Specht modules and Kazhdan--Lusztig cells in type $B_n$
SPECHT MODULES AND KAZHDAN–LUSZTIG CELLS IN TYPE Bn
MEINOLF GECK, LACRIMIOARA IANCU AND CHRISTOS PALLIKAROS
Abstract. Dipper, James and Murphy generalized the classical Specht module theory to Hecke
algebras of type Bn. On the other hand, for any choice of a monomial order on the parameters
in type Bn, we obtain corresponding Kazhdan–Lusztig cell modules. In this paper, we show that
the Specht modules are naturally isomorphic to the Kazhdan–Lusztig cell modules if we choose
the dominance order on the parameters, as in the “asymptotic case” studied by Bonnafé and the
second named author. We also give examples which show that such an isomorphism does not exist
for other choices of monomial orders.
1. Introduction
LetHn be the generic Iwahori–Hecke algebra of type An−1 or Bn. For any partition or bipartition
λ of n, we have a corresponding Specht module S̃λ, as defined by Dipper–James [5] (in type An−1)
and Dipper–James–Murphy [6] (in type Bn). On the other hand, we have the cell modules arising
from the theory of Kazhdan–Lusztig cells; see Lusztig [13], [14]. Now McDonough–Pallikaros [15]
showed that, in type An−1, the Specht modules and Kazhdan–Lusztig cell modules are naturally
isomorphic. The main purpose of this paper is to prove an analogous result for type Bn. Note
that, contrary to the situation in type An−1, there are many different types of Kazhdan–Lusztig
cell modules in type Bn, depending on the choice of a monomial order on the two parameters in
type Bn. We will show that it is precisely the “asymptotic case” studied in [3] which yields an
isomorphism with the Specht modules of Dipper–James–Murphy.
In Theorem 3.6, we show the existence of a canonical isomorphism between a Specht module and
a Kazhdan–Lusztig left cell module in the “asymptotic case” (where both of them are labelled by
the appropriate bipartition of n). Both the Specht modules and the Kazhdan–Lusztig cells have
certain standard bases. We show that, for a suitable ordering of these bases, the matrix of the
canonical isomorphism is triangular with 1 on the diagonal. Our proof essentially relies on the
combinatorial description [3] of the left cells in the “asymptotic case”. This allows us to determine
explicitly (in terms of reduced expressions of elements) certain distinguished left cells for every
bipartition of n; see Proposition 2.6.
In Section 4, we give examples which show that the Specht modules are not isomorphic to
Kazhdan–Lusztig cell modules for choices of the monomial order which are different from the
“asymptotic case”.
2. Kazhdan–Lusztig bases and cells
In this section, we recall the basic definitions concerning Kazhdan–Lusztig bases and cells, fol-
lowing Lusztig [13], [14]. We also recall some of the main results of [3], [4], [8] concerning the
“asymptotic case” in type Bn. This will allow us, see Proposition 2.6, to describe explicit reduced
expressions for the elements in certain distinguished left cells in type Bn.
Date: June, 2007.
2000 Mathematics Subject Classification. Primary 20C08; Secondary 20G40.
http://arxiv.org/abs/0704.1846v2
2 Geck, Iancu and Pallikaros
2.A. Basic definitions.
In [14], an Iwahori–Hecke algebra with possibly unequal parameters is defined with respect to an
integer-valued weight function on W . Following a suggestion of Bonnafé [4], we can slightly modify
Lusztig’s definition so as to include the more general setting in [13] as well.
Let Γ be an abelian group (written additively) and let A = Z[Γ] be the free abelian group
with basis {εγ | γ ∈ Γ}. There is a well-defined ring structure on A such that εγεγ
= εγ+γ
all γ, γ′ ∈ Γ. (Hence, if Γ = Z, then A is nothing but the ring of Laurent polynomials in an
indeterminate ε.) We write 1 = ε0 ∈ A. Given a ∈ A we denote by aγ the coefficient of ε
γ , so that
γ∈Γ aγε
γ . We say that a function
L : W → Γ
is a weight function if L(ww′) = L(w) + L(w′) whenever we have ℓ(ww′) = ℓ(w) + ℓ(w′) where
ℓ : W → N is the usual length function. (We denote N = {0, 1, 2, . . .}.) Let H = H(W,S,L) be the
generic Iwahori–Hecke algebra over A with parameters {vs | s ∈ S} where vs := ε
L(s) for s ∈ S.
The algebra H is free over A with basis {Tw | w ∈W}, and the multiplication is given by the rule
TsTw =
Tsw if ℓ(sw) > ℓ(w),
Tsw + (vs − v
s )Tw if ℓ(sw) < ℓ(w),
where s ∈ S and w ∈W .
Now assume that there is a total order 6 on Γ compatible with the group structure. (In the
setting of [14], Γ = Z with the natural order.) The following definitions will depend on the choice
of this total order. We denote by A>0 the set of Z-linear combinations of elements ε
γ where γ > 0.
