
(a) OjX -xak;
(b) ajx = xa[, for some nonzero integer r;
(c) ajx = xa[, for every integer r\
(d) Xjx =xrk;
(e) TJX = x rtr, for some nonzero integer r;
(f) x has a (possibly singular) (J,
k)-band.
By the equivalence of (a) with (f) we deduce that a singular braid has a (j, &)-band
if and only if it can be represented by an element of Z (j, k).
The method of [12] to identify elements of the centraliser of a generator in the
singular braid monoid relies on the geometric realisation of the Artin type An as
braids. There are no known geometric realisations for types other than type A, so the
method of (j, Jfc)-bands cannot be extended in an obvious manner to the other types.
The algebraic method provided here covers all types: By combining Theorem 1.5,
Proposition 4.3, Lemma 4.4 and Proposition 4.5, we can extend this theorem of Fenn,
Rolfsen and Zhu above to arbitrary type (that is, not just type A):
THEOREM 4.12. An element x of yM is in Z(j, k) if and only if aj x «s xa[ for
some non-zero integer p, if and only if
TJX
SS
xx[ for some non-zero integer r. For
any x, this is calculable.
In [12], the authors remark that 'it may not be obvious, from a presentation as a
word in the generators, whether a word has a (j, jfc)-band\ However, Proposition 4.5
and Proposition 4.8 provide ways of determining inclusion in Z(j, k) and Zs(j, k)
respectively.
Birman [5] conjectured that the monoid homomorphism rj from the singular braid
monoid to the group algebra of the braid group defined by rjiaf11) = cr,*1 and rj(r,) =
cr,
—
a~x is injective. Among many other interesting results about braids and singular
braids, the Fenn et al.'s paper [12] confirms certain cases of Birman's conjecture
[21] Conjugacy in singular Artin monoids
203
using results about centralisers.
It is
hoped that the results
of
this paper may help
in determining the truth or otherwise
of
Birman's conjecture
in
general. In the final
section, the singular braid monoid and its relationship with singular links is discussed.
Next, we define
parabolic submonoids
and describe some results about conjugacy of
parabolics.
5. Parabolic submonoids of singular Artin monoids
Recall that
M is a
Coxeter matrix over
/, a
finite indexing set. For
J c /, it is
clear that the submatrix Mj of M containing the entries indexed by
J
is also a Coxeter
matrix. We use the following notation:
Sj
=
[oj
|j e J),
SJ1
= [a'1
\
j e J), Tj =
{TJ
\
j e J}.
Recall that
&
denotes
the
defining relations
of
«^J. Denote
by
fflj
the
defining
relations
of
5?^,
•
The next two observations follow from the definitions
of
these
relations:
(1)
Stj c SP.
(2)
If
(X,
Y) e ^
and
X is a
word over
Sj
U 7>, then
Y
is
a
word over
Sj
U
Tj
also,
and
(X,Y)e&j.
Suppose that V and
W
are words over Sj
U
7).
If
V
and
W
represent the same element
of
y^tl
write
V
~/
W. Note that the relations
~
and
~/
are identical; in this section
we will use
~/ for
clarity. The observation (1) above ensures that
if V~y
W, then
V ~/ W. Conversely, suppose that V ~/ W. Since V and W only contain letters from
Sj U
TJ,
the second observation says that the only relations from
^E
which can
be
used to transform
V
into W are those which lie in
£%f
anyway; thus V ~y
W.
This
proves
PROPOSITION
5.1.
y^
embeds
in
yjj.
Now suppose that V and W are words over
Sj
U 5J1 U 7). The notation
V^j
W
(respectively,
V
&,
W) implies V and W represent the same element of
yMj
(respec-
tively,
S^M).
It is clear that
if V^j
W, then
V&j
W. However, to prove the stronger
result that
V %/
W implies
V
^j
W, we use the Embedding Theorem
of
[9], which
has only been proved for finite types.
Suppose that
M is of
finite type. Then
A, the
least common multiple
of
5/,
is
defined,
and is a
common multiple
of Sj, a
subset
of 5/.
