An isomorphism Theorem for Yokonuma--Hecke algebras and applications to link invariants

0.45 MB·English

by  N. Jacon

An isomorphism Theorem for Yokonuma–Hecke algebras
and applications to link invariants
N. Jacon and L. Poulain d’Andecy
5
1 Abstract
0
2 We develop several applications of the fact that the Yokonuma–Hecke algebra of the general linear group GL is
isomorphic to a direct sum of matrix algebras associated to Iwahori–Hecke algebras of type A. This includes a
t
c description of the semisimple and modular representation theory of the Yokonuma–Hecke algebras of GL and a
O
completeclassification ofalltheMarkovtracesforthem. Finally,fromtheseMarkovtraces, weconstruct3-variables
polynomials which are invariants for framed and classical knots and links, and investigate their properties with the
7
help of theisomorphism.

]
T
R 1 Introduction
.
h
t 1.1. The Yokonuma–Hecke algebras have been introduced by Yokonuma in [22] as centraliser algebras of
a
the permutation representation of Chevalley groups G with respect to a maximal unipotent subgroup of G.
m
TheyarethusparticularcasesofunipotentHeckealgebrasandtheyadmitanaturalbasisindexedbydouble
[
cosets (see [21] for more details on general unipotent Hecke algebras). For the Yokonuma–Hecke algebras,

3 thenaturaldescriptionhasbeentransformedintoasimplepresentationwithgeneratorsandrelations[9,11].
v Assumethatqisapowerofaprimenumberthen,fromthispresentation,onecanobservethattheYokonuma–
9 Hecke algebra of G=GL (F ) (sometimes called the Yokonuma–Hecke algebra of type A) is a deformation
n q
8
of the group algebra of the complex reflection group of type G(d,1,n), where d=q 1.
63 In this paper, we will consider the generic Yokonuma–Hecke algebras Yd,n (n −Z 0) depending on two
0 indeterminatesuandvandapositiveintegerd,overtheringC[u±1,v]. Thealgebra∈Yd,≥nisalsoadeformation
. of the groupalgebra of the complex reflection group of type G(d,1,n), for any d, and the Yokonuma–Hecke
1
0 algebra of GLn(Fq) is obtained by considering the specialization u2 =q, v =q 1 and the case d=q 1.
− −
5 There exist others well-known deformations of complex reflection groups of type G(d,1,n) which have
1 been intensively studied during the last past years : the Ariki–Koike algebras. These algebras turn out to
:
v haveadeeprepresentationtheory(inbothsemisimpleandmodularcases)whichisnowquitewell-understood
i (see for example [6] for an overview).
X
Generalizationsof Yokonuma–Heckealgebrasassociatedto Chevalley groupsappeared in the work of G.
r
a Lusztig on character sheaves [19]. In this work, it is proved that these algebras are isomorphic to certain
directsumsofmatrixalgebrasoverclassicalIwahori–Heckealgebras. FortheYokonuma–HeckealgebrasY
d,n
under considerationhere, it turns out that the Iwahori–Heckealgebrasappearingin the matrix algebrasare
all of type A.
Coherentlywiththisresult,thesetofsimplemodulesforY hasbeenexplicitlydescribedinthesemisim-
d,n
ple situation in combinatorialterms in [3] (see also [20] for general results on the semisimple representation
theory of unipotent Hecke algebras), and a criterion of semi simplicity has been deduced. In addition, a
certain symmetrizing form has been defined and the associated Schur elements (which control a part of
the representation theory of the algebra) have been calculated directly. They appear to have a particular
simple form, namely products of Schur elements of Iwahori–Hecke algebras of type A. This suggests an
interpretation of this symmetrizing form and its Schur elements with the help of the isomorphism.
In another way, a motivation for studying the Yokonuma–Hecke algebra comes from topology and more
precisely the theory of knot and link invariants. Indeed, the algebra Y is naturally a quotient of the
d,n
1
framed braid group algebra,and in turn can be used to search for isotopy invariants for framed links in the
same spirit as the Iwahori–Hecke algebra of type A is used to obtain an invariant for classical links (the
HOMFLYPT polynomial).