Similarly, we define A>0, A60 and A<0. We assume throughout that L(s) > 0 for all s ∈ S. Having
fixed a total order on Γ, we have a corresponding Kazhdan–Lusztig basis {Cw | w ∈W} of H. The
element Cw is self-dual with respect to a certain ring involution of H, and we have
Cw = Tw +
P ∗y,w Ty ∈ H,
where < denotes the Bruhat–Chevalley order on W and P ∗y,w ∈ A<0 for all y < w in W ; see [13,
§6]. (In the framework of [14], the polynomials P ∗y,w are denoted py,w and the basis elements Cw
are denoted cw.) Given x, y ∈W , we write
CxCy =
hx,y,z Cz where hx,y,z ∈ A.
We have the following more explicit multiplication rules (see [13, §6]): for w ∈ W and s ∈ S, we
TsCw =
Csw − v
s Cw +
M sz,w Cz if sw > w,
vsCw if sw < w,
where the elements M sz,w ∈ A are determined as in [13, §3].
We recall the definition of the left cells of W and the corresponding left cell representations of
H (see [13] or [14]). Note again that these depend on the choice of a total order on Γ.
We write z ←L y if there exists some s ∈ S such that hs,y,z 6= 0, that is, Cz occurs in CsCy (when
expressed in the C-basis). Let 6L be the pre-order relation on W generated by ←L, that is, we
have z 6L y if there exist elements z = z0, z1, . . . , zk = y such that zi−1 ←L zi for 1 6 i 6 k. The
equivalence relation associated with 6L will be denoted by ∼L and the corresponding equivalence
classes are called the left cells of W .
Specht modules and Kazhdan–Lusztig cells 3
Similarly, we can define a pre-order 6R by considering multiplication by Cs on the right in the
defining relation. The equivalence relation associated with 6R will be denoted by ∼R and the
corresponding equivalence classes are called the right cells of W . We have
x 6R y ⇔ x
This follows by using the anti-automorphism ♭ : H → H given by T ♭w = Tw−1 ; we have C
w = Cw−1
; see [14, 5.6]. Thus, any statement concerning the left pre-order relation 6L has an equivalent
version for the right pre-order relation 6R, via ♭.
Finally, we define a pre-order 6LR by the condition that x 6LR y if there exists a sequence
x = x0, x1, . . . , xk = y such that, for each i ∈ {1, . . . , k}, we have xi−1 6L xi or xi−1 6R xi. The
equivalence relation associated with 6LR will be denoted by ∼LR and the corresponding equivalence
classes are called the two-sided cells of W .
Each left cell C gives rise to a representation of H. This is constructed as follows (see [13, §7]).
IC = 〈Cy (y ∈W ) | y 6L w for some w ∈ C〉A,
ÎC = 〈Cy (y ∈W ) | y 6L w for some w ∈ C and y 6∈ C〉A.
These are left ideals in H. Hence [C]A = IC/ÎC is a left H-module; it is free over A with basis
{ew | w ∈ C} where ew denotes the class of Cw modulo ÎC. Explicitly, the action of H on [C]A is
given by
Cw.ex =
hw,x,y ey for all x ∈ C and w ∈W.
2.B. The “asymptotic case” in type Bn.
Now let Γ = Z2; then A = Z[Γ] is nothing but the ring of Laurent polynomials in two independent
indeterminates V = ε(1,0) and v = ε(0,1). Let W = Wn be the Coxeter group of type Bn (n > 2),
with generators, relations and weight function L : Wn → Γ given by the following diagram:
✐ ✐ ✐ · · · ✐
where a, b ∈ Γ. Let Hn be the corresponding generic two-parameter Iwahori–Hecke algebra over
A = Z[Γ], where we set
V := vt = ε
b and v := vs1 = · · · = vsn−1 = ε
(Note that any Hecke algebra of type Bn can be obtained from Hn by “specialisation”; see also
Remark 3.8 below.) In order to obtain Kazhdan–Lusztig cells and the corresponding cell modules,
we have to specify a total order 6 on Γ. Note that there are infinitely many such total orders:
For example, we have all the weighted lexicographic orders, given by (i, j) < (i′, j′) if and only if
xi+ yj < xi′ + yj′ or xi+ yj = xi′ + yj′ and i < i′, where x, y are fixed positive real numbers.
Here, we shall take for 6 the lexicographic order on Γ such that
(i, j) < (i′, j′)
⇐⇒ i < i′ or i = i′ and j < j′.
This is the set-up originally considered by Bonnafé–Iancu [3]; it is called the “asymptotic case” in
type Bn. We shall need some notation from [3]. Given w ∈Wn, we denote by ℓt(w) the number of
occurrences of the generator t in a reduced expression for w, and call this the “t-length” of w.
4 Geck, Iancu and Pallikaros
The parabolic subgroup Sn = 〈s1, . . . , sn−1〉 is naturally isomorphic to the symmetric group on
{1, . . . , n}, where si corresponds to the basic transposition (i, i + 1). For 1 6 l 6 n − 1, we set
Σl,n−l := {s1, . . . , sn−1} \ {sl}. For l = 0 or l = n, we also set Σ0,n = Σn,0 = {s1, . . . , sn−1}. Then
we have the Young subgroup
Sl,n−l = 〈Σl,n−l〉 = S{1,...,l} ×S{l+1,...,n}.
Let Yl,n−l be the set of distinguished left coset representatives of Sl,n−l in Sn. We have the
parabolic subalgebra Hl,n−l = 〈Tσ | σ ∈ Sl,n−l〉A ⊆ Hn.
