Table Of ContentON THE HECKE ALGEBRAS AND
THE COLORED HOMFLY POLYNOMIAL
6
0
0 XIAO-SONG LIN AND HAO ZHENG
2
n
Abstract. The coloredHOMFLY polynomial is the quantum invariant of ori-
a
J ented links in S3 associated with irreducible representations of the quantum
1 groupUq(slN). Inthispaper,usinganapproachtocalculatequantuminvariants
1 oflinksviacabling-projectionrule,wederiveaformulaforthecoloredHOMFLY
polynomial in terms of the characters of the Hecke algebras and Schur polyno-
]
A mials. The technique leads to a fairly simple formula for the colored HOMFLY
polynomial of torus links. This formula allows us to test the Labastida-Marin˜o-
Q
Vafa conjecture,whichrevealsa deeprelationshipbetweenChern-Simons gauge
.
h theory and string theory, on torus links.
t
a
m
[
1. Introduction
1
v
In the abstract of his seminal paper [7], V. Jones wrote: “By studying repre-
7
6 sentations of the braid group satisfying a certain quadratic relation we obtain a
2
polynomial invariant in two variables for oriented links. ...The two-variable poly-
1
0 nomial was first discovered by Freyd-Yetter, Lickorish-Millet, Ocneanu, Hoste, and
6 Przytycki-Traczyk.” ThistwovariablelinkpolynomialP (t,ν), commonlyreferred
L
0
to as the HOMFLY polynomial for an oriented link L in S3, is characterized by
/
h
the following crossing changing formula:
t
a
m (1.1) P (t,ν) = 1,
unknot
v: (1.2) ν−1/2PL+(t,ν)−ν1/2PL−(t,ν) = (t−1/2 −t1/2)PL0(t,ν).
i
X
Since then, this two variable link polynomial has been generalized to the quan-
r tum invariant associated with irreducible representations of the quantum group
a
U (sl ), with the variables t1/2 = q−1 and ν1/2 = q−N. We will refer to this
q N
generalization as the colored HOMFLY polynomial.
Despite the fact that the theory of quantum invariants of links is by now well
developed, the computation of colored HOMFLY polynomial is still extremely
challenging. Besides the trivial links, a general formula seems to exist in the
mathematics literature only for the Hopf link [14]. In the physics literature, Wit-
ten’s Chern-Simons path integral with the gauge group SU [21] offers an intrinsic
N
but not rigorous definition of the colored HOMFLY polynomial. There is a con-
jectured relationship between the 1/N expansion of Chern-Simons theory and the
Gromov-Witten invariants of certain non-compact Calabi-Yau 3-folds. See [5][16]
The first author is supported in part by NSF grants DMS-0404511.
1
2 XIAO-SONGLIN AND HAOZHENG
for example. Motivated by this conjectured relationship, Labastida, Marin˜o and
Vafa proposed a precise conjecture about the structure of their reformulation of
the colored HOMFLY polynomial [11][12]. See Section 5. A formula of the colored
HOMFLY polynomial for torus knots is given in [10], which was used to test the
Labastida-Marin˜o-Vafa conjecture on torus knots.
In this paper, using an approach to calculate quantum invariants of links via
cabling-projectionrule,wederiveaformulaforthecoloredHOMFLYpolynomialin
terms of the characters of the Hecke algebras and Schur polynomials. See Theorem
4.3. Animportant feature of this formula is that the character of theHecke algebra
is free of the variable ν and the Schur polynomial is independent of the link L. We
think that this separation of the variable ν and the link L might be important for
a possible proof of the Labastida-Marin˜o-Vafa conjecture.
Our technique leads to a fairly simple formula for the colored HOMFLY poly-
nomial of torus links. See Theorem 5.1. Using our formula, the Labastida-Marin˜o-
Vafa conjecture can be test on several infinite families of torus links. Our calcula-
tionalsosuggestsanewstructureofthereformulatedcoloredHOMFLYpolynomial
of torus links: it is equivalent to a family of polynomials in Z[t±1] invariant under
the transformation t → t−1. See Conjecture 6.2 and the examples following it.
Acknowledgments. The authors would like to thank Professors Kefeng Liu and
Jian Zhou for their interest in this work.
2. Link invariants from quantum groups
In this section, we give a brief review of the quantum group invariants of links.
