On the Markov trace for Temperley--Lieb algebras of type $E_n$
arXiv:0704.0283v1  [math.QA]  2 Apr 2007
ON THE MARKOV TRACE FOR
TEMPERLEY–LIEB ALGEBRAS OF TYPE En
R.M. Green
Department of Mathematics
University of Colorado
Campus Box 395
Boulder, CO 80309-0395
E-mail: rmg@euclid.colorado.edu
Abstract. We show that there is a unique Markov trace on the tower of Temperley–
Lieb type quotients of Hecke algebras of Coxeter type En (for all n ≥ 6). We
explain in detail how this trace may be computed easily using tom Dieck’s calculus
of diagrams. As applications, we show how to use the trace to show that the diagram
representation is faithful, and to compute leading coefficients of certain Kazhdan–
Lusztig polynomials.
1. Introduction
In the paper [17], Jones introduced a certain Markov trace on the tower of
Hecke algebras H(An−1) associated to the Coxeter groups Sn = W (An−1), which
are the symmetric groups. When Jones’ trace is restricted to one of the algebras
H = H(An−1), it is degenerate, but its radical is an ideal, J , of H and so we obtain
a generically nondegenerate trace on the algebra H/J , which is the Temperley–Lieb
algebra TLn occurring in statistical mechanics [25] (the trace is the matrix trace
of a transfer matrix algebra).
In [19], Kazhdan and Lusztig introduced a remarkable polynomial Px,w(q) for
any elements x, w in a Coxeter group W . These polynomials have important ap-
plications in representation theory. Although the polynomials have an elementary
1991 Mathematics Subject Classification. 20C08, 20F55, 57M15.
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http://arxiv.org/abs/0704.0283v1
2 R.M. GREEN
definition, the only obvious way to compute them is using a rather complicated
recurrence relation. One of the main obstructions to computing the polynomials
efficiently is a fast way to compute the integer µ(x, w), which is the coefficient of
q(ℓ(w)−ℓ(x)−1)/2 in Px,w(q). In [12], the author showed how Jones’ trace can be used
to compute the leading coefficients µ(x, w) ∈ Z in the case where x and w are fully
commutative elements of W (in the sense of [24]). In this paper, we will investigate
the analogous phenomenon in Coxeter type En. This includes Coxeter groups of
types A and D as special cases.
The algebras TLn may be defined in terms of generators and relations in a
way that generalizes readily to Coxeter systems of other types. These generalized
Temperley–Lieb algebras have been studied for Coxeter type En by a number of
people [2, 3, 7]. Although the Coxeter groups of type En are infinite for n > 8, the
Hecke algebra quotient TL(En) in this case is still finite dimensional. In [2], tom
Dieck constructed a diagrammatic representation of TL(En), although the question
of whether this is a realisation—a faithful representation—is not tackled. In §9, we
will prove
Theorem 1.1. The diagrammatic representation of TL(En) given in [2] is injec-
tive.
The closing remarks of [2] state without proof that this representation can be
used to define a Markov trace on the tower of algebras TL(En). In Theorem 8.11,
we will prove this claim and furthermore we will show that there is a unique such
Markov trace. Although this is similar to what happens in type A, the analogous
claim for Coxeter type D is false.
This trace is also remarkable for other reasons: after suitable rescaling, it is a
tabular trace in the sense of [10], and a generalized Jones trace in the sense of [12].
The fact that the trace is tabular implies that it is (generically) nondegenerate on
the algebras TL(En). The fact that we have a generalized Jones trace will lead
to the following theorem (proved in §9) where the monomial basis elements bw are
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 3
defined in §3.
Theorem 1.2. Let {bw : w ∈ Wc} be the monomial basis of TL(En) indexed by the
fully commutative Coxeter group elements, and let tr be the unique Markov trace on
the tower of algebras TL(En). If x, y ∈ Wc, then the coefficient of v−1 in tr(bxby−1)
(after expansion as a power series) is µ̃(x, y), where
µ̃(x, y) =
µ(x, y) if x ≤ y,
µ(y, x) if x 6≤ y,
and µ(a, b) is the integer defined in [19].
We will also show in §9 how µ̃(x, y) may be evaluated non-recursively using the
diagram calculus.
2. Traces and Markov traces
By a trace on an R-algebra A, we mean an R-linear map t : A −→ R such that
t(ab) = t(ba) for all a, b ∈ A. The radical of the trace is the set of all a ∈ A such
that t(ab) = 0 for all b ∈ A. The radical is always an ideal of A, and if it is trivial,
the trace is said to be nondegenerate. In any case, if I is the radical of t, then t
induces a nondegenerate trace on the quotient algebra R/I.
The set of traces on an R-algebra A has a natural R-module structure. In the
special case where ρ is a representation of an R-algebra A, then the matrix trace
associated to ρ is a trace in the above sense, which means that, if A is semisimple,
the Grothendieck group of A gives a Z-lattice in the space of traces, generated by
the traces of the simple modules.
We will be particularly concerned with algebras where the base ring R is obtained
by extending scalars from the ring of Laurent polynomials A = Z[v, v−1] to some
ring F ⊗ A. This has the effect of specializing the parameter v to an invertible
element of F . In this situation, a trace is called generically nondegenerate if it is
nondegenerate as a trace over A, and if it also remains nondegenerate as a trace
over F ⊗A for all but finitely many specializations of v.
4 R.M. GREEN
Suppose now that R is an integral domain and {An : n ≥ N} is a family of unital
R-algebras such that An is a subalgebra of An+1 for all n ≥ N . Let A∞ be the
associated direct limit. Suppose also that there is a set of elements {gn : n ∈ N}
such that gn+1 ∈ An+1\An for all n and such that {gn : n ≤ M} is an algebra
generating set for AM . Following [5, §4], we may now introduce the notion of
Markov trace.
Definition 2.1. Maintain the above notation, and let F be a field containing R.
A Markov trace on A∞ with parameter z ∈ F is an F -linear map τ : A∞ −→ F
satisfying the following conditions:
(i) τ(1) = 1;
(ii) τ(hbn+1) = zτ(h) for n ≥ N and h ∈ An;
(iii) τ(hh′) = τ(h′h) for all h, h′ ∈ A∞.
Jones [17] proved that there is a unique Markov trace with parameter z on the
tower of Hecke algebras of type An, and that the only one of these traces that
passes to the Temperley–Lieb quotient is the one with parameter z = (v + v−1)−1.
This is an important observation in the construction of the Jones polynomial, be-
cause conditions (ii) and (iii) for the trace are what is needed to ensure that the
polynomial is invariant under the two types of “Markov move”.
Some other notable work on Markov traces includes that of Geck and Lam-
bropoulou [4], who classified the Markov traces in Coxeter types B and D, using a
suitable extension of the above definition. Lambropoulou [20] extended this work
(in type B) to generalized and cyclotomic Hecke algebras of type B.
For the purposes of studying Temperley–Lieb type quotients of Hecke algebras,
a better definition of Markov traces seems to be one that appears in work of Seifert
[22] and recent work of Gomi [6, Definition 3.7]. In this case, one retains conditions
(i) and (iii) of Definition 2.1 and replaces condition (ii) by the requirement that
τ(aTs) = zsT (a)
whenever we have a ∈ H(WI) for some parabolic subgroup WI corresponding to
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 5
I ⊆ S\{s}. (In other words, we require condition (ii) to hold for all generators of
An+1, not just one particular generator.) Here, zs is an indeterminate depending
on the conjugacy class of s in W .