We denote by 6L,l the Kazhdan–Lusztig (left) pre-order relation on Sl,n−l and by ∼L,l the cor-
responding equivalence relation. The symbols 6R,l, 6LR,l, ∼R,l and ∼LR,l have a similar meaning.
Furthermore, as in [3, §4], we set a0 = 1 and
al := t(s1t)(s2s1t) · · · (sl−1sl−2 · · · s1t) for l > 0.
Then, by [3, Prop. 4.4], the set Yl,n−lal is precisely the set of distinguished left coset represen-
tatives of Sn in Wn whose t-length equals l. Furthermore, every element w ∈ Wn has a unique
decomposition
w = awalσwb
w where l = ℓt(w), σw ∈ Sl,n−l and aw, bw ∈ Yl,n−l;
see [3, 4.6]. We call this the Clifford normal form of w.
Theorem 2.1 (Bonnafé–Iancu [3, §7]). Assume that we are in the “asymptotic case” defined above.
Let x, y ∈Wn. Then x ∼L y if and only if l := ℓt(x) = ℓt(y), bx = by and σx ∼L,l σy.
Example 2.2. Let l ∈ {0, . . . , n} and C be a left cell of Sl,n−l. Since this group is a direct product,
we can write C = C1 · C2 where C1 is a left cell in S{1,...,l} and C2 is a left cell in S{l+1,...,n}. Now
Theorem 2.1 implies that
(a) Yl,n−l al C is a left cell of Wn (in the “asymptotic case”).
Now recall from [3, 4.1] that al = wlσl where wl is the longest element of the parabolic subgroup
Wl = 〈t, s1, . . . , sl−1〉 (of type Bl) and σl is the longest element of S{1,...,l}. Since wl is central in
Wl and conjugation with σl preserves the left cells of S{1,...,l}, we conclude that alC1al is a left cell
of S{1,...,l}, too. Furthermore, al commutes with all elements of S{l+1,...,n} and so alCal is a left
cell of Sl,n−l. Applying (a) now yields that
(b) Yl,n−l C al is a left cell of Wn (in the “asymptotic case”).
This example will be useful in the proof of Proposition 2.6 below.
2.C. Bitableaux.
Let Λn be the set of all bipartitions of n. We write such a bipartition in the form λ = (λ1|λ2)
where λ1 and λ2 are partitions such that |λ1| + |λ2| = n. For λ ∈ Λn, let T(λ) be the set of all
standard λ-bitableaux. (Whenever we speak of bitableaux, it is understood that the filling is by
the numbers 1, . . . , n.) The generalized Robinson–Schensted correspondence of [3] is a bijection
T(λ)× T(λ), w 7→ (P (w), Q(w)).
Thus, to each element w ∈Wn, we associate a pair of λ-bitableux for some λ ∈ Λn; in this case, we
also write w  λ and say that w is type λ.
The following result provides an explicit combinatorial description of the left, right and two-sided
cells in the “asymptotic case” in type Bn.
Theorem 2.3. Assume we are in the “asymptotic case” defined in §2.B. Let x, y ∈Wn.
Specht modules and Kazhdan–Lusztig cells 5
(a) (Bonnafé–Iancu [3, §7]) We have x ∼L y if and only if Q(x) = Q(y). Furthermore, x ∼R y
if and only if P (x) = P (y).
(b) (Bonnafé [4, §3 ]) We have x ∼LR y if and only if all of P (x), P (y), Q(x) and Q(y) have
the same shape.
Now let C be a left cell of Wn. We shall say that C is of type λ ∈ Λn if the bitableaux Q(x)
(where x ∈ C) have shape λ.
Theorem 2.4 (Geck [8, Theorem 6.3]). Let C and C′ be left cells of Wn (in the “asymptotic
case”) which have the same type λ ∈ Λn. Then the left cell modules [C]A and [C
′]A are canonically
isomorphic. In fact, there is a bijection C↔ C′ which induces an Hn-module isomorphism [C]A
[C′]A.
The above results show that, in order to study the left cell modules of Hn, it is sufficient to
exhibit one particular left cell of type λ, for each given λ ∈ Λn. For this purpose, we shall need
some further combinatorial notions from Dipper–James–Murphy [6, §3].
So let us fix a bipartition λ = (λ1|λ2) ∈ Λn, where l = |λ1| and 0 6 l 6 n. Let t
λ be the
“canonical” standard bitableau of shape λ defined in [6, p. 508]. Thus, tλ is a pair consisting of the
“canonical” λ1-tableau t
λ1 (obtaining by filling the rows in order from left to right by the numbers
1, . . . , l) and the “canonical” λ2-tableau t
′λ2 (obtained by filling the rows in order from left to right
by the numbers l + 1, . . . , n).
The symmetric group Sn acts (on the left) on bitableaux by permuting the entries. If t is any
bitableau of shape λ, denote by d(t) the unique element of Sn which sends t
λ to t. Thus, we have
d(t).tλ = t for any λ-bitableau t. Now let Tr(λ) denote the set of all row-standard λ-bitableaux.
Y λ := {d(t) | t ∈ Tr(λ)}
is the set of distinguished left coset representatives of the parabolic subgroup Sλ in Sn; see [6,
p. 509]. Applying this to the bipartition ((l), (n − l)), we find that
Yl,n−l = Y
((l),(n−l)).