See [8][17][19] for details.
Let g be a complex simple Lie algebra and let q be a nonzero complex number
which is not a root of unity. Let U (g) denote the quantum enveloping algebra of
q
g. The ribbon category structure of the set of finite dimensional complex repre-
sentations of U (g) provides the following objects.
q
1. Associated to each pair of U (g)-modules V,W is a natural isomorphism (the
q
braiding) Rˇ : V ⊗W → W ⊗V such that
V,W
Rˇ = (Rˇ ⊗id )(id ⊗Rˇ ),
U⊗V,W U,W V U V,W
(2.1)
ˇ ˇ ˇ
R = (id ⊗R )(R ⊗id )
U,V⊗W V U,W U,V W
hold for all U (g)-modules U,V,W. The naturality means
q
(2.2) (y ⊗x)Rˇ = Rˇ (x⊗y)
V,W V′,W′
for x ∈ Hom (V,V′), y ∈ Hom (W,W′). These equalities imply the braiding
Uq(g) Uq(g)
relation
ˇ ˇ ˇ
(R ⊗id )(id ⊗R )(R ⊗id )
V,W U V U,W U,V W
(2.3)
= (id ⊗Rˇ )(Rˇ ⊗id )(id ⊗Rˇ ).
W U,V U,W V U V,W
HECKE ALGEBRA AND COLORED HOMFLY POLYNOMIAL 3
2. There exists an element K ∈ U (g) (the enhancement of Rˇ, here ρ means
2ρ q
the half-sum of all positive roots of g) such that
(2.4) K (v ⊗w) = K (v)⊗K (w)
2ρ 2ρ 2ρ
for v ∈ V, w ∈ W. Moreover, for every z ∈ End (V ⊗W) with z = x ⊗y ,
Uq(g) Pi i i
x ∈ End(V), y ∈ End(W) one has the (partial) quantum trace
i i
(2.5) tr (z) = tr(y K )·x ∈ End (V).
W X i 2ρ i Uq(g)
i
3. Associated to each U (g)-module V is a natural isomorphism (the ribbon
q
structure) θ : V → V satisfying
V
(2.6) θ±1 = tr Rˇ±1 .
V V V,V
The naturality means
(2.7) x·θ = θ ·x
V V′
for x ∈ Hom (V,V′).
Uq(g)
W W V W
V
L β
With these objects, one constructs the quantum group invariants of links as
follows. Let L be an oriented link with the components L ,...,L labeled by the
1 l
U (g)-modules V ,...,V , respectively. Choose a closed braid representative βˆ of
q 1 l
L with β ∈ B being an n-strand braid. Assign to each positive (resp. negative)
n
crossing ofβ anisomorphism Rˇ (resp. Rˇ−1 )where V,W aretheU (g)-modules
V,W W,V q
labeling the two outgoing strands of the crossing.
V W V W
Rˇ Rˇ−1
V,W W,V
Then the braid β gives rise to an isomorphism
(2.8) h (β) ∈ End (V′ ⊗···⊗V′),
V1′,...,Vn′ Uq(g) 1 n
4 XIAO-SONGLIN AND HAOZHENG
where V′,...,V′ are the U (g)-modules labeling the strands of β, and the quantum
1 n q
trace
(2.9) tr h (β)
V′⊗···⊗V′ V′,...,V′
1 n 1 n
defines a framing dependent link invariant of L.
Example 2.1. The link shown in above figure has two components, labeled by W
and V respectively. It is the closure of β = σ−1σ−1σ ∈ B , which gives rise to an
1 2 1 3
isomorphism
(2.10) h (β) = (Rˇ−1 ⊗id )(id ⊗Rˇ−1 )(Rˇ ⊗id ).
W,V,W W,V W V W,W W,V W
Thus the link invariant is
(2.11) tr (Rˇ−1 ⊗id )(id ⊗Rˇ−1 )(Rˇ ⊗id ).
W⊗V⊗W W,V W V W,W W,V W
To eliminate the framing dependency, one should require the modules V ,...,V
1 l
be irreducible, hence the isomorphisms θ ,...,θ are multiples of identity and
V1 Vl
may be regarded as scalars. Let w(L ) be the writhe of L in β, i.e. the number of
i i
positive crossings minus the number of negative crossings. Then the quantity
(2.12) I = θ−w(L1)···θ−w(Ll)tr h (β)
L;V1,...,Vl V1 Vl V1′⊗···⊗Vn′ V1′,...,Vn′
defines a framing independent link invariant.