Thus
a
least common
multiple
of Sj
exists, so
Mj
is
of
finite
type. (Or alternatively, one may observe that
the list of Coxeter graphs of Figure
1
is closed under taking full subgraphs). We will
204 Ruth Corran
[22]
denote by Ay the fundamental element of
S^M,,
and by
fy
its corresponding central
element (either Ay or Ay according to (5) of Remark 1.8).
Let V and W be words over Sj U SJ1 U 7). Then there exists an integer m such
that V «y £7 V+ and W«y £7 W+ where V+ and W+ are words over
SJUTJ.
By the
comments following the proposition, we have that V
«s
7
£7 ^+ and
W
«/ £7 W+-
Suppose that V«/ W. Multiplying through by i;jm, we have V+ «/ W+.
By
the Embedding Theorem ([9, Theorem
20]),
V+ ~/ W+. Proposition 5.1 then ensures
V+
~j
W+, and so V+ «y W+. Finally, multiplying through by £7 results in V wy W.
Thus
PROPOSITION 5.2. Suppose that
M
is
of
finite type. Then
yMl
embeds in
yM.
The submonoid
Pj of yM
generated by
Sj
U
Sj{
U 7>
is
called the parabolic
submonoid defined by
J.
The image
of
the natural embedding described
in the
previous proposition is precisely Pj. Thus we have
THEOREM 5.3.
Parabolic
submonoids of
singular Artin
monoids
of
finite
type are
(isomorphic
to)
singular Artin
monoids.
The Artin group of
type
M is
the subgroup
of yM
generated by
5
U S~l.
The
previous result was first proved for parabolic subgroups
of
finite type Artin groups
in [7] and [10]. The set of types for which
it
holds for Artin groups was gradually
extended via various techniques
of
proof,
eventually
to
include all types
in
[11].
Paris [22] provides an alternative proof via CW-complexes.
It is
not immediately
clear how to generalise this to singular Artin monoids; although the author suspects
that Theorem 5.3 does hold for arbitrary types of singular Artin monoids.
The rest of this section is devoted to investigating when parabolics are conjugate.
Our first step is to generalise Paris' definition of
conjugators
of parabolics of Artin
groups ([22]), and to do so we must introduce some notation. Let Si and E2 be sets
of
words.
Then
Si ~
E2 means that the sets of
~
equivalence classes of elements of
Sj and S2 respectively coincide,
SiS2
denotes the set {X Y \
X €
Si,
Y e
S2}, and
if W is a word then WE^ — {W}S]. Using the terminology of
Paris,
given subsets J
and K of /, we define a (J, K)-conjugator to be a word V over S
U
5"1 U T such that
V5y
~
SK V. Thus Z(i,
j)
is the set of all ({j
},
(i})-conjugators.
Suppose that V is a (7, JO-conjugator. Then V defines abijection/v :
J ->
K by
VOJ
% <Tfv(j)
V. Well-definedness is assured by right cancellativity, and injectivity by
left cancellativity. For each
j e J,
V lies in Z(fv(j),j)', and thus
Vef)Z(f
v
(j)J).
Conversely, if /
: J
—>
K
is a bijection, and some word W lies in the intersection
of Z(f(j),j) for all
; e J,
then Wcrj
«
afU)W for all
j e J,
and so W
is a
[23] Conjugacy in singular Artin monoids 205
(7,
A")-conjugator. Since membership of Z(i,j) is calculable (Proposition 4.5), we
can determine when a word is a conjugator.
THEOREM 5.4. Suppose that J and K are subsets of I. Let F — [bijections f :
7 -*• K}. The set of all (7, K) conjugators is
\J
f€F
Ojej z(f
(J)J)-
Membership
of this set is calculable.
If 7 is a subset of /, then we denote by TMj the full subgraph of VM on the vertices
labelled by 7—that is, the graph whose vertices are 7, and for which there is an edge
between vertices
j\
andj2
in rMj precisely when there is one between j
j
and72 in TM.
PROPOSITION 5.5. Suppose that 7 and K are subsets of I. If the set of (7, K)-
conjugators is not empty, then VM] and VMK are isomorphic.
PROOF. Suppose that Vis a (7, K) -conjugator; so V defines a bijection/ : 7
—>
K,
as described already. Thus |7| = \K\. It remains to show that my = m/(0/(/) for each
1
and j in 7.