In this paper, we will restrict our attention to the tower of algebras TL(En),
and in this case, the above definitions happen to agree; however, they do not agree
in the corresponding question for type Dn. In the latter case, it can be shown that
the Seifert–Gomi formulation produces a unique Markov trace, and Definition 2.1
does not.
3. The algebras TL(En)
Let X = X(En) be a Coxeter graph of type En, where n ≥ 6. Following [3], we
label the vertices of X by 0, 1, . . . , n− 1 in such a way that 1, 2, 3, . . . , n− 1 lie in
a straight line, and such that 3 is the unique vertex of degree 3, which is adjacent
to 2, 4 and 0. Figure 1 shows the case n = 6.
Figure 1. Coxeter graph of type E6
 4  5 3 2 1
Let W (En) be the associated Coxeter group with distinguished set of generating
involutions
S(En) = {si : i is a vertex of X(En)}.
In other words, W = W (En) is given by the presentation
W = 〈S(En) | (st)m(s,t) = 1 for m(s, t) < ∞〉,
where m(s, s) = 1, m(s, t) = 2 if s and t are not adjacent in X , and m(s, t) = 3
if s and t are adjacent in X . The elements of S = S(En) are distinct as group
elements, and m(s, t) is the order of st. Denote by Hq = Hq(En) the Hecke algebra
6 R.M. GREEN
associated to W . This is a Z[q, q−1]-algebra with a basis consisting of (invertible)
elements Tw, with w ranging over W , satisfying
TsTw =
Tsw if ℓ(sw) > ℓ(w),
qTsw + (q − 1)Tw if ℓ(sw) < ℓ(w),
where ℓ is the length function on the Coxeter group W , w ∈ W , and s ∈ S. If
n > 8, the group W is infinite and Hq has infinite rank as an A-algebra.
For the applications we have in mind, it is convenient to extend the scalars of
Hq to produce an A-algebra H, where A = Z[v, v−1] and v2 = q, and to define a
scaled version of the T -basis, {T̃w : w ∈ W}, where T̃w := v−ℓ(w)Tw. We will write
A+ and A− for Z[v] and Z[v−1], respectively.
A product w1w2 · · ·wn of elements wi ∈ W is called reduced if
ℓ(w1w2 · · ·wn) =
i ℓ(wi). We reserve the terminology reduced expression for
reduced products w1w2 · · ·wn in which every wi ∈ S. We write
L(w) = {s ∈ S : ℓ(sw) < ℓ(w)}
R(w) = {s ∈ S : ℓ(ws) < ℓ(w)}.
The set L(w) (respectively, R(w)) is called the left (respectively, right) descent set
of w.
Call an element w ∈ W complex if it can be written as a reduced product
x1wss′x2, where x1, x2 ∈ W and wss′ is the longest element of some rank 2 parabolic
subgroup 〈s, s′〉 such that s and s′ correspond to adjacent vertices in the Coxeter
graph En. Denote by Wc(En) the set of all elements of W that are not complex.
The elements of Wc = Wc(En) are the fully commutative elements of [24]; they
are characterized by the property that any two of their reduced expressions may be
obtained from each other by repeated commutation of adjacent generators.
Let J(En) be the two-sided ideal of H generated by the elements
T1 + Ts + Tt + Tst + Tts + Tsts,
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 7
where (s, t) runs over all pairs of elements of S for which m(s, t) = 3. Follow-
ing Graham [7, Definition 6.1], we define the generalized Temperley–Lieb algebra
TL(En) to be the quotient A-algebra H(En)/J(En). We denote the corresponding
epimorphism of algebras by θ : H(En) −→ TL(En). Let tw (respectively, t̃w) de-
note the image in TL(En) of the basis element Tw (respectively, T̃w) of H. If s ∈ S,
we define bs ∈ TL(En) by bs = v−11 + t̃s.
A more convenient description of TL(En) for the purposes of this paper is by
generators and relations (as in [3, §2.2]). Since the Laurent polynomial v + v−1
occurs frequently, we denote it by δ.
Proposition 3.1. As a unital A-algebra, TL(En) is given by generators {bs : s ∈
S} and relations
b2s = δbs,
bsbt = btbs if m(s, t) = 2,
bsbtbs = bs if m(s, t) = 3.
The following basis theorem will be used freely in the sequel.
Theorem 3.2 [3, 7].
(i) The set {t̃w : w ∈ Wc} is a free A-basis for TL(En).
(ii) If w ∈ Wc and w = si1si2 · · · sir is reduced, then the element
bw = bsi
· · · bsir
is a well-defined element of TL(En).
(iii) The set {bw : w ∈ Wc} is a free A-basis for TL(En).
Proof. Part (i) is due to Graham [7, Theorem 6.2]. Parts (ii) and (iii) are stated
by Fan in [3, §2.2], and more details may be found in [13, Proposition 2.4]. �
Definition 3.3 [3, §2.3]. Let P = P (n) denote the set of subsets of the Coxeter
graph En that consist of non-adjacent vertices. We allow P to include the empty set,
8 R.M. GREEN
∅. For any A ∈ P , let i(A) be the product of the elements of S(En) corresponding
to the vertices in A (with i(∅) = 1); note that the order of the product is immaterial
since the vertices in A correspond to commuting generators. Let A,B ∈ P . We say
that A and B are neighbours if and only if 1 + #(A ∩ B) = #A = #B, and the
two vertices in (A∪B)\(A∩B) are adjacent in En. Define an equivalence relation
on P by taking the reflexive and transitive closure of the relation A ∼ B if A and
B are neighbours. Let P̄ denote the set P/ ∼ .
Example 3.4. In type E7, let A = {0, 2, 4, 6} and B = {0, 1, 4, 6}. In this case,
i(A) = b0b2b4b6 and i(B) = b0b1b4b6, A and B are neighbours, and the equivalence
class of A is precisely {A,B}.
Definition 3.5 [3, §6.3]. Let n ≥ 6.
If n is odd, we define P ′ = P ′(n) to be the subset of P (n) consisting of the sets
{(n− 1)− 2j : 0 ≤ j ≤ N} : 0 ≤ N ≤ n− 1
together with the set
{n− 1, n− 3, n− 5, . . . , 4} ∪ {0}
and the empty set.
If n is even, we define P ′ = P ′(n) be the subset of P (n) consisting of the sets
{(n− 1)− 2j : 0 ≤ j ≤ N} : 0 ≤ N ≤ n− 2
together with the empty set.
Example 3.6. In type E6, we have
P ′ = {{5}, {5, 3}, {5, 3, 1}, ∅} .
In type E7, we have
P ′ = {{6}, {6, 4}, {6, 4, 2}, {6, 4, 2, 0}, {6, 4, 0}, ∅} .
The importance of the set P ′ comes from the following
Proposition 3.7 (Fan, [3, Lemma 8.1.2]). The set P ′ constitutes a complete set
of equivalence class representatives for P with respect to ∼. �
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 9
4. Cells and the a-function
In §4, we recall the definitions of the a-function and cells arising from the mono-
mial basis. Most of this material comes from the papers [3] and [10], or is implicit
in them.
Definition 4.1 [3, Definition 2.3.1]. The a-function a : Wc −→ Z≥0 is defined by
a(w) := max
{#A : w = xi(A)y is reduced}
for w ∈ Wc.