Now we also define Trl (λ) to be the set of all t = (t1|t2) ∈ T
r(λ) where t1 is filled by the numbers
1, . . . , l and t2 is filled by the numbers l + 1, . . . , n. Then, by the same argument as above,
Y λl := {d(t) | t ∈ T
l (λ)}
is the set of distinguished left coset representatives of the parabolic subgroup Sλ inside Sl,n−l.
Hence, considering the chain of parabolic subgroups Sλ ⊆ Sl,n−l ⊆ Sn, we obtain a decomposition
Y λ = Yl,n−l · Y
where ℓ(yd(t)) = ℓ(y) + ℓ(d(t)) for all y ∈ Yl,n−l and t ∈ T
l (λ).
Now we have the following purely combinatorial result.
Lemma 2.5. In the above setting, let s ∈ Tr(λ), t ∈ Trl (λ) and y ∈ Yl,n−l be such that d(s) = y d(t).
Then s is a standard bitableau if and only if t is a standard bitableau.
Proof. We have s = d(s).tλ = (yd(t)).tλ = y.(d(t).tλ) = y.t. The permutation y ∈ Yl,n−l has the
property that y(i) < y(i + 1) for 1 6 i < l and y(i) < y(i + 1) for l 6 i < n. Now it is an
easy combinatorial exercise to see that s is standard if and only if t is standard; we omit further
details. �
6 Geck, Iancu and Pallikaros
Proposition 2.6. Let λ = (λ1|λ2) ∈ Λn and l = |λ1|. Let σλ ∈ Sλ be the longest element and
Cλ be the left cell (with respect to the “asymptotic case”) containing σλal ∈Wn. Then Cλ has type
(λ∗2|λ1) and we have
Cλ = {d(t)σλ al | t ∈ T(λ)},
where ℓ(d(t)σλal) = ℓ(d(t)) + ℓ(σλal) for all t ∈ T(λ).
Proof. By relation (♠) in the proof of [10, Prop. 5.4], the element alσλ has type (λ
2|λ1). Now since
σλal = (alσλ)
−1 it follows that σλal also has type (λ
2|λ1). Now, by [15, Lemma 3.3] (extended to
the direct product of two symmetric groups), the set
C := {d(t)σλ | t ∈ Tl(λ)}
is the left cell of Sl,n−l containing σλ, where Tl(λ) is the set of all standard λ-bitableaux in T
l (λ).
Hence, by Example 2.2(b), we have
Cλ = {y d(t)σλ al | y ∈ Yl,n−l, t ∈ Tl(λ)}.
Furthermore, ℓ(y d(t)σλ al) = ℓ(y d(t)) + ℓ(σλ al). Now it remains to use Lemma 2.5. �
Remark 2.7. In the above setting, it is not difficult to prove the following related result. Let x ∈Wn
and l := ℓt(x). Then we have:
x 6L σλal ⇐⇒ x = d(s)σλal where s is a row-standard λ-bitableaux.
This follows from the properties of the Clifford normal form of the elements in Wn established in [3,
§7] and the refinement obtained in [8, Theorem 5.11]. As we do not need this result in this paper,
we omit further details.
3. Specht modules
We keep the setup of the previous section, where we consider the Iwahori–Hecke algebra Hn of
type Bn, defined over a polynomial ring A = Z[V
±1, v±1] in two independent indeterminates. We
now consider the Specht modules defined by Dipper–James–Murphy [6]. The definition is based on
the construction of a new basis of Hn, which is of the form
{xst = Td(s) xλ Td(t)−1 | λ ∈ Λn and s, t ∈ T(λ)}
where the element xλ is defined in [6, 4.1]; note that the definition of xλ does not rely on the choice
of a total order on Γ. (An explicit description of xλ will be given in Lemma 3.2 below.)
Let Nλ ⊆ Hn be the A-submodule spanned by all xst where s and t are standard µ-bitableaux
such that λ E µ. Here, E denotes the dominance order on bipartitions, which is defined as follows;
see Dipper–James–Murphy [6, §3]: Let λ = (λ1|λ2) and µ = (µ1|µ2) be bipartitions of n, with parts
λ1 = (λ
1 > λ
1 > · · · > 0), λ2 = (λ
2 > λ
2 > · · · > 0),
µ1 = (µ
1 > µ
1 > · · · > 0), µ2 = (µ
2 > µ
2 > · · · > 0).
Then λ E µ if
1 (∀j) and |λ1|+
2 6 |µ1|+
2 (∀j).
By [6, Cor. 4.13], Nλ is a two-sided ideal of Hn. Since the basis elements Tw (w ∈Wn) are invertible
in Hn, we conclude that
µ∈Λn;λEµ
HnxµHn.
Specht modules and Kazhdan–Lusztig cells 7
Similarly, we have the two-sided ideal N̂λ spanned by all xst where s and t are standard µ-bitableaux
such that λ ⊳ µ (that is, λ E µ but λ 6= µ).
Definition 3.1 (Dipper–James–Murphy [6, Def. 4.19]). Let λ ∈ Λn. The corresponding Specht
module is defined by
S̃λ := Mλ/(Mλ ∩ N̂λ) where Mλ = Hnxλ.
By [6, Theorem 4.20], S̃λ is free over A, with standard basis {xs | s ∈ T(λ)} where xs denotes the
class modulo Mλ ∩ N̂λ of the element x
Our task will be to identify these Specht modules with certain Kazhdan–Lusztig left cells modules.