In [10], Juyumaya introduced on Y an analogue of the Ocneanu trace of the Iwahori–Heckealgebra of
d,n
type A. This trace was subsequently used by Juyumaya and Lambropoulou to produce isotopy invariants
for framed links [12, 15]. Remarkably, they also produced isotopy invariants for classical links and singular
links [13, 14]. Even though the obtained invariants for classical links are different from the HOMFLYPT
polynomial (excepted in some trivial cases), all the computed examples seem to indicate that the invariants
for classical links obtained from Y so far are topologically equivalent to the HOMFLYPT polynomial [2].
d,n
In fact, if we restrict to classical knots, such an equivalence has been announced in [1].
Again, a natural strategy is to seek for an explanation of this fact starting from the isomorphism con-
necting the algebra Y with Iwahori–Heckealgebras of type A.
d,n
1.2. Inthispaper,wegiveseveralanswersandresultsinbothdirections: therepresentationtheoryandthe
knots and links theory. After recalling several results and detailing the structure of the algebras (in Section
2),inthethirdsection,weprovideexplicitformulasforanisomorphism,overtheringC[u 1,v],betweenthe
±
Yokonuma–Hecke algebra Y and a direct sum of matrix algebras over tensor products of Iwahori–Hecke
d,n
algebras of type A. The direct sum is naturally indexed by the set of compositions of n with d parts. This
result is in fact a special case of a result by G. Lusztig [19, §34]. For the subsequent developments, it is
convenient for us to start from the presentation of Y adapted to framed links theory. Therefore, for later
d,n
use and in order to be self-contained, we give full details. The explicit formulas for the isomorphism (and
its inverse) will allow us to concretely translate questions and problems from one side to the other.
Then,wefirstdevelopinSection4theapplicationsoftheisomorphismtheoremconcerningrepresentation
theory. Indeed,theisomorphismcanberephrasedbysayingthattheYokonuma–HeckealgebraY isMorita
d,n
equivalent to a direct sum of tensor products of Iwahori–Hecke algebras of type A. As the result is valid
in the generic situation (over the ring C[u 1,v]), it passes to the specializations of the parameters u and
±
v. This implies that both the semisimple and the modular representation theories of Y can be deduced
d,n
from the corresponding ones of the Iwahori–Hecke algebra of type A, which are well-studied (see e.g. [7]).
In particular, the classification of simple modules of Y and the decomposition matrices (in characteristic
d,n
0) follow.
In addition, the isomorphism theorem provides a natural symmetrizing form on Y derived from the
d,n
canonical symmetrizing form of the Iwahori–Hecke algebra of type A. As a first application of the explicit
formulas,weshowthatthis symmetrizingformactuallycoincideswith thesymmetrizingformdefinedin[3],
which provides a direct proof and an explanation of the form of the Schur elements.
1.3. Another class of applications of the isomorphism theorem concerns the theory of classicaland framed
knots and links (Sections 5 and 6). Indeed, we obtain a complete classification of the Markov traces on
the family, on n, of the Yokonuma–Hecke algebras Y (Theorem 5.3). This is done in two steps. First we
d,n
translate, with the help of the isomorphism theorem, the Markov trace properties into properties of traces
on tensor products of Iwahori–Hecke algebras of type A; then we fully characterize these traces using the
knownuniquenessofthe Markovtraceonthe Iwahori–Heckealgebrasoftype A.Inparticular,weshowthat
all the Markov traces on Y are related with the unique Markov trace on the Iwahori–Hecke algebras of
d,n
type A.
