Table Of ContentModular Representations of
Hecke Algebras and Related Algebras
by
John J. Graham
A thesis submitted in fulfilment of the requirements for the degree of
Doctor of Philosophy in Pure Mathematics at the University of Sydney.
September 1995
ii
Abstract
The representations of various specialisations of Iwahori-Hecke algebras
and related algebras are studied in the context of “Cellular algebras”. The
latter are associative algebras with a special basis. The thesis contains a thor-
ough study of the representations of a cellular algebra over a field, including
a simple construction for the irreducibles. The motivating examples are the
Hecke algebra of type A with the Kazhdan-Lusztig basis and the Temperley-
Lieb algebra. For certain parameters, the Hecke algebra of TypeB is shown to
be cellular, using the “φ-weighted” Kazhdan-Lusztig basis defined by Lusztig
in[L3]. Theproofmakesuseofa“relative” formulationofLusztig’safunction,
defined with respect to a parabolic subgroup. Another class of algebras which
are shown to be cellular is the class of “generalised Temperley-Lieb algebras”,
which are defined as canonical quotients of the Hecke algebras. These may
be finite dimensional even when the Hecke algebra is infinite dimensional, and
we classify the finite dimensional ones completely. They are examples of a
still wider class of algebras called “projection algebras”, many of which have a
cellular structure, and whose representation theory may therefore bediscussed
in this context.
iii
Introduction
The motivation for this thesis is a systematic understanding of the repre-
sentations of (specialisations of) an Iwahori-Hecke algebra. We concentrate on
the non-semisimple specialisations over a field. Their study is related to the
modular representation theory of reductive algebraic groups over finite fields.
(See [DJ3].)
Let A be a commutative ring with identity. A “cellular algebra” (See 4.1)
is an associative A-algebra with a special basis, whose properties reflect the
structure of A as a left A-module. The axioms are modelled on the Hecke
algebra of type A with theKazhdan-Lusztig basis, the Temperley-Lieb algebra
and the cyclic algebra A[X]/f(X) where f(X) is a monic polynomial in A[X].
The datum defining a cellular algebra is essentially unchanged by specialisa-
tion. We study the representations of a cellular algebra over a field. A short
version of this study appears in [GL] together with applications to the Ariki-
Koike-Hecke algebra, the Brauer centraliser algebra and various subalgebras
of the latter defined by topological conditions. (This paper is joint work with
Gus Lehrer.) In this thesis, we treat the Hecke algebra of type B (for certain
parameters) with the “φ-weighted” Kazhdan-Lusztig basis defined in [L3] and
“projection algebras” in this setting.
The structure of the thesis is as follows. The goal of the first three chap-
ters is to exhibit the Hecke algebras of types A and B as cellular algebras.
Chapter one is an introduction to the Kazhdan-Lusztig basis of a Hecke alge-
bra. Well known results of [KL] are extended to the “φ-weighted” formulation
of Lusztig. We also present a “relative” a function whose definition depends
on a parabolic subgroup of the corresponding Hecke algebra. Apart from its
principal application in the proof of Theorem 3.2, this tool is also used to in-
vestigate the action of T˜ on A as a left A-module where w is the longest
wJ J
element of a parabolic subgroup. (This is completed in chapter five.) Chapter
two outlines well known combinatorics associated with the symmetric group
and the hyperoctahedral group (the Coxeter groups of types A and B). In
Chapter three, we determine the isomorphisms between cell representations.
The main result, Theorem 3.2, is the analogue for type B of Theorem 1.4 of
[KL].
A detailed analysis of the representations of a cellular algebra over a field
is carried out in Chapter four. With the exception of Proposition 4.12, the
iv
results from 4.1 to 4.22 appear in [GL]. We construct the irreducible modules
(4.17) and present a simple minded algorithm (4.28) to compute a basis for
the radical and a complete set of primitive idempotents. We also extend the
notions of blocks and decomposition numbers; these may also be computed in
a routine fashion.
The starting point for this research was a paper [DJ1] of Dipper and
James on the (modular) representations of the Hecke algebras of type A. In
chapter five, we sketch the relationship between their work and our approach
via cellular algebras. We also illustrate the usefulness of the Kazhdan-Lusztig
basis from a computational point of view, by computing the eigenvalues of the
action of T˜ on an irreducible module where x is a reflection.
x
The Hecke algebra of type B is a special case of the Ariki-Koike-Hecke
algebra. The cell datum constructed here is quite different from the one in
[GL]. Our treatment also differs from the recent work of Dipper, James and
Murphy [DJM]. In their language, our method applies only with parameters
Q = u2 or u6 and q = u4. However, this is precisely the case which arises in
applications to the study of modular representations of a unitary group over a
finite field.