When the link involved is the unknot, it is easy to see that
(2.13) I = tr id .
unknot;V V V
This quantity is regarded as the quantum version of the classical dimension of V,
referred to as the quantum dimension of V and denoted by dim V.
q
3. Centralizer algebra and cabling-projection rule
In general, the isomorphism Rˇ is very complicated when the dimensions of
V,W
V,W are larger, so it is not practical to compute the link invariants from their
definition. However, on the other hand, general representations of a simple Lie
algebra g (thus its quantum deformation U (g)) are often realized as components
q
of tensor products of some simple ones. For example, irreducible representations
of U (sl ) are always the components of some tensor products of the fundamental
q N
representation.
In this section, we follow this observation and develop a cabling-projection rule
to break down the complexity of general Rˇ. For this purpose we need the notion
of centralizer algebra.
The centralizer algebras of the modules of simple Lie algebras have played an
important role in representation theory. Parts of their quantum version were stud-
ied in [2][13][20]. In the case of U (sl ), the situation is desirable. The centralizer
q N
algebras are nothing but the subalgebras of the Hecke algebras of type A.
Let V be a U (g)-module. The centralizer algebra of V⊗n is defined as
q
(3.1) C (V) = End (V⊗n) = {x ∈ End(V⊗n) | xy = yx, ∀y ∈ U (g)}.
n Uq(g) q
HECKE ALGEBRA AND COLORED HOMFLY POLYNOMIAL 5
It is immediate from definition that C (V) is a finite dimensional von Neumann
n
algebra, i.e. the algebra is isomorphic to a direct sum of matrix algebras. Indeed,
if V⊗n admits the irreducible decomposition
(3.2) V⊗n = d ·V ,
M λ λ
λ∈Λ
by Schur’s lemma we have
(3.3) C (V) = C
n M λ
λ∈Λ
where C = End (d V ) is a full d × d matrix algebra. Since each matrix
λ Uq(g) λ λ λ λ
algebra admits a unique irreducible representation, via above decomposition the
irreducible representations of C (V) are naturally indexed by Λ.
n
Let ζλ denote the character of the irreducible representation of C (V) indexed
n
by λ ∈ Λ.
Lemma 3.1. For every x ∈ C (V) we have
n
(3.4) tr x = ζλ(x)·dim V .
V⊗n X q λ
λ∈Λ
Proof. Let π be the unit of C . Then π x is a matrix in C , whose normal trace
λ λ λ λ
trπ x is precisely ζλ(x). Therefore,
λ
(3.5) tr x = trπ x·tr id = ζλ(x)·dim V .
V⊗n X λ Vλ Vλ X q λ
λ∈Λ λ∈Λ
(cid:3)
A projection (or idempotent) of C (V) is an element p ∈ C (V) satisfying the
n n
idempotent equation p2 = p. By definition, an element p ∈ C (V) is a projection
n
if and only if it is, restricted on each C , diagonalizable and has the only possible
λ
eigenvalues 0 and 1. It is clear that for each projection p ∈ C (V),
n
(3.6) pV⊗n ∼= ζλ(p)·V .
M λ
λ∈Λ
A projection p ∈ C (V) is called minimal (or primitive) if pV⊗n ∼= V for some
n λ
λ ∈ Λ.
Let h be the homomorphism
V
(3.7) h : CB → C (V), σ 7→ id ⊗R ⊗id .
V n n i V⊗(i−1) V,V V⊗(n−i−1)
ˇ
The following lemma makes it possible to recover general R, θ from specific ones.
6 XIAO-SONGLIN AND HAOZHENG
∆2 χ3,4
5
Lemma 3.2. Let∆2 = (σ σ ···σ )n ∈ B and χ = n′ (σ σ ···σ ) ∈
n 1 2 n−1 n n,n′ i=1 i+n−1 i+n−2 i
B denote the full twist braid and the (n,n′)-crossingQbraid, respectively. Let
n+n′
p ∈ C (V), p′ ∈ C (V) be projections and U = pV⊗n, W = p′V⊗n′. We have
n n′
(3.8) h (χ )·(p⊗p′) = Rˇ ⊕0 ,
V n,n′ W,U Kerp⊗p′
(3.9) (θ )⊗n ·h (∆2)·p = θ ⊕0 .