Proposition 4.2. Let w ∈ Wc and let f ∈ A. Define the degree, deg f , of f to
be the largest integer n such that vn occurs with nonzero coefficient in f , with the
convention that deg 0 = −∞. Denote the structure constants with respect to the
monomial basis by gx,y,z ∈ A, namely
bxby =
gx,y,zbz.
(i) The structure constant gx,y,z is either zero or a nonnegative power of δ, and,
given x and y, we have gx,y,z 6= 0 for a unique z.
(ii) If s ∈ S and gs,y,z 6∈ Z, then gs,y,z = δ, ℓ(sy) < ℓ(y) and y = z. Similarly, if
gx,s,z 6∈ Z, then gx,s,z = δ, ℓ(xs) < ℓ(x) and x = z.
(iii) We have a(w) = maxx,y∈Wc deg gx,y,w.
(iv) We have a(w) = maxx,y∈Wc deg gw,x,y.
Proof. Parts (i) and (ii) are well known and follow easily from [3, Proposition 5.4.1].
Part (iii) is proved in [10, Proposition 4.2.3] using the results of [3].
The proof of [3, Theorem 5.5.1] shows that
deg gw,x,y ≤ min(a(w), a(x)),
which means that
x,y∈Wc
deg gw,x,y ≤ a(w).
10 R.M. GREEN
Conversely, [3, Lemma 5.2.6] shows that
bwbw−1 = (v + v
−1)a(w)bd
for some d ∈ Wc, so taking x = w−1 and y = d, we find that
x,y∈Wc
deg gw,x,y ≥ a(w),
which completes the proof of (iv). �
Definition 4.3 [3, Definition 4.1].
For any w,w′ ∈ Wc, we say that w′ ≤L w if there exists bx such that gx,w,w′ 6= 0,
where g is as in Proposition 4.2.
For any w,w′ ∈ Wc, we say that w′ ≤R w if there exists bx such that gw,x,w′ 6= 0.
For any w,w′ ∈ Wc, we say that w′ ≤LR w if there exist bx and by such that
bxbwby = cbw′ for some c 6= 0.
We write w ∼L w′ to mean that both w′ ≤L w and w ≤L w′. Similarly, we
define w ∼R w′ and w ∼LR w′.
The relation ∼L (respectively, ∼R, ∼LR) is an equivalence relation, and the
corresponding equivalence classes of Wc are called the left (respectively, right, two-
sided) cells.
It is clear from the definitions and the fact that the identity element is a monomial
basis element that two-sided cells are unions of left cells, and also unions of right
cells.
Proposition 4.4.
(i) Let w ∈ Wc. If we have w = xi(A)y reduced for some A such that #A = a(w),
then i(A) ∼LR w and w ∼R xi(A).
(ii) The a-function is constant on left, right, and two-sided cells.
(iii) If w,w′ ∈ Wc are such that w′ ≤R w and w′ 6∼R w, then a(w′) > a(w). An
analogous statement holds for left cells and two-sided cells.
(iv) The right cell containing i(A) is precisely the set
{w ∈ Wc : w = i(A)x reduced, a(w) = #A}.
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 11
(v) A left cell and a right cell contained in the same two-sided cell intersect in a
unique element.
Proof. Statement (i) is proved during the argument establishing [3, Theorem 4.5.1.].
The fact that the a-function is constant on two-sided cells is implicit in the proof
of [3, Theorem 4.5.1]. Since two-sided cells are unions of left (or right) cells, part
(ii) follows.
Suppose now that w,w′ ∈ Wc are such that w′ ≤R w and w′ 6∼R w. An inductive
argument using the definition of ≤R reduces the problem to the case where there
is some s ∈ S such that bwbs is a multiple of bw′ , so let us assume that this is
the situation. By [3, Corollary 4.2.2], the assumption that w′ ≤R w implies that
a(w′) ≥ a(w). The statement follows unless a(w′) = a(w), so suppose we are in
this case.
Let us write w = xi(A)y as in statement (i). Now [3, Lemma 4.2.5], applied to
the element xi(A) and the sequence of generators corresponding to ys, shows that
we have w′ = xi(A)y′ reduced. By part (i), we find that w′ ∼R xi(A), and thus
that w′ ∼R w, a contradiction.
The statement for left cells follows by a symmetrical argument, and the statement
for two-sided cells follows from the previous claims and the fact that if w′ ≤LR w,
then there is a chain
w′ = w1, w2, w3, . . . , wk = w
where, for each 1 ≤ i < k, we have either wi ≤L wi+1 or wi ≤R wi+1. This
completes the proof of (iii).
Part (iv) is [3, Proposition 4.4.3].
Part (v) is well known and follows from the proof of [3, Theorem 6.1.2]. �
Remark 4.5. For finite and affine Weyl groups, the a-function defined above is
known by [23, Theorem 3.1] to be the restriction of Lusztig’s more general a-
function [21] restricted to the subset Wc.
Although it is not true that each of the monomial cells studied above is a cell
12 R.M. GREEN
in the sense of Kazhdan–Lusztig [19], it can be shown fairly easily that each left
(respectively, right, two-sided) monomial cell is a subset of some left (respectively,
right, two-sided) Kazhdan–Lusztig cell.
5. Traces on the algebras TL(En)
In §5, we will extend scalars and deal with a K-form of TL(En), where K is
a field containing A and a square root of δ. (The existence of
δ is needed for
compatibility with [3], but can ultimately be removed; see Remark 6.4.) We write
TLK(En) := K ⊗A TL(En). We aim to classify the traces, τ : TLK(En) −→ K,
that is, linear functions τ with the property that τ(ab) = τ(ba) for all a, b ∈
TLK(En). It is clear that the set of all traces on TLK(En) is a K-vector space
(dependent in principle on K and δ). The main result of §5 is that there is a basis
for this vector space in natural bijection with the set P ′ of §3.
The next result shows how τ naturally induces a function P/ ∼ −→ K.
Lemma 5.1. Maintain the notation of Definition 3.3. Suppose A,B ∈ P are such
that A ∼ B, and let τ : TLK(En) −→ K be a trace. Then τ(i(A)) = τ(i(B)).
Proof. The proof immediately reduces to the case where A and B are neighbours.
Let s (respectively, t) be the element of S corresponding to the unique element
of A\B (respectively, B\A). It is immediate from the definitions that i(A) =
bsi(A ∩B) = i(A ∩B)bs and i(B) = bti(A ∩B) = i(A ∩B)bt. We then have
τ(i(A)) = τ(bsi(A ∩B)) = τ(bsbtbsi(A ∩B))
= τ(btbsi(A ∩B)bs) = τ(btbsbsi(A ∩B))
= δτ(btbsi(A ∩B))
= τ(btbtbsi(A ∩B)) = τ(btbsi(A ∩B)bt)
= τ(btbsbti(A ∩B)) = τ(bti(A ∩B))
= τ(i(B)),
as required. �
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 13
Lemma 5.2. Any trace τ : TLK(En) −→ K is determined by its values on the set
{i(A) : A ∈ P}.
Proof. Suppose the values of τ(i(A)) are known for each A ∈ P . We will show how
to compute the value of τ(bw), where w ∈ Wc is arbitrary.