A projection algebra (defined in chapter six) is an algebra analogue of a
Coxeter group. Under mild hypotheses, a basis is constructed and the struc-
turalcoefficients ofmultiplicationaredetermined. Inchapterseven,weclassify
projectionalgebras forwhichthisbasisisfinite. Thefinalchapterisdevoted to
the construction of cell data for certain finite dimensional projection algebras.
This makes use of an analogue of the Robinson-Schensted correspondence,
which is examined in chapter eight.
A Hecke algebra of arbitrary type possesses a canonical quotient which
we call a “generalised Temperley Lieb algebra” and which turns out to be a
projectionalgebra. Whenfinitedimensional(see7.1),ageneralisedTemperley-
Lieb algebra is cellular (9.7). Thus, one may study the representations of the
Hecke algebra which factor through this quotient algebra using the results of
chapterfour. ForaHecke algebrawithasimplylacedCoxetergraph,wesketch
the relationship between our cell datum for this projection algebra and the the
Kazhdan-Lusztig basis of the Hecke algebra. For type A, the quotient is the
Temperley-Lieb algebra whose representations are well known; see [GW]. The
relationship with the Kazhdan-Lusztig basis allows us to recover some results
of Lascoux and Schu¨tzenberger [LS].
v
Our work suggests some topics for further research. Is a Hecke algebra
associatedwithanarbitraryWeylgroup,cellular? Howistheintimategeomet-
ric relationship [L3] between the Hecke algebra of type A (resp. A ) and
2n 2n−1
the Hecke algebra of type B (with their weighted Kazhdan-Lusztig bases),
n
reflected in their modular representations? Find cellular structures for the q-
SchuralgebraandtheBirman-Wenzlalgebra. Extendtheframeworkofchapter
four to treat the modular representations of Quantum groups.
This thesis contains no material which has been accepted for the award
of another degree or diploma at any university. The material presented here
is believed original, with the exception of the results of chapters one and two
and other cases where due attribution is given.
Acknowledgements
I would like to thank my supervisorGus Lehrer for hisfriendly advice and
support in all aspects of my candidature. I appreciate the time that he has
devoted to this project. The maturation of these ideas has been influenced by
numerous discussions. I am particularly indebted to Jie Du, Gerhard Ro¨hrle
andAndrewMathas. Aspecialthanksis duetoBrigitte BrinkandGusLehrer
for their comments on drafts of the manuscript. I also thank King-Fai Lai and
Karl Wehrhahn for help with some background material. I am grateful for
the encouragement of my friends and family. This work was partially sup-
ported by an Australian Postgraduate Research Award and the A.R.C. grant
for the project “Group Representation Theory and Cohomology of Algebraic
Varieties” at the University of Sydney (No. A69330390). This document was
typeset by AMS-TEX
vi
Contents
Chapter Page
1. The Kazhdan-Lusztig Basis of a Hecke Algebra 1
2. The Symmetric and Hyperoctahedral Groups 11
3. Explicit Isomorphisms between Cell Representations 21
4. Representations of Cellular Algebras 31
5. The Hecke Algebra of Type A 46
6. Generalised Temperley-Lieb Algebras and Projection Algebras 52
7. Classification of Finite Dimensional Examples 74
8. An Analogue of the Robinson-Schensted Algorithm 88
9. A Cellular Structure for the Generalised Temperley-Lieb Algebras 100
References 113
1
Chapter 1
The Kazhdan-Lusztig Basis of a Hecke Algebra
In this thesis, ring means commutative ring with identity and algebra
means associative algebra with identity. We denote the integers by Z, the
positive integers by N and the non-negative integers by N .
0
Let W be a Coxeter group generated by a set R of simple reflections. A
reduced expression for y ∈ W is a sequence s ,s ,...,s of simple reflections
1 2 l
s ∈ R of minimal length l(y) = l such that the product s s ...s = y. (The
i 1 2 l
empty sequence is the reduced expression for the identity 1 of W.) An element
x ∈ W is less than y in the Bruhat (partial) order (denoted x ≤ y) iff there
is a subsequence of s ,s ,...,s which is a reduced expression for x. This is
1 2 l
independent of the choice of reduced expression for y [H, 5.10].
Let A be the ring Z[u,u−1] of Laurent polynomials in an indeterminate u.