We note thatwe use a differentdefinitionofa Markovtraceon Y than in[10,12,13, 14,15]. Inthere,
d,n
thestandardapproachinitiatedbyJonesforclassicallinkswasfollowed(see[8]andreferencestherein). The
first step is the construction on Y of an analogue of the Ocneanu trace by Juyumaya [10]. Additional
d,n
conditions were imposed in [10] in order to obtain the existence and unicity of this trace. Then, a rescaling
procedure is necessaryto construct invariants and, as it turned out, the trace does not rescale directly as in
the classical case. A non-trivial rescaling procedure was implemented by Juyumaya and Lambropoulou in
[13, 14, 15] by means of the so-called "E-system" and led to further restrictions on the parameters.
2
In the definition we use here for the Markov trace on Y , the imposed conditions are the minimal ones
d,n
allowingto obtainlink invariants,namely,the centralityandthe so-calledMarkovcondition(see Subsection
5.2). This will allow us to avoid any kind of rescaling procedure during the construction of invariants. This
approach is explained in [7, §4.5] in the classical setting of the Iwahori–Hecke algebras of type A.
With the definition used here, the space of Markov traces has a structure of C[u 1,v 1]-module. Our
± ±
approachviathe isomorphismtheoremprovidesadistinguishedbasis,whichisindexedbythe setofcompo-
sitions into d parts with all parts equal to 0 or 1. Thus the space of Markovtraces on the Yokonuma–Hecke
algebras has dimension 2d 1.
−
1.4. Finally, the last section is devoted to the construction and the study of invariants for classical links
and framed links. Following Juyumaya and Lambropoulou, we realize Y as a quotient of the framed
d,n
braidgroup. Actually we do more: we introduce a one-parameterfamily of homomorphisms from the group
algebra of the framed braid group to the algebra Y (deforming the canonical homomorphism). Then, for
d,n
each homomorphism of this family and each Markov trace on Y , we construct an invariant for classical
d,n
and framed links. As mentioned above, no rescaling is needed and the invariant is simply obtained by the
composition of the homomorphism followed by the Markov trace.
TheobtainedinvariantsareLaurentpolynomialsinthreevariables: twoofthesevariablesaretheparam-
eters u and v in the definition of Y , and the third one is the parameter appearing in the homomorphism
d,n
from the group algebra of the framed braid group to Y . Among these invariants, we recover the ones
d,n
obtained by Juyumaya and Lambropoulou by taking a particular value for this third parameter and some
specific Markov traces.
Restricting to classical links, we use our construction via the isomorphism theorem to prove two results
devoted to the comparison of the obtained invariants with the HOMFLYPT polynomial.
First we show that the HOMFLYPT polynomial is contained in them. More precisely, among the 2d
−
1 basic Markov traces, there are d of them whose associated invariants coincide with the HOMFLYPT
polynomial. ThesebasicMarkovtracesaretheonesindexedbycompositionsintodpartswithonlyonepart
equal to 1 and all the others equal to 0 (in the particular case of the Juyumaya–Lambropoulouinvariants,
this result corrresponds to [2, Corollary 1]).
Then, we show that the invariants obtained from the others basic Markov traces are always equal to 0
when applied to a classical knot. For classical knots, this solves completely the study of these invariants,
whicharethusshowntobetopologicallyequivalenttotheHOMFLYPTpolynomial. Thisgives,inparticular,
a different proof of results of [1] about the Juyumaya–Lambropoulouinvariants.
Notations.
We fix an integer d 1, and we let ξ ,...,ξ be the set of roots of unity of order d. We will often
1 d
• use without mention≥ing that d1 0 s{d 1ξasξb−s}is equal to 1 if a=b and is equal to 0 otherwise.
≤ ≤ −
Let be an algebra defined ovPer a commutative ring R. If R is a commutative ring with a given
′
• A
ring homomorphism θ : R R, we will denote the specialized algebra R := R where the
→ ′ θ′A ′⊗R A
tensor product is defined via θ. In particular, if R is a commutative ring containing R as a subring,
′
we denote simply by R :=R the algebra with ground ring extended to R.