Let us write w = xi(A)y reduced as in Proposition 4.4 (i). Using a reverse
induction, we will assume that the values of τ(bw′) for a(w
′) > a(w) = #A, if
such w′ exist, have been determined. By the defining relations of TL(En), we have
bi(A)bi(A) = δ
#Abi(A), and so we have
τ(bw) = τ(bxbi(A)by)
= δ−#Aτ(bxbi(A)bi(A)by)
= δ−#Aτ(bi(A)bybxbi(A)).
Now i(A)y and xi(A) lie in Wc because w does, and Proposition 4.4 (i) and (ii)
shows that a(w) = a(xi(A)). By Proposition 4.2 (i), we have
bi(A)bybxbi(A) = bi(A)ybxi(A) = δ
for some z ∈ Wc, and it is clear from the definitions that z ≤L xi(A). By Proposi-
tion 4.4 (ii) and (iii), we see that
a(z) ≥ a(xi(A)) = a(w) = #A.
If a(z) > #A then our inductive hypothesis determines the value of τ(δcbz),
which in turn determines the value of τ(bw). We may therefore assume that a(z) =
#A. To complete the proof, it is enough to show that z = i(A), because the value
of τ(bz) will then have been determined by our assumptions.
Let s ∈ A. Since bsbi(A) = δbi(A) by the defining relations, the definition of bz
shows that bsbz = δbz. By Proposition 4.2 (ii), this means that ℓ(sz) < ℓ(z), and
it follows that A ⊆ L(z). Because A is a set of commuting generators, standard
14 R.M. GREEN
properties of Coxeter groups show that we can write z = i(A)z′ reduced. Applying
Proposition 4.4 (iv) to the fact that a(z) = #A shows that z ∼R i(A). A symmet-
rical argument then shows that we have z ∼L i(A). By Proposition 4.4 (v), this
can only happen if z = i(A). �
Theorem 5.3. For each Ā ∈ P̄ (as in Definition 3.3), there is a unique trace
τĀ : TLK(En) −→ K such that for each B ∈ P we have
τĀ(i(B)) =
1 if B ∈ Ā,
0 otherwise.
The set
{τĀ : Ā ∈ P̄}
is a K-basis for the set of all traces τ : TLK(En) −→ K.
Proof. It is clear from the definition of trace that the traces from TLK(En) to K
form a K-vector space. Lemmas 5.1 and 5.2 show that this space has dimension at
most the size of P̄ .
Fan [3, Theorem 5.6.1] shows that TLK(En) is semisimple and that is then a
direct sum of |P̄ | matrix rings. This proves that the dimension of the space of traces
is at least the size of P̄ , and thus that the space has the claimed dimension.
A dimension count, together with another application of lemmas 5.1 and 5.2,
then shows that there are unique traces τĀ with the properties claimed, and that
they form a basis. �
We now come to the central definition of the paper.
Definition 5.4. The trace tr : TLK(En) −→ K is defined by
Ā∈P̄
δ−#AτĀ,
where τĀ is as in Theorem 5.3.
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 15
Corollary 5.5. Any trace τ : TLK(En) −→ K satisfies τ(bw) = τ(bw−1) for all
w ∈ Wc.
Proof. It follows from Proposition 3.1 that there is a unique A-linear antiautomor-
phism ∗ : TL(En) −→ TL(En) fixing the generators bs. We may extend this to a
K-linear antiautomorphism ∗ : TLK(En) −→ TLK(En). If a ∈ TLK(En), let us
write a∗ for ∗(a). Note that if A ∈ P , then i(A) is invariant under ∗, because i(A)
is a product of commuting generators bs.
Given a trace τ : TLK(En) −→ K, the K-linear map τ ′ : TLK(En) −→ K
defined by τ ′(a) = τ(a∗) is also a trace. Since τ and τ ′ agree on all elements i(A)
for A ∈ P , Lemma 5.2 shows that τ = τ ′, and the assertion follows. �
Remark 5.6. The trace tr will turn out to induce the Markov trace of the title.
Note that the definition makes sense because Ā, B̄ ∈ P̄ implies #A = #B.
Traces on Hecke algebras of finite Coxeter groups are known have a property
similar to that given in Corollary 5.5; see [5, Corollary 8.2.6] for more details.
6. Cellular structure and the a-funtion
In §6, we explain how the trace tr is particularly compatible with the structure
of TL(En) as a cellular algebra, in the sense of [8]. We will not recall the complete
definition of a cellular algebra here, but we summarize below the properties of the
cellular structure that are important for our purposes.
Definition 6.1. Let Λ be the set of two-sided cells for TL(En), equipped with the
partial order induced by ≤LR. For each λ ∈ Λ, let M(λ) be an indexing set for
the left cells contained in λ; note that the inversion map on the Coxeter group W
induces a bijection between the set of left cells in λ and the set of right cells in λ
(see the remarks at the end of [3, §4.4]).
Proposition 6.2. Maintain the above notation.
(i) Let T, U ∈ M(λ) for some fixed λ ∈ Λ. Then T ∩ U contains a unique element,
w, and we define CT,U = bw.
16 R.M. GREEN
(ii) The A-algebra anti-automorphism ∗ : TL(En) −→ TL(En) defined by ∗(bw) =
bw−1 satisfies ∗(CT,U ) = CU,T . In particular, we have w2 = 1 if and only if
bw = CT,T for some T .
(iii) Suppose that CP,Q and CR,S are arbitrary monomial basis elements, and define
CT,U by the condition
CP,QCR,S = δ
aCT,U
(which makes sense by Proposition 4.2 (i)). If P,Q,R, S, T and U all belong
to the same two-sided cell, then P = T and S = U ; if, furthermore, we have
Q = R, then a = a(CT,U ). If it is not the case that P = T , S = U and Q = R,
then we have a < a(CT,U ).
Proof. Parts (i) and (ii), which are originally due to Graham [7], are proved in [10,
Proposition 4.2.1]. Part (iii) is proved in [10, propositions 4.2.1 and 4.2.3] using
the results of [3]. �
Proposition 6.3. For all w ∈ Wc, we have tr(bw) = δa, where a = −a(w) if
w2 = 1, and a < −a(w) otherwise.
Proof. Let λ be the two-sided cell containing w. We will prove the statement by
induction on the partial order on two-sided cells given in Definition 6.1. Writing
w = CT,U for T, U ∈ M(λ), as in Proposition 6.2 (i), and applying Proposition 6.2
(ii), we see that the condition w2 = 1 is equivalent to T = U .
By Proposition 4.4, there exists a product of a(w) commuting generators, i(A), in
λ. Define V ∈ M(λ) by the condition CV,V = bi(A). Since tr is a trace, Proposition
6.2 (iii) shows that
tr(CT,U ) = δ
−a(w)tr(CT,V CV,U ) = δ
−a(w)tr(CV,UCT,V ).
By Proposition 4.2 (i), we have
CV,UCT,V = δ
bCX,Y
for some b ≥ 0 and some basis element CX,Y . There are now two cases to consider.
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 17
The first possibility is that CX,Y comes from the two-sided cell λ. (If T = U ,
this case must occur by Proposition 6.2 (iii).) In this case, we have X = Y = V ,
and thus CX,Y = bi(A). Proposition 6.2 (iii) then shows that b = a(w) if T = U ,
and b < a(w) otherwise. Since we have tr(bi(A)) = δ
−a(w) by definition of tr, we
have tr(CT,U ) = δ
−a(w)+b−a(w), and the result follows.