There is a unique ring involution x7→ x¯ of A which takes u to its inverse. Let
φ: W → Z be a function such that φ(s) > 0 if s ∈ R and φ(xy) = φ(x)+φ(y)
if l(xy) = l(x)+l(y) (x,y ∈ W). (The length function is an example.) The
Hecke algebra H = H is the free A-module with basis {T˜ | y ∈ W},
W,R,φ y
identity 1 = T˜ and associative A-linear multiplication determined by
1
(T˜ +u−φ(s))(T˜ −uφ(s))=0 if s ∈ R, and
s s
T˜ =T˜ T˜ if l(xy) = l(x)+l(y).
xy x y
This basis is related to the more common {T | y ∈ W} found in [H] by the
y
equation T˜ = q−φ(y)/2T (y ∈ W,q = u2). The first equation shows that T˜ is
y y s
invertibleifs ∈ R. Thereisaringinvolutionh 7→ h¯ ofHwhichtakes a T˜
w w w
to a¯ T˜−1 . In the important case when φ is the length function,PKazhdan
w w w−1
P
and Lusztig [KL] construct a basis of H. In our study of the representations
of H, we require the following weighted version which is due to Lusztig.
Theorem 1.1. (Proposition 2, [L3]) Let H be a Hecke algebra as
W,R,φ
above. If y ∈ W, the ring involution h7→ h¯ of H fixes a unique element
C′ = P˜ T˜ (P˜ ∈ A),
y x,y y x,y
Xx≤y
whose coefficients satisfy P˜ = 1 and P˜ ∈u−1Z[u−1] if x <y.
y,y x,y
Ananti-involutionofanA-algebraisanA-linearfunctionA → A: x7→ x∗
satisfying (xy)∗ = y∗x∗ and (x∗)∗ = x for x,y ∈A.
2 Chapter1
Corollary 1. Let h 7→ h∗ be the anti-involution of H extending T˜ 7→
x
T˜x−1. If y ∈ W, then (Cy′)∗ = Cy′−1.
Proof. Theinvolutionsh7→ h∗ andh7→ h¯ commute. Hence(C′)∗ satisfies
y
the conditions which characterise C′ by 1.1.
y−1
The structure coefficients of left translation by the generators C′ = T˜ +
s s
u−φ(s) (s ∈ R) may be determined as follows. Let y ∈ W and s ∈ R such that
y < sy. For x∈ W such that sx< x <y defineLaurentpolynomials Ms ∈ A
x,y
inductively by Ms = Ms and
x,y x,y
P˜ Ms −uφ(s)P˜ ∈ u−1Z[u−1].
x,z z,y x,y
sx≤Xsz<z<y
When φ is the length function (and sx < x < y < sy), Ms is the coefficient
x,y
µ(x,y) of u−1 in P˜ ; see [KL].A key example is thecase whenl(y) = l(x)+1.
x,y
We obtain a reduced expression for xby deleting some reflection t in a reduced
expression for y. Underthese circumstances, Lusztighas shown(Proposition 5
of [L3]) that P˜ = u−φ(t) and
x,y
0 if φ(s) < φ(t),
(1.1.1) Ms = 1 if φ(s) = φ(t),
x,y
uφ(s)−φ(t) +uφ(t)−φ(s) if φ(s) > φ(t).
Proposition 1.2. If s ∈ R and y ∈W, then
C′ + Ms C′ if sy > y,
C′C′ = sy sx<x<y x,y x
s y (cid:26) (uφ(s)+Pu−φ(s))C′ if sy < y.
y
This is Proposition 4 of [L3].
It follows that the definition of Ms may be uniquely extended to all
x,y
triples in R×W ×W so that
(1.2.1) C′C′ = Ms C′ and C′C′ = Ms C′.
s y x,y x y s x−1,y−1 x
xX∈W xX∈W
We next present a modified form of Proposition 2.4 of [KL].
Corollary 1. Let x,y ∈ W and s ∈R such that Ms 6= 0.
x,y
(1) If r ∈ R and yr < y, then xr < x.
(2) Assume φ(s) = 1. If t ∈ R and ty < y, then tx < x unless x= sy or ty.
Proof.
(1) The proposition shows that (C′C′)C′ is a linear combination of terms
s y r
C′ such that zr < z. However the coefficient (uφ(r)+u−φ(r))Ms of C′ in the
z x,y x
product C′(C′C′), is nonzero.
s y r
TheKazhdan-Lusztig Basis 3
(2) Without loss of generality, assume sx < x < y. Under the assumption,
the recurrence for Ms reduces to the coefficient of u−1 in P˜ . The propo-
x,y x,y
sition shows that P˜ = u−φ(t)P˜ . Therefore the coefficient of uφ(t)−1 in
x,y tx,y
Laurent polynomial P˜ , is nonzero. It follows that φ(t) = 1, tx = y and
tx,y
Ms = 1.
x,y
The aim of this thesis is to find analogues of our next result, and to
examine applications to the study of (modular) representations of H. The
case when φ is the length function, is established in [KL].