Fix / and
j,
and suppose that my < mf
(/)/(,•).
Then
VCT,
&
"/(o^.
a°d similarly
for
j,
and so
(ff/wW"* V* Vtocj)1" « Via-jOi)-* « (ff/(/)a/{0)"* V.
After cancelling V from the right, we obtain (o/(o0/(/))mi' ^
(
a
f<j)
<J
f(i))'"
l
'>
an^ so
the Embedding Theorem implies (o/doO/o))"1" ~ (°/y)cr/(o)m|'' ^y me Reduction
Lemma (which is Lemma 1.1 of this paper), (0/(,)0/o))m/")/l/> divides (o/(0oy0))mi';
and so we have my = m/0)/(/). The case when my > mfWfQ) is analogous. D
We say that two submonoids Q\ and Q2 of 5^M are conjugate if there exists a
word V over 5 U S"1 U 7 such that VQi ^ Q2 V. Suppose that J and K are subsets
of /, and that V is a (7, K>conjugator with corresponding bijection / : J -*• K.
Thus V e Z(f(j)J) f°r eacn j £ J, and so, by Lemma 4.4, Vo, « o/y)^'
VCT"1
% oy"^, V and Vr, « r/0) V. Thus if {/ is any word over Sj U 5J1 U 7>, then,
since K = / (7), Vf/ «
C/'
V, by pushing each letter of U through V one at a time,
for some U' over S* U 5^' U TK. Similarly, for any word W over
S
K
^S^UT
K
,
there
is a word W over Sj U SJ1 U 7) satisfying W V % VW. Thus V/»y « Pjf V, and we
have:
LEMMA 5.6. If the set of (7, K) conjugators is not empty, then Pj and PK are
conjugate.
In Theorem 5.8, we will show that the converse of this lemma holds, although
the converse of the preceding proposition does not. To see why, we will exploit
206 Ruth Corran [24]
results about, and connections between, Coxeter and Artin groups and singular Artin
monoids. Recall that throughout, M is a finite type Coxeter matrix over /.
The Artin group of type M, denoted SfM, is the group generated by 5U S"1 subject to
the braid relations
(piOj)
1
""
= (cr,<7,)""' and the free group relations
cr,
erf1 =ar'a, = 1.
Suppose that V and W are words over 5 U S~l U T such that V «s W. If V may be
transformed into W using only the relations from above, we will write V %s W. In
particular, if V and W are r-free (that is, are words over 5 U S"1) with V s» W, then
the only relations which can be used to transform V into W are the r-free defining
relations of yM, which are precisely those above, so V ^j W, Thus the Artin group
&M embeds in the singular Artin monoid yM.
The Artin group &M is also a homomorphic image of yM, by a map~ which we
define first on generators, and then show it can be extended homomorphically. Define
~from 5 U S~l U T to S U 5"1 to be the identity on 5 U S"1, and such that
%
- a,
for each i e /. Extend"to words over 5 U 5"' U T in the obvious way. Observe
that if Pi = p2 is any defining relation of yM, then pi = p2 is one of the relations
above. Thus ~
is
a well-defined map from « equivalence classes of words in yM to
ssj equivalence classes of words in &M; so V % W implies V %5 W.
The Coxeter group of type M, denoted by 'S'M, is a quotient of ^M obtained by
identifying each element of 5 with its inverse, that is, 'S'M is generated by S U S"1
subject to the braid relations and the relations of — 1 for each i e /. If V and W are
words over 5
U
5"' such that V may be transformed into W using only these relations,
then write V = W. For any word V over S U S'\ denote by V the word obtained
by replacing each occurrence of
CT,~'
in V with a,. Then V = V, showing that every
element of
<£M
may be expressed as a word over S. (We could have chosen 5 as a
set of monoid generators for &M-) A word over 5 is said to be reduced if it is of
shortest length amongst all words representing the same element of <$ M. We will use
the following standard results (see [6] or [20]):
(1) The set of all reduced words is precisely £}c?(S).
(2) If V and W are reduced words such that V = W, then V ~ W (that is, V may
be transformed into W using only the braid relations).