The other possibility is that CX,Y comes from a two-sided cell λ
′ with λ′ < λ,
and T 6= U . In this case, Proposition 4.4 (iii) shows that a(CX,Y ) > a(w). By
the inductive hypothesis, we know that tr(CX,Y ) = δ
a′ , where a′ ≤ −a(CX,Y ) <
−a(w). This means that tr(CT,U ) = δ−a(w)+b+a
. By propositions 4.2 (iii) and 4.4
(ii), we have b ≤ a(w), and thus tr(CT,U ) = δa for a < −a(w), as required. �
Remark 6.4. The above proposition shows that we do not actually need
δ ∈ k to
define tr. From now on, we need only assume that K is a field containing A.
Proposition 6.5. If K is the field of fractions of the power series ring Z[[v−1]],
then tr is a nondegenerate trace on TLK(En), and
tr(CP,QCR,S)− δQRδPS ∈ v−1Q[[v−1]],
where δQR and δPS are the Kronecker delta.
Proof. An element x of K is uniquely representable in the form
where λi ∈ Q for all i. If x 6= 0, we define deg x to be the largest integer j such
that λj 6= 0. If x, y 6= 0 then deg(xy) = deg x + deg y, so the facts that deg δ = 1
and deg 1 = 0 imply that deg δa = −a.
The second assertion follows from the fact that deg δa = −a combined with
Proposition 4.4 (ii), Proposition 6.2 (iii) and Proposition 6.3.
We will now show that for any nonzero a ∈ TLK(En), we have tr(aa∗) 6= 0, from
which the assertion follows. We have
λwbw,
18 R.M. GREEN
and by clearing denominators (thus multiplying a by a nonzero scalar), we may
assume that we have λw ∈ A for all w ∈ Wc. Choose w′ with λw′ 6= 0 and
N(w′) := deg λw′ maximal, and let cw′ be the (integer) coefficient of v
N(w′) in λw′ .
Setting aw′ = v
−N(w′)λw′bw′ , we then have
tr(aw′a
w′) = c
2 mod v−1Q[[v−1]].
If λw′′ 6= 0 but deg λw′′ is not maximal, we may again define aw′′ = v−N(w
′′)λw′′bw′′ ,
but then
tr(aw′′a
w′′) ∈ v−1Q[[v−1]].
Since the integers c2 are strictly positive, it follows that
tr((v−N(w
′)a)(v−N(w
′)a)∗) 6∈ v−1Q[[v−1]],
which completes the proof. �
Proposition 6.6. Let K be the field of fractions of the power series ring Z[[v−1]],
and let K ′ be the subfield of K consisting of the field of fractions of Z[[v−2]].
(i) The field TLK(En) has a unique structure as a Z2-graded algebra over K
which vn has degree n mod 2 and K ′ is precisely the set of elements of degree 0
mod 2.
(ii) The algebra TLK(En) has a unique structure as a Z2-graded algebra over K
in which vn has degree n mod 2 and the generators bs have degree 1 mod 2.
We denote the even subalgebra consisting of elements of degree 0 mod 2 by
TLK′(En).
(iii) Let τ : TLK(En) −→ K be any trace. Then there are unique K ′-linear maps
τ(0), τ(1) : TLK′(En) −→ K ′ such that τ(0) + vτ(1) is the restriction of τ to
TLK′(En), and furthermore, τ(0) and τ(1) are themselves traces.
Proof. Recall from the proof of Proposition 6.5 that K = Q((v−1)) = Q[v][[v−1]],
so that each element x ∈ K has a unique expression of the form
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 19
where qi ∈ Q and N ∈ Z depends on x. Similar reasoning shows that the subfield
K ′ of K then consists precisely of those elements for which qi = 0 whenever i is
odd. Part (i) is a consequence of this construction.
The assertion of (ii) is immediate from the observation that the defining relations
of Proposition 3.1 respect the given grading.
Let π : K −→ K ′ be the map
where
q′i =
qi if i is even,
0 otherwise.
Our description ofK ′ shows that π is a K ′-linear map. Denoting the restriction of τ
to TLK′(En) by τ
′, it follows that π ◦τ ′ is a trace on TLK′(En). Since τ(0) = π ◦τ ′,
the maps τ(0), vτ(1) = τ
′ − τ(0) and τ(1) are also traces, completing the proof of
(iii). �
Note that any trace from TLK′(En) to K
′ extends uniquely to a trace from
TLK(En) to K by tensoring by K ⊗K′ −.
Lemma 6.7. The trace tr : TLK(En) −→ K arises from a trace
tr′ : TLK′(En) −→ K ′
by extension of scalars.
Proof. We use the notation of §5. Note that if A ∈ P , then i(A) is an element of
TLK(En) of degree #A mod 2. We also have tr(i(A)) = δ
#A, which is an element
of K of degree #A mod 2.
Recall that TLK′(En) is a K
′-subalgebra of TLK(En) and note that if y, z are
homogeneous elements of TLK(En), then yz and zy have the same degree. The
argument of Lemma 5.2 now shows that if x is an element of TLK′(En), we have a
relation
tr(x) = tr
Ā∈P̄
λĀi(A)
20 R.M. GREEN
where for each Ā ∈ P̄ , we have λĀ(i(A)) ∈ TLK′(En). By the first paragraph of
the proof, λĀ must be homogeneous of degree #A mod 2, and tr(λĀi(A)) ∈ K ′.
The proof is completed by the observation that any x ∈ TLK(En) is uniquely
expressible as x(0) + vx(1) for x(0), x(1) ∈ TLK′(En) (compare with Proposition 6.6
(iii)). �
Corollary 6.8. If w ∈ Wc and tr(bw) = δa as in Proposition 6.3, then a ≡ λ(w)
mod 2.
Proof. By Lemma 6.7, we have deg tr(bw) = ℓ(w) mod 2, so the assertion follows
from the fact that deg δ = 1. �
§7. tom Dieck’s diagram calculus
In [2], tom Dieck introduced a diagram calculus for the algebras TL(En). To
give a rigorous definition of tom Dieck’s diagram calculus, as we do here, we first
need to recall the graphical definition of the Temperley–Lieb algebra. We start by
recalling Jones’ formalism of k-boxes [18], following the approach of Martin and
the author in [15]. For further details and references, the reader is referred to [11,
Definition 7.1. Let k be a nonnegative integer. The standard k-box, Bk, is the
set {(x, y) ∈ R2 : 0 ≤ x ≤ k + 1, 0 ≤ y ≤ 1}, together with the 2k marked points
1 = (1, 1), 2 = (2, 1), 3 = (3, 1), . . . , k = (k, 1),
k + 1 = (k, 0), k + 2 = (k − 1, 0), . . . , 2k = (1, 0).
Definition 7.2. Let X and Y be embeddings of some topological spaces (such
as lines) into the standard k-box. Multiplication of such embeddings to obtain a
new embedding in the standard k-box shall, where appropriate, be defined via the
following procedure on k-boxes. The product XY is the embedding obtained by
placing X on top of Y (that is, X is first shifted in the plane by (0, 1) relative to Y ,
so that marked point (i, 0) in X coincides with (i, 1) in Y ), rescaling vertically by a
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 21
scalar factor of 1/2 and applying the appropriate translation to recover a standard
k-box.