For this purpose, assume from now on that we have chosen a total order on Γ such that we are in
the “asymptotic case” defined in §2.B. Our first result, which is based on Bonnafé [4], identifies xλ
in terms of the corresponding Kazhdan–Lusztig basis of Hn.
Lemma 3.2. Let λ = (λ1|λ2) ∈ Λn and l = |λ1|. Then
V lvl(l−1)−ℓ(σλ) xλ = Tσl Calσλ = CσλalTσl ,
where the elements σl, al and σλ are defined in §2.
Proof. In [6, 4.1], the element xλ is defined as the product of three commuting factors u
l , xλ1 , xλ2 .
Bonnafé’s formula [4, Prop. 2.5] shows that
V lvl(l−1) u+l = CalTσl = TσlCal .
Furthermore, by Lusztig [14, Cor. 12.2], we have xλ1xλ2 = v
ℓ(σλ)Cσλ . Finally, by [4, Prop. 2.3], we
have CalCσλ = Calσλ and CσλCal = Cσλal . This yields the desired formulas. �
Corollary 3.3. Let λ ∈ Λn. Then M
λ = HnCalσλ = HnCσλalTσl .
Proof. Clear by Lemma 3.2; just note v, V and Tσl are invertible in Hn. �
Proposition 3.4. Let λ ∈ Λn. Then we have
Nλ = 〈Cy (y ∈Wn) | y  (ν1|ν2) where (λ1|λ2) E (ν2|ν
1 )〉A(a)
⊇ 〈Cy (y ∈Wn) | y 6LR alσλ〉A,
N̂λ = 〈Cy (y ∈Wn) | Cy ∈ N
λ and y 6∼LR alσλ〉A.(b)
Proof. (a) The equality is proved in [10, Theorem 1.5]. Now let y ∈ Wn be such that y 6LR alσλ.
Assume that y  (µ1|µ2). Then Proposition 2.6 and [10, Prop. 5.4] show that (µ1|µ
2) E (λ
or, equivalently, (λ1|λ2) E (µ2|µ
1). Thus, we have Cy ∈ N
λ, as required.
(b) Since N̂λ is the sum of all Nµ where µ ∈ Λn and λ ⊳ µ, the equality in (a) also implies that
N̂λ = 〈Cy (y ∈Wn) | y  (ν1|ν2) where (λ1|λ2) ⊳ (ν2|ν
1)〉A.
So (b) follows from the description of the two-sided cells in Theorem 2.3(b). �
Now we are ready to construct a canonical homomorphism from a Specht module to a certain
Kazhdan–Lusztig cell module.
Lemma 3.5. Let λ = (λ1|λ2) ∈ Λn and l = |λ1|. Let Cλ be the left cell of Wn containing σλal
(with respect to the “asymptotic case”); see Proposition 2.6. Then there is a unique Hn-module
homomorphism ϕλ : S̃
λ → [Cλ]A which sends the class of xλ ∈M
λ in S̃λ to the class of Cσλal ∈ Iλ
in [Cλ]A.
8 Geck, Iancu and Pallikaros
Proof. Recall that [Cλ]A = Iλ/Îλ, where
Iλ = 〈Cy | y ∈Wn such that y 6L σλal〉A,
Îλ = 〈Cy | y ∈Wn such that y 6L σλal and y 6∈ Cλ〉A.
We define ζλ := V
−(l−1) vℓ(σλ)−l(l−1) T−1σl ∈ Hn. (Note that any element of the T -basis is invertible
in Hn.) Then the map
ρλ : Hn →Hn, h 7→ h ζλ,
(that is, right multiplication by ζλ) is a left Hn-module isomorphism. By Lemma 3.2, Corollary 3.3
and the definition of 6L, we have
ρλ(xλ) = Cσλal and ρλ(M
λ) = HnCσλal ⊆ Iλ.
Now, by Proposition 3.4, we certainly have Iλ ∩ N̂
λ ⊆ Îλ and so
λ ∩ N̂λ) ⊆ HnCσλal ∩ N̂
λ ⊆ Iλ ∩ N̂
λ ⊆ Îλ.
Hence, recalling also that S̃λ = Mλ/Mλ∩ N̂λ, we obtain a well-defined Hn-module homomorphism
ϕλ : S̃
λ → [Cλ]A, m+ (M
λ ∩ N̂λ) 7→ mζλ + Îλ,
having the desired properties. The unicity of ϕλ is clear since S̃
λ is generated, as an Hn-module,
by the class of xλ. �
Next, we would like to obtain more detailed information about the matrix of ϕλ : S̃
λ → [Cλ]A
with respect to the standard bases of the two modules. The aim will be to show that this matrix
is triangular with 1 on the diagonal; in particular, this will show that ϕλ is an isomorphism.
Recall that the Specht module S̃λ has a standard basis {xs | s ∈ T(λ)}; see Definition 3.1. On
the other hand, by the definition of cell modules and Proposition 2.6, [Cλ]A has a standard basis
{ed(s)σλal | s ∈ T(λ)} where ed(s)σλal denotes the class modulo Îλ of the element Cd(s)σλal ∈ Iλ. So,
for any t ∈ T(λ), we write
ϕλ(xt) =
s∈T(λ)
gs,t ed(s)σλal where gs,t ∈ A.