If J C /, and Wj = Pj is the subgroup of
<$
M
generated by Sj. Suppose that J and
K are subsets of / for which there is a word V over S such that V Wj = WK V. There
is always an element v of minimal length in the coset VWj, and Howlett [19] has
shown:
(3) If vWj = WKv with v a minimal length coset representative, then vSj — SKv.
The properties (1), (2) and (3) above allow us to refer to Coxeter groups to determine
conjugacy of parabolic submonoids of singular Artin monoids, due to the following:
THEOREM 5.7. Suppose that M is a finite type Coxeter matrix over I, and that J
[25] Conjugacy in singular Artin monoids 207
and K are subsets of I. Then Pj and PK are conjugate in yM if and only if Wj
and WK are conjugate in & M.
PROOF. Suppose that there is_a_word
V_
over S U 5"' U 7 such that VPj « PK V.
Then VPj_*s PKV, and so VPj = PKV. Let U = V, andjecall that Wj is
precisely Pj\ thus UWj = WK U, so Wj and WK are conjugate in <£M.
Conversely, suppose that V Wj = WK V for some word V over 5. Then there exists
a minimal length word v (a word over S) in the coset VWj, so for this element
(a) vWj = WKv,
(b) v is reduced, and
(c) v is not right divisible by any word in Wj.
Then by (3) above, vSj = 5A:V,
SO
there is a bijection / : J -> K defined by
VOJ
=af{j)v. The points (b) and (c) ensure that
vcxj
is reduced, and so by (2) above,
we have vOj ~ o/y)^, for ally e J. Thus the word v is a (J, AO-conjugator, and so
by Lemma 5.6, Pj and PK are conjugate. •
Notice, as a scholium, that if Wj and WK are conjugate, then we can find a (7, K)-
conjugator v which lies in
THEOREM 5.8. Suppose J and K are subsets of I. The following statements are
equivalent.
(1) 3&{S) contains a (J, K) conjugator.
(2) The set of(J, K) conjugators is not empty.
(3) Pj and PK are conjugate.
PROOF. Clearly (1) implies (2). Lemma 5.6 is precisely that (2) implies (3). Finally,
(3) implies Wj and
W%
are conjugate by the previous theorem, and then the comments
immediately above imply (1). •
The equivalence of (1) and (3) allows us to calculate whether Pj and PK are
conjugate. Let F denote the bijections from J to K. Recall that the set of (J, K)-
conjugators is
\J
feF
C\
j£j
Z(f(j),j), so Pj and PK are conjugate if and only if
for some/ € F, there exists a word in £2&{S) which lies in Z(f (j),j) for each
j e J. Since J, F and £1&{S) are finite, and by Proposition 4.5, membership of
each Z(f(j),j) is calculable, we have only a finite number of possibilities which we
need to check, to determine whether or not Pj and PK are conjugate.
THEOREM 5.9. Let Pj and PK be parabolic submonoids of the singular Artin
monoid of type M, where M is of finite type. It is calculable whether Pj and PK are
conjugate.
208 Ruth Con-an [26]
Theorem 5.8 provides the desired converse of Lemma 5.6. The converse of The-
orem 5.5 turns out not to hold, that is, there are occasions when the Coxeter graphs
of J and K are isomorphic, but Pj and PK are not conjugate. This fact comes directly
from the same result of Howlett [19] for parabolics in Coxeter
groups.
For example,
in type D4, there are three distinct A3 parabolics, none of which are conjugate to one
another. The idea
is,
if TMj and TMK are isomorphic subgraphs of TM, the correspond-
ing parabolics are conjugate if and only if one subgraph may be shifted along TM
one edge at a time, via isomorphic subgraphs, to eventually coincide with the second
subgraph. Since A 3 -type subgraphs of VDl cannot be shifted in any way to produce
an
A3
subgraph, the three distinct subgraphs correspond to non-conjugate parabolics.
Howlett [19], and Brink and Howlett [8] give complete description of how parabolic
subgroups of Coxeter groups with isomorphic graphs can fail to be conjugate, and by
Theorem 5.7, their list for finite types completely describes the situation for singular
Artin monoids of finite types also.