Definition 7.3. Let k be a nonnegative integer. Consider the set of smooth em-
beddings of a single curve (which we usually call an “edge”) in the standard k-box,
such that the curve is either closed (isotopic to a circle) or its endpoints coincide
with two marked points of the box, with the curve meeting the boundary of the
box only at such points, and there transversely.
By a smooth diffeomorphism of this curve we mean a smooth diffeomorphism of
the copy of R2 in which it is embedded, that fixes the boundary, and in particular
the marked points, of the k-box, and takes the curve to another such smooth
embedding. (Thus, the orbit of smooth diffeomorphisms of one embedding contains
all embeddings with the same endpoints.)
A concrete Brauer diagram is a set of such embedded curves with the property
that every marked point coincides with an endpoint of precisely one curve. (In
examples we can represent this set by drawing all the curves on one copy of the
k-box. Examples can always be chosen in which no ambiguity arises thereby.)
Two such concrete diagrams are said to be equivalent if one may be taken into
the other by applying smooth diffeomorphisms to the individual curve embeddings
within it.
There is an obvious map from the set of concrete diagrams to the set of pair
partitions of the 2k marked points. It will be evident that the image under this
map is an invariant of concrete diagram equivalence.
The set Bk(∅) is the set of equivalence classes of concrete diagrams. Such a class
(or any representative) is called a Brauer diagram.
Let D1, D2 be concrete diagrams. Since the k-box multiplication defined above
internalises marked points in coincident pairs, corresponding curve endpoints in
D1D2 may also be internalised seamlessly. Each chain of curves concatenated in
this way may thus be put in natural correspondence with a single curve. Thus
the multiplication gives rise to a closed associative binary operation on the set of
22 R.M. GREEN
concrete diagrams. It will be evident that this passes to a well defined multiplication
on Bk(∅). Let R be a commutative ring with 1. The elements of Bn(∅) form the
basis elements of an R-algebra PBn (∅) with this multiplication.
A curve in a diagram that is not a closed loop is called propagating if its endpoints
have different y-values, and non-propagating otherwise. (Some authors use the
terms “through strings” and “arcs” respectively for curves of these types.)
Note that in a Brauer diagram drawn on a single copy of the k-box it is not gen-
erally possible to keep the embedded curves disjoint. Let Tk(∅) ⊂ Bk(∅) denote the
subset of diagrams having representative elements in which the curves are disjoint.
Representatives of this kind are called Temperley–Lieb diagrams.
It will be evident that PBn (∅) has a subalgebra with basis the subset Tk(∅). (That
is to say, the disjointness property is preserved under multiplication.) We denote
this subalgebra Pn(∅)
Because of the disjointness property there is, for each element of Tk(∅), a unique
assignment of orientation to its curves that satisfies the following two conditions.
(i) A curve meeting the r-th marked point of the standard k-box, where r is odd,
must exit the box at that point.
(ii) Each connected component of the complement of the union of the curves in the
standard k-box may be oriented in such a way that the orientation of a curve
coincides with the orientation induced as part of the boundary of the connected
component.
Note that the orientations match up automatically in composition. If D1 and
D2 are equivalent concrete Temperley–Lieb diagrams, the diffeomorphisms that
give rise to the equivalence set up a bijection between the connected components
of D1 and those of D2.
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 23
Figure 2. A pillar diagram corresponding to an element of T8(∅)
1 2 3 4 5 6 7 8
910111213141516
Definition 7.4. A pillar diagram consists of a pair (D, f), where D ∈ Tk(∅) is a
Temperley–Lieb diagram and f is a function from the connected components of D
to Z≥0, such that any component with anticlockwise orientation is mapped to zero.
On the diagram D, we indicate the values of f on the clockwise connected com-
ponents either by writing in the appropriate integer, or by inserting k disjoint discs
(the “pillars” of [2]).
The set of pillar diagrams arising from the set Tk(∅) will be denoted Tk(•).
Example 7.5. Let k = 8. A pillar diagram corresponding to an element of Tk(•)
is shown in Figure 2. Note that there are 10 connected components, precisely 7 of
which inherit a clockwise orientation. The values of f on these 7 components are
3, 2, 2, 1, 0, 0, 0.
We define an algebra Pn(•), analogous to Pn(∅), with the set Tk(•) as a basis.
The multiplication is k-box multiplication with the added convention that function
values on the connected components are additive. (This is natural if one represents
the function values with pillars as in Figure 2.)
For our purposes, we need to apply an equivalence relation on the concrete
diagrams of Tk(•). Locally, this is given by the relation shown in Figure 3.
24 R.M. GREEN
Figure 3. A topological reduction rule
In the notation where clockwise regions are labelled by nonnegative integers, the
relation of Figure 3 is that shown in Figure 4.
Figure 4. Alternative notation for the topological reduction
If the regions labelled k and l are connected to each other, Figure 3 shows that
we have k = l > 1 and p = k − 1. On the other hand, if the regions labelled k
and l are genuinely distinct, that is, the arcs shown on the left hand side of figure
3 are not sections of some longer arc, then we have p = k+ l− 1 ≥ 1. In the latter
case, it is not possible for any regions labelled by the integer zero to be created or
destroyed by the topological reduction. Note that the other partial regions shown
in figures 2 and 3 have anticlockwise orientation, and as such they are labelled by
the integer 0.
Definition 7.6. If L is a closed loop in a concrete diagram of Tk(•), we define
m(L) to be the integer label of the region immediately interior to L; in particular,
we have m(L) = 0 if L has anticlockwise orientation.
Let R be a commutative ring with 1. The R-algebra PEn (•) is the quotient of
the R-algebra Pn(•) obtained by applying the following three relations:
(i) for each closed loop L whose immediate interior is labelled 1 and whose imme-
diate exterior is necessarily labelled 0, relabel the immediate interior of L by 0
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 25
and remove L;
(ii) for each closed loop L whose immediate interior is labelled 0 and whose imme-
diate exterior is labelled k, relabel the immediate interior of L by k, remove L
and multiply by δ;
(iii) for each region R labelled by k ≥ 2 (whether or not R is a closed loop), decrease
the label of R by 1 and multiply by δ.
A basis for PEn (•) may be obtained by using the notion of “reduced” diagrams
given in [2, §2] and Bergman’s diamond lemma [1]. However, we do not pursue this
because we do not need it for our purposes.
Definition 7.7. Suppose n > 1 and 1 ≤ k < n.
The diagram Enk of PEn (•) is the one where each point i is connected by a
propagating edge to point 2n+1− i, unless i ∈ {k, k+1, 2n−k, 2n+1−k}. Points
k and k + 1 are connected by an edge, as are points 2n − k and 2n + 1 − k. All
regions are labelled by 0.
The diagram Bnk of PEn (•) is the one where each point i is connected by a
propagating edge to point 2n+ 1− i, and all regions are labelled by 0, except the
rectangular region bounded by k, k+1, 2n− k and 2n+1− k, which is labelled by
Proposition 7.8. There is a unique homomorphism ρ : TL(En) −→ PEn (•) of
unital A-algebras sending b0 to Bn3 and bs to Ens for i ∈ {1, 2, . . . , n − 1}, where
the numbering of generators is as in §3.
Proof. This is a routine (but important) exercise using the presentation of Propo-
sition 3.1, and is essentially the same as the proof of [2, Theorem 2.5]. �
We shall see later that ρ is in fact a faithful representation. We will not determine
the image of ρ, but this can be done by an inductive combinatorial argument similar
to those in [9, §5].