Thus, Gλ :=
s,t∈T(λ)
is the matrix of ϕλ with respect to the standard bases of S̃
λ and [Cλ]A,
respectively. Now we can state the main result of this paper.
Theorem 3.6. The map ϕλ : S̃
λ → [Cλ]A constructed in Lemma 3.5 is an isomorphism. More
precisely, the following hold. For any s, t ∈ T(λ), we have
gt,t = 1 for all t ∈ T(λ),
gs,t = 0 unless d(s) 6 d(t),
gs,t ∈ v
Z[v−1] if s 6= t;
here, 6 denotes the Bruhat–Chevalley order. Thus, the matrix Gλ has an upper unitriangular shape
for a suitable ordering of the set T(λ).
Proof. We begin with the following computation inside the parabolic subgroup Sn ⊆ Wn. Let
t ∈ Tr(λ). By the multiplication rules for the Kazhdan–Lusztig basis, Td(t)Cσλ equals Cd(t)σλ plus
a Z[v, v−1]-linear combination of terms Cx where x ∈ Sn, x 6L,n σλ and x < d(t)σλ. Now, the
condition x 6L,n σλ implies that x can be written as x = d(s)σλ for some s ∈ T
r(λ) (see, for
Specht modules and Kazhdan–Lusztig cells 9
example, [15, 2.9]). Then the condition x = d(s)σλ < d(t)σλ implies that d(s) < d(t) (see [14,
9.10(f)]). Thus, we obtain
(∗) Td(t) Cσλ =
s∈Tr(λ)
s,tCd(s)σλ for any t ∈ T
r(λ),
where g′
s,t ∈ Z[v, v
−1] for all s, t ∈ Tr(λ); furthermore, g′
t,t = 1 and g
s,t = 0 unless d(s) 6 d(t) and
d(s)σλ 6L,n d(t)σλ.
To pass from Sn to Wn, we use the following argument. First note that al is a distinguished
right coset representative of Sn in Wn. By [4, Prop. 2.3], we have Cσal = CσCal for any σ ∈ Sn.
Hence, multiplying (∗) on the right by Cal , we obtain
Td(t) Cσλal =
s∈Tr(λ)
s,tCd(s)σλal for any t ∈ T
r(λ).
Now assume that t ∈ T(λ). Let s ∈ Tr(λ) be such that g′
s,t 6= 0. Then d(s)σλ 6L,n d(t)σλ and so
d(s)σλal 6L d(t)σλal; see [14, Prop. 9.11]. Hence, using Proposition 2.6, we find that
Td(t) Cσλal ≡
s∈T(λ)
s,tCd(s)σλal mod Îλ.
Passing to the quotient Iλ → [Cλ]A = Iλ/Îλ, we obtain
Td(t).eσλal =
s∈T(λ)
s,t ed(s) σλal for any t ∈ T(λ).
Now note that ϕλ(xt) = ϕλ(Td(t).x̄λ) = Td(t).ϕλ(x̄λ) = Td(t).eσλal , where x̄λ denotes the class of xλ
in S̃λ. Thus, we see that g′
s,t = gs,t for all s, t ∈ T(λ). Consequently, the coefficients gs,t have the
property that gt,t = 1 and gs,t = 0 unless d(s) 6 d(t). Hence, for a suitable ordering of the rows
and columns, the matrix Gλ is unitriangular and ϕλ is an isomorphism.
It remains to prove that gs,t ∈ v
Z[v−1] for s 6= t. We will actually show that g′
s,t ∈ v
Z[v−1]
for all s, t ∈ Tr(λ) such that s 6= t. This is seen as follows. We can invert the equations (∗) and
obtain
Cd(t)σλ =
s∈Tr(λ)
g̃s,tTd(s) Cσλ for any t ∈ T
r(λ),
where the g̃s,t’s are the entries of the inverse of the matrix
s,t∈Tr(λ)
. A comparison with [7,
Prop. 3.3] shows that
g̃s,t = p
d(s)σλ,d(t)σλ
∈ v−1Z[v−1] if s 6= t.
Hence we also have g′
s,t ∈ v
Z[v−1] for s 6= t. �
Remark 3.7. Let λ ∈ Λn and C be any left cell such that C and Cλ are contained in the same
two-sided cell. Then, by Theorem 2.4, [C]A and [Cλ]A are canonically isomorphic as Hn-modules.
Hence, in combination with Theorem 3.6, we conclude that S̃λ ∼= [C]A. Thus, any left cell module
of Hn is canonically isomorphic to a Specht module.
Remark 3.8. The above results also hold for specialized algebras. More precisely, let R be any
commutative ring (with 1) and fix two invertible elements Q, q ∈ R which admit square roots Q1/2
and q1/2 in R. Then we have a unique ring homomorphism θ : A→ R such that θ(V ) = Q1/2 and
θ(v) = q1/2. We can extend scalars from A to R and set
Hn,R = R⊗A Hn, S̃
R = R⊗A S̃
λ, [C]R = R⊗A [C]A,
10 Geck, Iancu and Pallikaros
for any λ ∈ Λn and any left cell C of Wn. Then S̃
R precisely is the Specht module of the algebra
of type Bn with parameters Q, q, as considered by Dipper–James–Murphy [6]. By Theorem 3.6
and Remark 3.7, we have an induced canonical isomorphism ϕ̃R : S̃
−→ [C]R whenever C is in the
same two-sided cell as Cλ.