6. Markov moves and the singular braid monoid
Singular braids may be defined geometrically in a similar way to the geometric
braids of Artin [3]. The important amendment to the definition is that the strings of a
singular braid may intersect—although at most two strings may intersect at any one
point, and there may be at most a finite number of such intersections. The notion
of equivalence of singular braids is rigid vertex isotopy (see [5, Section 5]). The
set of (equivalence classes of) singular braids on n strings forms a monoid under
concatenation, often denoted SBn. Baez [4] first presented this monoid, and Birman
first showed that the relations suffice ([5, Lemma 3]). Their presentation is precisely
that of the singular Artin monoid of type An_\ defined above. See Figure 1 for the
Coxeter graph of this type. For this section, we will use the notation:
Sk
=
(CTI,
CT
2
,
• •
•. °k}, S^1 =
{a~\a~x
a"1}- Tk = {r,,
r
2
,...,
rk}.
Thus the presentation
of SBn
([4, 5]) has generators
Sn_i
U
S~\ U Tn_u
and relations:
1
=
cr.cr"1
= o~xo. I
\ for
all
i,
with
1 < i <n- 1,
CT,T,
=
T/CT,
j
idi = O.Oi I
' ' \ whenever
11
-j | > 2,
OiTj = TjOj J
<7,<7;
<T,
=
(Xj
CFjCFj
I
> whenever
|i
—
j | = 1 .
[27] Conjugacy in singular Artin monoids
209
As in Artin's well known presentation of the braid group, the generator
CT,
represents
the braid with just a single crossing of the ith string over the (i
+
l)th string, and a~x
represents the braid with just the
(i +
l)th string crossing over the ith string.
The
generator
r,
represents the braid with no crossings but
a
single intersection between
the ith string and the
(i +
l)th string. The permutation associated to each
of
these
singular braids is the transposition
(i i +
1).
The singular braid monoid
SBn
arose in the context of knot theory and Vassiliev
invariants of
knots.
A singular braid
fi
may be closed by associating the corresponding
endpoints, to produce
a
singular link
fi.
Birman showed that every singular link
is
equivalent
to a
closed singular braid
fi on n
strings,
for
some
n
([5, Lemma 2]).
(See also [2] for
a
generalisation
of
this result.) The question becomes, when
do
inequivalent singular braids close to produce equivalent singular links? Gemein [14]
answered this with his 'singular version' of Markov's theorem, stated below. He uses
the notation (fi, n) to indicate that
fi
is
a
singular braid on n strings.
Proposition 5.2 implies that there is an injective homomorphism from the singular
Artin monoid
of
type An_i
to
the singular Artin monoid
of
type
An,
type
A
being
in the list
of
finite types. This
is a
map from SBn
to
SBn+u and may be described
geometrically as follows: given
a
singular braid (fi, n), append an extra string in the
(n
+
l)th place which has no interaction with the other strings of the braid. The new
braid is denoted (fi,
n +
1). A word over 5n_i U
S~\
U Tn^x representing (fi, n) will
also serve to represent (fi, n + 1). The additional fact that this map is injective means
that
if
fi
and
y
are singular braids on
n
strings such that (fi, n + 1)
=
(y, n
+
1), then
(fi, n) = (y, n).
Given
a
singular braid (fi, n), the notation (fion,
n +
1) now makes sense—first
apply the map 'adding
a
string' to obtain ($,
n +
1), and then concatenate this new
braid with an. We now state Gemein's result [14].
MARKOV'S THEOREM (SINGULAR VERSION). Let (/J, n) and (/*', «') be two braids.
Let
K
and
K'
be the two associated closed braids.
K
and
K'
are equivalent as links
if and only
if
(ft, n) and (/*', n') are related by a sequence of the following algebraic
operations, called Markov's moves:
(1)
(a) djfp<j?\n)*~*{fi,n);
(b)
(T^,/i)«w(^Tl-,n);
(2) (0,n)«~* <fi<j±\n+l).
Both parts of (1) may be combined into the one statement:
(x0, n) «~» (fix, n) for*
e
S«_, U
S;\
U Tn_x.