V V n U Kerp
Proof. Applying the identities (2.1), (2.3) and (2.6) inductively, we have
(3.10) Rˇ = h (χ ),
V⊗n′,V⊗n V n,n′
(3.11) θ = tr Rˇ = (θ )⊗n ·h (∆2).
V⊗n V⊗n V⊗n,V⊗n V V n
Then from the naturality of Rˇ and θ the lemma follows. (cid:3)
ˇ
As an easy consequence of (3.8) and the naturality of R, we have the cabling-
projection rule
Lemma 3.3. Let β ∈ B be a braid and p ∈ C (V), i = 1,...,m be projections
m i ni
such that p = p whenever the i-th strand of β ends at j-th point. Moreover, let
i j
V = p V⊗ni, n = n +···+n and β(n1,...,nm) ∈ B be the braid obtained by cabling
i i 1 m n
the i-th strand of β to n parallel ones. Then
i
(3.12) h (β(n1,...,nm))·(p ⊗···⊗p ) = h (β)⊕0 ,
V 1 m V1,...,Vm Kerp1⊗···⊗pm
thus
(3.13) trV1⊗···⊗VmhV1,...,Vm(β) = trV⊗n hV(β(n1,...,nm))·(p1 ⊗···⊗pm).
With above lemmas, one is able to re-express the link invariant (2.12), by choos-
ing a suitable U (g)-module V, in terms of much more accessible objects: the char-
q
acters and projections of the centralizer algebras C (V) and the quantum traces
n
of U (g)-modules. In the next section, we present a detailed realization of this
q
approach for the case g = sl .
N
4. Hecke algebras and colored HOMFLY polynomial
In the rest part of this paper we will extensively apply the facts concerning
the Hecke algebras, the quantum enveloping algebras U (sl ) and symmetric func-
q N
tions. The facts are well known and most of them can be found, for example, in
[3][8][9][15][18].
HECKE ALGEBRA AND COLORED HOMFLY POLYNOMIAL 7
The Hecke algebra H (q) of type A is the complex algebra with generators
n n−1
g ,g ,...,g and relations
1 2 n−1
g g = g g , |i−j| ≥ 2,
i j j i
(4.1) g g g = g g g , |i−j| = 1,
i j i j i j
(g −q)(g +q−1) = 0, i = 1,2,...,n−1.
i i
Note that, when q = 1, the Hecke algebra H (q) is nothing new but the group
n
algebra CΣ of the symmetric group. In fact, if q is nonzero and not root of
n
unity we still have the isomorphism H (q) ∼= CΣ and H (q) also canonically
n n n
decomposes as
(4.2) H (q) = H (q)
n M λ
λ⊢n
with each H (q) being a matrix algebra.
λ
Here we fix several notations of combinatorics. A composition µ of n, denoted
by µ |= n, is a sequence of nonnegative integers (µ ,µ ,...) such that µ = n.
1 2 i i
P
The length ℓ(µ) of µ is the maximal index i with µ nonzero. If, in addition,
i
µ ≥ µ ≥ ··· then µ is also called a partition and one writes µ ⊢ n and |µ| = n.
1 2
Itisastandardresultthatthecentralizer algebrasofsl -modulesarecanonically
N
subalgebras of CΣ . So it is not surprising to see that the centralizer algebras of
n
U (sl )-modules are realized as subalgebras of H (q), the quantum deformation of
q N n
CΣ .
n
Now fix g = sl and let V be the module of the fundamental representation of
N
U (sl ). With suitable basis {v ,...,v } of V and generators {K±1,E ,F | 1 ≤
q N 1 N i i i
i ≤ N −1} of U (sl ), the fundamental representation is given by the matrices
q N
K 7→ qE +q−1E + E ,
i ii i+1,i+1 j6=i jj
P
(4.3) E 7→ E ,
i i,i+1
F 7→ E ,
i i+1,i
where E is the N ×N matrix with 1 in the (i,j)-position and 0 elsewhere. We
ij
also have
(4.4) q1/Nθ = qN ·id ,
V V
(4.5) K (v ) = qN+1−2iv ,
2ρ i i
and
qv ⊗v , i = j,
i j
(4.6) q1/NRˇ (v ⊗v ) = v ⊗v , i < j,
V,V i j j i
v ⊗v +(q −q−1)v ⊗v , i > j.