4. Counterexample
Recall that Hn is defined over the ring of Laurent polynomials A = Z[V
±1, v±1] in two indepen-
dent indeterminates. In the previous sections, we considered the Kazhdan–Lusztig cell modules
of Hn with respect to the “asymptotic case” [3], that is, assuming that the group of monomials
{V ivj | i, j ∈ Z} is endowed with the pure lexicographic order where V ivj < vi
if and only
if i < i′ or i = i′ and j < j′. But there are many other monomial orders, each giving rise to a
Kazhdan–Lusztig basis of Hn and corresponding cell modules.
The aim of this section is to show that, in general, the Dipper–James–Murphy Specht modules
S̃λ cannot be identified with cell modules for these other choices of a monomial order. We do
this in two ways: (1) by a concrete example in type B3 and (2) by a general argument involving
non-semisimple specialisations of Hn.
4.A. An example in type B3.
Let n = 3; then W3 = 〈t, s1, s2〉. Let λ = ((1), (2)) ∈ Λ3 and consider the corresponding Specht
module S̃λ. By Theorem 3.6, it is isomorphic to [Cλ]A, where Cλ is a left cell with respect to the
“asymptotic case”. We have l = 1 and σλal = s2t. Using Proposition 2.6, we find that
Cλ = {s2t, s1s2t, s2s1s2t}.
The corresponding left cell representation ρλ : H3 →M3(A) is given by
Tt 7→
V V v−1V −1v V v−2 + V −1v2
0 −V −1 0
0 0 −V −1
Ts1 7→
−v−1 0 0
1 v 0
0 0 v
 , Ts2 7→
v 1 0
0 −v−1 0
0 1 v
Now let us choose a different monomial order on {V ivj | i, j ∈ Z}, namely, the weighted lexico-
graphic order where
V ivj < V i
′ def
⇐⇒ i+ j < i′ + j′ or i+ j = i′ + j′ and i < i′.
(In particular, we have v < V < v2.) By an explicit computation, one can show that, in this
case, the Kazhdan–Lusztig cell modules are all irreducible over K (the field of fractions of A), in
accordance with [2, Conjecture A+]. (We are in the case r = 1 of that conjecture.) Furthermore,
there are precisely three left cells C1,C2,C3 such that [Ci]K ∼= S̃
K ; they are given as follows:
C1 = {s1s2s1, s1ts1s2s1, ts1s2s1},
C2 = {s1s2s1t, s1ts1s2s1t, ts1s2s1t},
C3 = {s1s2s1ts1, s1ts1s2s1ts1, ts1s2s1ts1}.
Specht modules and Kazhdan–Lusztig cells 11
The corresponding left cell representations are all identical and given by:
Tt 7→
−V −1 0 0
0 −V −1 0
1 V v−1 + V −1v V
Ts1 7→
v 0 0
0 v 1
0 0 −v−1
 , Ts2 7→
v V v−2 + V −1v2 0
0 −v−1 0
0 1 v
Denote this representation by ρ : H3 → M3(A). Now one checks that Pρλ(Ts) = ρ(Ts)P for
s ∈ {t, s1, s2} where
0 0 V v−2 + V −1v2
0 1 0
1 0 0
Thus, P defines a non-trivial module homomorphism between ρλ and ρ. Since these representations
are irreducible over K, the matrix P is uniquely determined up to scalar multiples. But we see
that there is no scalar λ ∈ K such that λP ∈ M3(A) and det(λP ) ∈ A
×. Hence, S̃λ will not be
isomorphic to any Kazhdan–Lusztig cell module with respect to the above weighted lexicographic
order.
4.B. General cell modules.
Let k be a field and fix an element ξ ∈ k×. Let a, b ∈ Z>0 and consider the specialisation A→ k
such that V 7→ ξb and v 7→ ξa. Let Hn,k = k ⊗A Hn be the corresponding specialized algebra. As
in Remark 3.8, we also have corresponding Specht modules S̃λk for Hn,k. Now, for each λ ∈ Λn,
there is a certain Hn,k-invariant bilinear form φλ : S̃
k × S̃
k → k; see [6, §5]. Let rad(φλ) be the
radical of that form and set Dλ = S̃λk/rad(φλ). Then D
λ is either 0 or an absolutely irreducible
Hn,k-module; furthermore, we have
Irr(Hn,k) = {D
µ | µ ∈ Λ♣} where Λ♣ = {λ ∈ Λ | Dλ 6= 0};
see Dipper–James–Murphy [6, Theorem 6.6]. The conjecture in [6, 8.13] about an explicit combi-
natorial description of Λ♣ has recently been proved by Ariki–Jacon [1].