26 R.M. GREEN
§8. Existence and uniqueness of the Markov trace
There is a well-known embedding ιn : TL(En) −→ TL(En+1) sending bs to bs
for each generator of TL(En) (see [3, §6.3]). This means that the tower of algebras
TL(En), equipped with the generators bs, fits into the framework of Markov traces
defined in §2. We recall the definition in order to fix some notation.
Definition 8.1. Let K be a field containing A. A Markov trace on TLK(E∞) with
parameter z ∈ K is a K-linear map τ : TLK(E∞) −→ K satisfying the following
conditions:
(i) τ(1) = 1;
(ii) τ(hbn) = zτ(h) for n ≥ 6 and h ∈ TLK(En);
(iii) τ(hh′) = τ(h′h) for all n ≥ 6 and h, h′ ∈ TLK(En).
Remark 8.2. Note that in condition (ii), bn is the unique generator in TL(En+1)
that does not lie in TL(En). As mentioned in [3, §2.2], the algebras TL(En) are
quotients of the Hecke algebras of the Coxeter groups W (En), and bs = q
−1/2(Ts+
1), where the Ts are the usual generators for the Hecke algebra as given in [16, §7].
This means that the Markov trace can also be regarded as a trace on a tower of
Hecke algebras.
Proposition 8.3. If τ is a Markov trace on TLK(E∞), then the parameter z must
be equal to δ−1, and τ is unique. Restricted to TL(En), such a Markov trace must
agree with the trace tr.
Proof. Let n ≥ 6. Part (ii) of Definition 8.1 shows that τ(bn−1bn) = zτ(bn−1). On
the other hand, the defining relations and part (iii) of the definition show that
τ(bn−1bn) = δ
−1τ(bn−1(bn−1bn)) = δ
−1τ(bn−1bnbn−1) = δ
−1τ(bn−1),
proving the assertion about the parameter.
To prove the other assertions, it suffices to show that, regarding TLK(En) as
a subalgebra of TLK(E∞), we have τ(i(A)) = δ
−#A for A ∈ P = P (n). Choose
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 27
such an A. It follows from Definition 3.3 that for sufficiently large N ≥ n, and
identifying A in the obvious way with an element of P (N), we can find B ∈ P (N)
with A ∼ B and B∩{b0, b1, . . . , b5} = ∅. The first assertion together with repeated
applications of part (ii) of Definition 8.1 (and one application of part (i)) now show
that τ(i(B)) = δ−#B = δ−#A, and Lemma 5.1 completes the proof. �
To prove that the Markov trace on TLK(E∞) exists, we make use of the diagram
calculus, as hinted in [2, §6].
Definition 8.4. Let k be a nonnegative integer. The standard k-cone is obtained
from the standard k-box by identifying each pair of points {(x, 0), (x, 1)} for each
0 ≤ x ≤ k+1, and identifying all the points in the set {(k+1, y) : 0 ≤ y ≤ 1}. The
standard k-cone is homeomorphic to a closed disc.
Let D be a diagram in PEk (•). The trace diagram, D, of D is obtained by
identifying the boundary points of the k-box bounding D to form the standard
k-cone.
Figure 5. The trace diagram of the pillar diagram in Figure 2
Example 8.5. The trace diagram D corresponding to the diagram D of Figure 2
is shown in Figure 5.
Notice that the outer part of the trace diagram (regarded as a disc) will always
have an anticlockwise orientation and thus be labelled by 0. Consequently, any
28 R.M. GREEN
regions in the trace diagram not labelled by zero must be bounded by at least one
closed loop. (It is possible for the closed loops to be nested.)
Definition 8.6. Let g : Z≥0 −→ Z≥0 be given by
g(c) =
1 if c = 0,
c− 1 if c ≥ 1.
If D is a trace diagram for TL(En), we define the content, c(D), of D to be the
integer
g(f(L)),
where the sum is over all the connected components L of D that are interior to at
least one closed loop, and where f(L) is the integer assigned to L as in Definition
Example 8.7. The content of the trace diagram in Figure 5 is
g(2) + g(3) + g(3) = 5.
Lemma 8.8. The content of a trace diagram D is invariant under the topological
reduction rule shown in Figure 3.
Proof. Consider the application of the topological reduction rule to a diagram that
looks locally like the situation in Figure 6.
Figure 6. Labelling of points involved in the topological relation
 C  D
As in the discussion following Figure 4, there are two cases to consider, according
as the two pillar regions are connected or not in D.
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 29
There are four cases to consider, according as there is an oriented curve in D
from point A to point C, and (independently) according as there is an oriented
curve in D from point D to point B.
Suppose first that there is no oriented curve in D from point A to point C, and
also that there is no oriented curve in D from point D to point B. In this case,
the two pillar regions are genuinely distinct, and applying the topological relation
does not produce any new closed loops. We are then in the case p = k + l − 1 ≥ 1
of Figure 4, so the summands (k − 1) and (l − 1) appearing in Definition 8.6 are
replaced by a single ((k + l − 1)− 1), leaving the content unchanged.
We next deal with the case where there is an oriented curve from point A to
point C, but no oriented curve from point D to point B. In this case, the two pillar
regions are connected to each other, and the application of the topological rule
produces a new closed loop (labelled zero) from the curve originally connecting
point A to point C. We are now in the case k = l > 1 of Figure 4. This will change
one of the summands (k− 1) of Definition 8.6 to (k− 2), and a new summand of 1
will be produced, corresponding to the new closed loop. The content thus remains
unchanged.
Consideration of the case where there is an oriented curve from point D to point
B, but not from point A to point C, proceeds in exactly the same way. The last case,
in which both oriented curves exist, also works similarly, except that the oriented
curves shown in Figure 6 are already part of a closed loop. Application of the
topological relation splits this closed loop into two closed loops, again producing
an extra summand of 1 and changing a summand (k − 1) to (k − 2), leaving the
content unchanged. �
Lemma 8.9. There is a well-defined K-linear map
τ•n : PEn (•) −→ K
such that for each pillar diagram D, τ•n(D) = δ
c(D). If x, y ∈ PEn (•), we have
τ•n(xy) = τ
n(yx).
30 R.M. GREEN
Proof. For the first assertion, we need to check relations (a)–(c) of Definition 7.6.
Relation (iii) holds by Lemma 8.8.
In relation (i), we have D = D1, where D1 is the result of removing a loop
labelled 1 from D. Since c(D) = c(D1), we have τ
n(D) = τ
n(D1).
In relation (ii), we have D = δD2, where D2 is the result of removing a loop
labelled 0 from D. Since c(D) = c(D2) + 1, we have τ
n(D) = τ
n(D2).
By linearity, we only need check the second assertion in the case where x and y
are pillar diagrams, and this is immediate from the construction of trace diagrams
from pillar diagrams. �
It is not hard to see that there is an algebra embedding ι•n : PEn (•) −→ PEn+1(•)
analogous to the map ιn. Given a pillar diagram D of PEn (•), ι•(D) is the diagram
obtained by adding a vertical line on the right of the diagram.
Lemma 8.10. Let D be a pillar diagram of PEn (•).
(i) We have τ•n+1(ι
n(D)) = δτ
n(D).