j i i j
Itisstraightforwardtoverifythatthehomomorphismh : CB → C (V)factors
V n n
through H (q) via
n
(4.7) q1/Nσ 7→ g 7→ q1/Nh (σ ).
i i V i
8 XIAO-SONGLIN AND HAOZHENG
Therefore, V⊗n is a module of both U (sl ) and H (q), and the two algebras
q N n
act commutatively on V⊗n. For convenience, we introduce an N-independent
homomorphism
(4.8) h : CB → H (q), σ 7→ g .
n n i i
Let Sλ denote the irreducible module of H (q) indexed by the partition λ ⊢ n
n
and let ζλ denote its character. Fix a minimal projection p ∈ H (q) for each
λ λ
λ ⊢ n. Let V denote the irreducible U (sl )-module, whose highest weight vector
λ q N
v behaves like K (v) = qλi−λi+1v, if ℓ(λ) ≤ N and be 0 otherwise.
i
We state below two important facts concerning the U (sl )-module V⊗n. One
q N
is the irreducible decomposition of U (sl )-module
q N
(4.9) V⊗n = dimSλ ·V .
M λ
λ⊢n,ℓ(λ)≤N
in which the subspace dimSλ·V is H (q)-invariant and, as a H (q)-module, con-
λ n n
sists of only Sλ-components. Notice that the U (sl )-modules {V | λ ⊢ n, ℓ(λ) ≤
q N λ
N} are mutually inequivalent. Comparing (4.9), (4.2) with (3.2), (3.3), we have
immediately
(4.10) C (V) = H (q).
n M λ
λ⊢n,ℓ(λ)≤N
Moreover, for every partition λ ⊢ n,
(4.11) p V⊗n ∼= V .
λ λ
The other fact is the weight decomposition of U (sl )-module
q N
(4.12) V⊗n = Mµ
M
µ|=n,ℓ(µ)≤N
where
(4.13) Mµ = {v ∈ V⊗n | K (v) = qµi−µi+1v}.
i
Moreover, the dimensions of the weight spaces of V for λ ⊢ n
λ
(4.14) K = dim(p V⊗n ∩Mµ)
λµ λ
are encoded in Schur polynomial as
N
(4.15) s (z ,...,z ) = K · zµj.
λ 1 N X λµ Y j
µ|=n,ℓ(µ)≤N j=1
Indeed, Mµ isnothingbutthesubspaceofV⊗n spannedbythevectorsv ⊗···⊗v
i1 in
in which v appears precisely µ times. It is clear that Mµ is H (q)-invariant. In
i i n
the literature, Mµ is called permutation module and the integers K are referred
λµ
to as Kostka numbers.
Various choices of minimal projections of the Hecke algebras are available in
[1][4][6][15]. It is also shown
HECKE ALGEBRA AND COLORED HOMFLY POLYNOMIAL 9
Theorem 4.1 (Aiston-Morton [1, Theorem 5.5]). For each partition λ ⊢ n with
ℓ(λ) ≤ N, one has
(4.16) θ = qκλ+nN−n2/N ·id
Vλ Vλ
where
ℓ(λ) λi
(4.17) κ = 2(j −i).
λ XX
i=1 j=1
The next proposition is a strong version of Lemma 3.1. Equation (4.19) holds
even for x 6∈ C (V).
n
Proposition 4.2. We have
(4.18) dim V = s qN−1,qN−3,...,q−(N−1)
q λ λ(cid:16) (cid:17)
thus for every x ∈ H (q),
n
(4.19) tr x = ζλ(x)·s qN−1,qN−3,...,q−(N−1) .
V⊗n X λ(cid:16) (cid:17)
λ⊢n
Proof. By (2.4) and (4.5), K acts as a scalar N q(N+1−2i)µj on Mµ. Therefore,
2ρ i=1
Q
it follows from identity (4.15) that for each λ ⊢ n,
N
(4.20) dim V = K · q(N+1−2i)µj = s qN−1,qN−3,...,q−(N−1) .
q λ X λµ Y λ(cid:16) (cid:17)
µ|=n,ℓ(µ)≤N i=1
(cid:3)
Now it is time to give our main result.