Now consider the Kazhdan–Lusztig basis {Cw} of Hn,k with respect to the weight function
L : Wn → Z such that L(t) = b and L(si) = a for all i. Assume that Lusztig’s conjectures (P1)–
(P15) in [14, 14.2] on Hecke algebras with unequal parameters hold. (This is the case, for example,
in the “equal parameter case” where a = b; see [14, Chap. 15].) Using these properties, it is shown
in [9] that Hn,k has a natural “cellular structure” in the sense of Graham–Lehrer [12]. The elements
of the “cellular basis” are certain linear combinations of the basis elements {Cw}. Then, by the
general theory of “cellular algebras”, for any λ ∈ Λn, we have a “cell module” Wk(λ) for Hn,k and
this cell module is naturally equipped with an Hn,k-invariant bilinear form gλ : Wk(λ)×Wk(λ)→ k.
Let rad(gλ) be the radical of that form and set L
λ = Wk(λ)/rad(gλ). Then, again, L
λ is either 0
or an absolutely irreducible Hn,k-module; furthermore, we have
Irr(Hn,k) = {L
µ | µ ∈ Λ♠} where Λ♠ = {λ ∈ Λ | Lλ 6= 0};
see Graham–Lehrer [12, §3] and [9, Example 4.4].
Given these two settings, it is natural to ask if S̃λk
∼= Wk(λ) and, subsequently, if Λ
♣ = Λ♠ ? In
the case where Hn,k is semisimple, it is shown in [9, Example 4.4] that S̃
∼= Wk(λ); furthermore,
by the general theory of “cellular algebras” [12] and the results in [6], we have Λ♣ = Λ♠ = Λ in
12 Geck, Iancu and Pallikaros
this case. However, if Hn,k is not semisimple, then the answer to these questions is negative, as
can be seen from the fact that Λ♣ 6= Λ♠ in general; see [11] and the references there.
By [11, Theorem 2.8] it is true, however, that Λ♣ = Λ♠ if b > (n − 1)a > 0 which corresponds
precisely to the “asymptotic case” discussed in this paper. Indeed, by [8, Corollary 6.3], the basis
{Cw} of Hn,k is cellular under this assumption on a, b, and by Theorem 3.6, we have Wk(λ) ∼= S̃
Acknowledgements. The final form of this paper grew out of several discussions which the three authors
could hold thanks to the hospitality of various institutions. Part of this work was done while all three authors
enjoyed the hospitality of the Bernoulli Center at the EPFL Lausanne (Switzerland) in 2005. C.P. thanks
the University of Aberdeen (Scotland) for an invitation in November 2006; M.G. and L.I. would like to thank
the University of Cyprus at Nicosia for an invitation in March 2007.
References
[1] S. Ariki and N. Jacon, Dipper–James–Murphy’s conjecture for Hecke algebras of type B, preprint; available
at math.RT/0703447.
[2] C. Bonnafé, M. Geck, L. Iancu and T. Lam, On domino insertion and Kazhdan–Lusztig cells in type Bn,
preprint; available at math.RT/0609279.
[3] C. Bonnafé and L. Iancu, Left cells in type Bn with unequal parameters, Represent. Theory 7 (2003),
587–609.
[4] C. Bonnafé, Two-sided cells in type B in the asymptotic case, J. Algebra 304 (2006), 216–236.
[5] R. Dipper and G. D. James, Representations of Hecke algebras of general linear groups, Proc. London Math.
Soc. 52 (1986), 20–52.
[6] R. Dipper, G. D. James and G. E. Murphy, Hecke algebras of type Bn at roots of unity, Proc. London
Math. Soc. 70 (1995), 505–528.
[7] M. Geck, On the induction of Kazhdan–Lusztig cells, Bull. London Math. Soc. 35 (2003), 608–614.
[8] M. Geck, Relative Kazhdan–Lusztig cells, Represent. Theory 10 (2006), 481–524.
[9] M. Geck, Hecke algebras of finite type are cellular, Invent. Math. (2007), to appear.
[10] M. Geck and L. Iancu, Lusztig’s a-function in type Bn in the asymptotic case. Special issue celebrating the
60th birthday of George Lusztig, Nagoya J. Math. 182 (2006), 199–240.
[11] M. Geck and N. Jacon, Canonical basic sets in type B. Special issue in honour of Gordon Douglas James,
J. Algebra 306 (2006), 104–127.
[12] J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), 1–34.
[13] G. Lusztig, Left cells in Weyl groups, Lie Group Representations, I (R. L. R. Herb and J. Rosenberg, eds.),
Lecture Notes in Math., vol. 1024, Springer-Verlag, 1983, pp. 99–111.
[14] G. Lusztig, Hecke algebras with unequal parameters, CRM Monographs Ser. 18, Amer. Math. Soc., Provi-
dence, RI, 2003.
[15] T. P. McDonough and C. A. Pallikaros, On relations between the classical and the Kazhdan–Lusztig
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133–144.
M.G. and L.I.: Department of Mathematical Sciences, King’s College, Aberdeen University, Ab-
erdeen AB24 3UE, Scotland, U.K.
E-mail address: m.geck@maths.abdn.ac.uk
E-mail address: l.iancu@maths.abdn.ac.uk
C.P.: Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678
Nicosia, Cyprus
E-mail address: pallikar@ucy.ac.cy
http://arxiv.org/abs/math/0703447
http://arxiv.org/abs/math/0609279
	1. Introduction
	2. Kazhdan–Lusztig bases and cells
	2.A. Basic definitions
	2.B. The ``asymptotic case'' in type Bn
	2.C. Bitableaux
	3. Specht modules
	4. Counterexample
	4.A. An example in type B3
	4.B. General cell modules
	References