Suppose that
V
and W are words over Sn_i U
S~\
U Tn_\ representing
xfi
and
fix
respectively. Then
V is
swap conjugate
to W
(defined
in
Section 3). Applying
Theorem 3.5, we immediately obtain the following
210 RuthCorran [28]
PROPOSITION 6.1. Let ft and y be singular braids on n strings, represented by the
words V and W over 5n_i U 5~J, U Tn_\ respectively. Then P and y are related by an
arbitrary sequence of singular Markov moves of type (1) if and only ifVx W.
Turning now to the second type of Markov move, we make use of the normal form
for the singular Artin monoid given in [9, Section 5]. An operator Nn on words over
Sn U S~l U Tn is defined, which has the properties that for all words V and W over
Sn U 5n"' U Tn,
(1) Wn(
WO
is calculable,
(2) 7Jn(W)^ W,and _ _
(3) W « V if and only if ~Nn(W) =
~Nn(
V) ([9, Theorem 23]).
Thus we get immediately from (2) and (3) the following:
PROPOSITION 6.2. Let (/}, n) and (y, n + 1) be singular braids represented by the
words V over 5n_, U S~\ U 7n_! and W over Sn U S~] U Tn respectively. Then fi and
y are related by a type (2) Markov move if and only if either Nn(W) = N
„(
Van) or
Combining this result with Proposition 6.1 and (1), we have
THEOREM 6.3. It is calculable when two singular braids differ by an application of
a Markov move.
We can consider positive braids—those which may be represented by positive
words, that is, words with no inverses. The positive Markov moves are defined to
be:
(1) (x0, n) *~» (0x, n) for x e 5n_i U Tn_u and
(2) <fi,ri)«~*(fian,n + \).
Given any positive braid P on n strings, and any positive integer Af, we will calculate
a set which contains all words representing positive braids which can be obtained
from P by up to N applications of positive Markov moves. Define the set
Q = {(W, n + 1) | n is a non-negative integer, and W is a word over Snl) Tn}.
Elements of
£2
are said to be allowable.
Let K be a set of allowables. For each n, define Kn = {W | (W, n) e K). The set
<p(Kn)
is the set of all words over 5n_i U 7n_i which are K-conjugate to elements of Kn.
By Lemma 3.3,
<p(Kn)
contains all words which are swap-conjugate to elements of Kn.
Thus (p(Kn) contains all words corresponding to the application of a positive Markov
move of type (1) to words in Kn.
[29] Conjugacy in singular Artin monoids 211
For any set K of allowables, we may define
K+ = {(Wan, n + l)\(W,n)e K], and
K_ = {(W, n) | (Wan, n + 1) 6 K and (W, n) is allowable).
Then K+
U
K_ contains all words corresponding to the application of a positive Markov
move of type (2) to words in K. We now define K' = K U K+ U
AT_,
and let
lxK = {(W,n)\ W€<p(K'n)}.
Then ixK contains only allowables, and is finite if K finite. Moreover, since K' is
calculable,
<p(K')
is calculable (Theorem 2.4) and so /xK is calculable.
THEOREM 6.4. Let (/3, n) and (y, n') be positive singular braids represented by
allowables (V, n) and (W, n') respectively. If (ft, n) is related to (y, «') by N pos-
itive Markov moves, then (V, n) lies in
fiN[(W,
n')}. Conversely, if (V, n) lies in
(i
N
[(W,
n')} for some positive integer N, then (/J, n) and (y, n') are related by a
sequence of positive Markov moves. Membership
of/j.N{(W,
n')} is calculable.
Acknowledgements
The author thanks David Easdown, Bob Howlett and the referee for many helpful
suggestions and discussions. In particular, the author would like to recognise the
referee's improvements to the article in strengthening Lemma 1.7, and in the proofs of
Lemma 1.7 and Theorem 1.5. The majority of the work in this article was undertaken
while the author was a research student in the School of Mathematics and Statistics at
the University of Sydney, supported by the Australian Postgraduate Award.
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Institut de Geometrie, Algebre et Topologie
Batiment BCH
Ecole Polytechnique Federate de Lausanne
CH-1015
Switzerland
e-mail:
ruth.corran@epfl.ch