Theorem 4.3. Let L be an oriented link with l components L ,...,L . Suppose
1 l
L is the closure of β ∈ B and the m strands of β are living on L ,...,L ,
m i1 im
respectively. Then for partitions λi ⊢ n , i = 1,...,l, we have
i
IL;Vλ1,...,Vλl = q−Pli=1(κλi+niN−n2i/N)w(Li)−w(β(ni1,...,nim))/N·
(4.21)
ζλ(x)·s qN−1,qN−3,...,q−(N−1) ,
X λ(cid:16) (cid:17)
λ⊢n
where n = ni1+···+nim, β(ni1,...,nim) ∈ Bn is the braid obtained by cabling the j-th
strand of β to nij parallel ones and x = h(β(ni1,...,nim))·(pλi1 ⊗···⊗pλim) ∈ Hn(q).
(cid:3)
Proof. Combine Lemma 3.3, Theorem 4.1 and Proposition 4.2.
One notices that, on the right hand side of (4.21), there is an explicit factor q1/N
to the power
l
(4.22) Xn2iw(Li)−w(β(ni1,...,nim)) = −2Xninjlk(Li,Lj)
i=1 i<j
10 XIAO-SONGLIN AND HAOZHENG
where lk(L ,L ) are the linking numbers. As in [11], we drop this insignificant
i j
factor and regard the remaining part as a rational function of q and qN.
Definition 4.4. The colored HOMFLY polynomial W (t,ν) with λi ⊢ n is
L;λ1,...,λl i
a rational function of t1/2,ν1/2 determined by
(4.23) WL;λ1,...,λl(t,ν)|t1/2=q−1,ν1/2=q−N = q2Pi<jninjlk(Li,Lj)/N ·IL;Vλ1,...,Vλl.
Note that the definition means the components of the link L are labeled by
partitions rather than U (sl )-modules. When the labeling partitions are trivial
q N
(theuniquepartitionof1),thecoloredHOMFLYpolynomial,uptoasimplefactor,
specializes to the HOMFLY polynomial:
t1/2 −t−1/2
(4.24) P (t,ν) = νlk(L) · ·W (t,ν)
L ν1/2 −ν−1/2 L;(1),...,(1)
Let s∗(t,ν) be defined by (see (5.14))
λ
(4.25) s∗(t,ν)| = s qN−1,qN−3,...,q−(N−1) .
λ t1/2=q−1,ν1/2=q−N λ(cid:16) (cid:17)
Corollary 4.5. In the same notations as Theorem 4.3, we have
(4.26) WL;λ1,...,λl(t,ν) = tPli=1κλiw(Li)/2 ·νPli=1niw(Li)/2 ·Xζλ(x)(cid:12)(cid:12)q=t−1/2 ·s∗λ(t,ν).
λ⊢n (cid:12)
5. Torus links
Let notations be the same as in the previous section. In this section, we derive
an explicit formula of the colored HOMFLY polynomial of torus links by applying
Corollary 4.5.
The torus link T(r,k) is defined to be the closure of (δ )k = (σ ···σ )k. They
r 1 r−1
form the family of link that can be put on the standardly embedded torus T ⊂ R3.
Some common links such as the trefoil knot T(2,3), the Hopf link T(2,2) are
included in this family.
Theorem 5.1. Let L be the torus link T(rl,kl) with r,k relatively prime. Let
λi ⊢ n , i = 1,...,l be partitions and n = n +···+n . Then
i 1 l
(5.1) WL;λ1,...,λl(t,ν) = tkrPli=1κλi/2 ·νk(r−1)n/2 · X cλλ1...λl ·t−kκλ/2r ·s∗λ(t,ν)
λ⊢rn
where cλ are the integers determined by the equation
λ1...λl
l
(5.2) s (xr,xr,...) = cλ ·s (x ,x ,...).
Y λi 1 2 X λ1...λl λ 1 2
i=1 λ⊢rn
The theorem is an easy consequence of following lemmas.
Lemma 5.2. For each partition λ ⊢ n we have
(5.3) h(∆2)·p = qκλ ·p .
n λ λ