(ii) Let En+1n be as in Definition 7.7. Then we have τ
n(D) = τ
n+1(ιn(D)En).
Proof. Part (i) follows from the observation that the trace diagram ι•n(D) differs
from the trace diagram D only in having a single extra closed loop, labelled 0.
A short calculation involving diagrams shows that the trace diagrams D and
ιn(D)En are equivalent, from which part (ii) follows. �
Theorem 8.11. Let τn : TLK(En) −→ K be the trace defined by
τn(x) = δ
−nτ•n(ρ(x)).
The family of traces {τn : n ≥ 6} is compatible with the direct limit of algebras
TLK(En) and gives the unique Markov trace on TLK(E∞). Furthermore, the
Markov trace agrees with the traces tr of Definition 5.4.
Proof. The maps τn are traces by Proposition 7.8 and Lemma 8.9. They are
compatible with the direct limit by Lemma 8.10 (i). Since τ•n(1) = δ
n, we have
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 31
τn(1) = 1. Condition (ii) of Definition 8.1 follows from part (ii) of Lemma 8.10.
Uniqueness of the Markov trace, and agreement with the traces tr, is given by
Proposition 8.3. �
9. Proofs and applications
Proof of Theorem 1.1. We need to show that the homomorphism ρ of Proposition
7.8 is injective, and there is no loss in passing to the field of fractions K of Z[[v−1]].
In this case, Proposition 6.5 and Theorem 8.11 show that the unique Markov trace
on TLK(En), which can be defined on Im(ρ), is nondegenerate on TLK(En). The
conclusion follows. �
Proposition 9.1. The linear map
(1 + v−2)nτn = v
−nτ•n ◦ ρ
restricted to TL(En) takes values in A. It is a tabular trace in the sense of [10],
and a positive generalized Jones trace in the sense of [12].
Proof. The first assertion comes from the fact that τ•n evaluated on a diagram (such
as an element of the form ρ(bw) for w ∈ Wc) yields a nonnegative integer power of
To check that (1+v−2)nτn is a tabular trace, we need to check that axiom (A5) of
[10, Definition 1.3.4] is satisfied. We have just shown that (1+v−2)nτn takes values
in A, and it is clear from Theorem 8.11 that (1+ v−2)nτn is a trace. We have seen
in Corollary 5.5 and Proposition 6.2 (ii) that (1 + v−2)nτn(x) = (1 + v
−2)nτn(x
for all x ∈ TL(En). All that remains to check is that
τ(va(CS,T )CS,T ) = δS,T mod v
−1A−.
This follows from propositions 6.2 (ii) and 6.3 once we observe that we have
(1 + v−2)n = 1 mod v−2Q[[v−1]],
32 R.M. GREEN
regarded as power series in Q[v][[v−1]].
To show that (1+ v−2)nτn is a generalized Jones trace (see [12, Definition 2.9]),
two further conditions must be checked. One of these is precisely that established
by Lemma 6.7; the other is that, for x, y ∈ Wc, we should have
(1 + v−2)nτn(cxcy−1) =
1 mod v−1A− if x = y,
0 mod v−1A− otherwise,
where {cw : w ∈ Wc} is the canonical basis of TL(En) defined by J. Losonczy
and the author in [14]. By [14, Theorem 3.6], this is nothing other than the
basis {bw : w ∈ Wc} in this case. The corresponding property for tr (instead of
(1+v−2)nτn) follows from Proposition 6.5, and the assertion for (1+v
−2)nτn follows
from the fact that (1 + v−2)n = 1 mod v−2A−.
A generalized Jones trace is positive if it sends canonical basis elements to el-
ements of N[v, v−1]. This holds for (1 + v−2)nτn by Proposition 6.3: in this case,
(1 + v−2)nτn(bw) = δ
b for some b ≥ 0, so that (1 + v−2)n ∈ N[v, v−1]. �
Remark 9.2. Proposition 9.1 corrects the proof of [10, Theorem 4.3.5], where the
proof that the tabular trace takes the same values on x and x∗ contains a gap.
Proof of Theorem 1.2. By [12, Theorem 7.10], the conclusion of Theorem 1.2 holds
for a generalized Jones trace if the underlying Coxeter group has “Property F”
and a bipartite Coxeter graph. Clearly the graphs En are bipartite, because they
contain no circuits. Property F holds by [12, Remark 3.5]; see [13, Lemma 5.6] for
a fuller explanation.
To complete the proof, we simply have to transfer the result from (1+v−2)nτn to
the Markov trace, which follows from the fact that (1+v−2)n = 1 mod v−2A−. �
The next result is an easier to use version of Theorem 1.2.
Corollary 9.3. Let x, y ∈ Wc(En). Then we have
µ̃(x, y) =
1 if τ•n ◦ ρ(bxby−1) = δn−1,
0 otherwise.
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 33
Proof. This follows from Theorem 1.2 together with the observation that bxby−1 =
δbbw for some b ≥ 0 and w ∈ Wc, and the fact that τ•n sends diagrams to positive
powers of δ. �
Remark 9.4. It follows from [13, Theorem 4.6 (iv)] and [14, Theorem 3.6] that the
monomial basis element bx is the projection of the Kazhdan–Lusztig basis element
C′x ∈ H(En). Regarding tr and τ•n ◦ ρ as traces on the Hecke algebra, Theorem
1.2 and Corollary 9.3 can be used to evaluate the trace on products of certain
Kazhdan–Lusztig basis elements, without evaluating the product (which would be
difficult). Another noteworthy property of these results is that they give non-
recursive formulae for certain of the integers µ(x, y).
Remark 9.5. In [7, §9], Graham showed that if x, w ∈ Wc for TL(En) then µ(x, y) ∈
{0, 1}, and also produced a nonrecursive method of finding all the x with µ(x, y) = 1
for a fixed y. (In [7], x and y are said to be “close” if µ̃(x, y) = 1.) However, unlike
the results above, this does not give an efficient way to compute µ(x, y) when both
of x and y are specified. Corollary 9.3 can therefore be regarded as a quick way to
tell if two elements are close or not.
Remark 9.6. It is possible to modify Theorem 1.2 and Corollary 9.3 so that they
provide a nonrecursive way to test whether two diagrams represent the same algebra
element. However, we do not pursue this here for reasons of space.
Example 9.7. Consider the Coxeter system of type En with n = 6, and generators
s0, . . . , s5 as numbered in Figure 1. Define y = s1s2s4s0s5 and
w = s1s2s3s4s0s3s5s2s4s1s3s2s0s3s4s5;
these are both reduced expressions for fully commutative elements. The diagrams
ρ(by) and ρ(bw) are shown in figures 7 and 8 respectively. To evaluate τ
n(bybw−1),
we invert the diagram for bw, compose it with by and identify boundary points to
produce a trace diagram. The trace diagram so obtained is shown in Figure 9 (up
34 R.M. GREEN
to equivalence), and by inspection, it has content 1 + 1 + 1 + (3− 1) = 5 = n − 1.
It follows from Corollary 9.3 that µ(y, w) = 1.
Figure 7. The diagram ρ(by) of Example 9.7
REFERENCES 35
Figure 8. The diagram ρ(bw) of Example 9.7
Figure 9. The trace diagram corresponding
to τ•6 ◦ ρ(bybw−1) of Example 9.7
Acknowledgement
I am grateful to P.P. Martin for helpful comments on an early version of this
paper.
References
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