Definition 1.3. LetW beaCoxetergroupwithsetRofsimplereflections
and φ : W → Z as above. Suppose s,t ∈ R such that st has order three. If
y ∈ W and sy < y < ty, then exactly one of sy and ty is an element ∗y ∈ W
such that t∗y < ∗y < s∗y. The pair (y,∗y) is called a Knuth relation of type
(s,t).
Proposition 1.4. Assume the notation and hypotheses of the previous
definition. Assume (φ(s) =)φ(t) = 1. If r ∈ R and x,y ∈ W are such that
sx < x < tx and sy < y < ty, then the coefficients defined by 1.2.1 satisfy
Mr = Mr .
x−1,y−1 (∗x)−1,(∗y)−1
Proof. The previous proposition ensures
Cs′Cy′ = (u+u−1)Cy′ and Ct′C∗′y = (u+u−1)C∗′y.
Secondly, the A-span I ⊆ H of {C′ | z ∈ W, sz,tz < z} is a right H-module.
z
Furthermore left multiplication by C′ or C′ takes I to itself. The corollary
s t
and Equation 1.1.1 show
Ct′Cy′ = C∗′y and Cs′C∗′y = Cy′ modulo I.
Therefore, if in addition r ∈ R and x ∈ W such that sx < x < tx, then when
C′C′C′ is expressed as a linear combination of the Kazhdan-Lusztig basis, the
t y r
coefficient of C∗′x is Mxr−1,y−1 = M(r∗x)−1,(∗y)−1.
The various symmetries of H yield related special bases. Because the A-
algebra involution θ : H → H : T˜ 7→ −T˜−1 commutes with h 7→ h¯, we have a
s s
basis C := (−1)l(y)θ(C′) indexed by y ∈ W such that
y y
(1.4.1) C = C¯ = (−1)l(y)−l(x)P˜ T˜ .
y y x,y x
Xx≤y
Assume W is finite. Then there exists a unique element w of maximal
length. In fact, l(w) = l(y)+l(wy) and x ≤ y ⇐⇒ wx ≥ wy if x,y ∈ W
4 Chapter1
[H, 5.6]. If α : W → W denotes conjugation by the involution w, we have
T˜ T˜ T˜−1 = T˜ . Therefore
w x w α(x)
T˜ C′T˜−1 = P˜ T˜
w y w x,y α(x)
xX≤y
satisfies the conditions characterising C′ (1.1). If z ∈ W, define
α(y)
(1.4.2) D′ := C T˜ = T˜ C = (−1)l(y)−l(z)P˜ T˜ .
z zw w w wz wy,wz y
Xz≤y
The Hecke algebra is a symmetric algebra with trace:
(1.4.3) τ : H → A: a T˜ 7→ a .
x x 1
Xx
The associated bilinear form hg,hi = τ(gh) on H is non-degenerate because
hT˜ ,T˜ i = δ , where δ is the Kronecker delta. Our next theorem shows that
y x yx,1
hD′,C′i= δ for z,v ∈ W.
z v zv,1
Theorem 1.5. Let W be a finite Coxeter group with longest element w
and H be its Hecke algebra. If x,y ∈ W, then
W,R,φ
(−1)l(y)−l(z)P˜ P˜ = δ
x,z wy,wz x,y
x≤Xz≤y
where δ is the Kronecker delta.
Proof. The case when φ is the length function, is Theorem 3.1 of [KL].
However its proof is valid for arbitrary φ.
Corollary 1. Assume the hypotheses and notation of the previous the-
orem. If s ∈ R and x,y ∈ W, then
Ms +(−1)l(y)−l(x)Ms = (uφ(s)+u−φ(s))δ .
x,y yw,xw x,y
Proof. The case when φ is the length function is Corollary 3.2 of [KL]. In
general, we evaluate both sides of
(1.5.1) hT˜wCwx−1Cs′,Cy′i= hT˜wCwx−1,Cs′Cy′i.
The right side of 1.5.1 is Ms by Proposition 1.2. Furthermore we have
x,y
Cwx−1Cs = − (−1)l(y)−l(x)Mysw,xwCwy−1.
Xy
As C′ = (uφ(s) +u−φ(s))+C , the left side of 1.5.1 is
s s
(uφ(s)+u−φ(s))δ −(−1)l(y)−l(x)Ms
x,y yw,xw
which yields the assertion.