′ ′ R ′
A ⊗ A
We will denote by M an elementary matrix with 1 in position (i,j) and 0 everywhere else (the size
i,j
•
of the matrix will always be given by the context).
Acknowledgements. The authors thank Cédric Bonnafé, Sofia Lambropoulou and Ivan Marin for useful
discussionsandcomments. The authorsaregratefulto GeorgeLusztigfor indicatingthe reference[19]. The
first author is supported by Agence National de la Recherche Projet ACORT ANR-12-JS01-0003.
3
2 Definitions and first properties
2.1 The Iwahori–Hecke algebra of type A
Letn Z andletuandvbeindeterminates. TheIwahori–Heckealgebra oftypeA istheassociative
1 n n 1
C[u 1∈,v]-≥algebra (with unit) with a presentation by generators : H −
±
T ,...,T ,
1 n 1
−
and relations :
T T = T T for all i,j =1,...,n 1 such that i j >1,
i j j i
− | − |
TiTi+1Ti = Ti+1TiTi+1 for all i=1,...,n 2, (1)
−
T2 = u2+vT for all i=1,...,n 1.
i i −
Note that =C[u 1,v]. It is convenient also to set :=C[u 1,v].
1 ± 0 ±
Let R bHe a ring and let θ : C[u 1,v] R be a specHialization such that θ(u2)=1 and θ(v) =0 then the
±
specialized algebra R is naturally iso→morphic to the group algebra RS of the symmetric group.
θ n n
H
Remark 2.1. Letqbeanindeterminate. Anotherusualpresentationof isobtainedbythespecialization
n
θ :C[u 1,v] C[q,q 1] given by θ(u2)=1 and θ(v)=q q 1. H
± − −
→ − △
Let w S and s ...s a reduced expression of w (where (i ,...,i ) 1,...,n 1 r and s S
∈ n i1 ir 1 r ∈ { − } i ∈ n
denotes the transposition (i,i + 1) for i = 1,...,n 1). Then, by Matsumoto’s lemma (see [7, §1.2]),
−
the element T ...T does not depend of the choice of the reduced expression of w and thus, the element
i1 ir
T := T ...T is well-defined. Then is free as a C[u 1,v]-module with basis T w S (see [7,
w i1 ir Hn ± { w | ∈ n}
Thm. 4.4.6]). In particular it has dimension n!.
We also set T :=u 1T , for i 1,...,n 1 and,
i − i
∈{ − }
e Tw :=u−ℓ(w)Tw =Ti1...Tir , for w ∈Sn . (2)
where ℓ(w) is the length of w.eThe set Tw w eSn ies also a C[u±1,v]-basis of n.
{ | ∈ } H
2.2 Compositions of n e
Let Comp (n) be the set of compositions of n with d parts (or d-compositions of n), that is the set of
d
d-tuples of non negative integers µ=(µ ,...,µ ) such that µ =n. The set of d-compositions of n
1 d 1 a d a
is denoted by Comp (n). We denote by µ :=n the size of the≤co≤mposition µ.
d | | P
For µ Comp (n), the Young subgroup Sµ is the subgroup S S of S , where S acts on
the letters∈ 1,...,dµ , S acts on the letters µ +1,...,µ , anµd1 ×so··o·n×. Thµed subgrnoup Sµ isµg1enerated
{ 1} µ2 { 1 2}
by the transposition (i,i+1) with i I := 1,...,n 1 µ ,µ +µ ,...,µ +...+µ .
µ 1 1 2 1 d 1
∈ { − }\{ − }
Similarly, we have an associated subalgebra µ of generated by T i I . This is a free
n i µ
H H { | ∈ }
C[u 1,v]-module with basis T w Sµ (or T w Sµ ). The algebra µ is naturally isomorphic
± w w
to ... , where th{e ten|sor∈produ}cts ar{e ove|r C[∈u 1,v}]. Note that theHdefining relations of µ in
Hµ1 ⊗ ⊗Hµr ± H
terms of the generators Ti with i Iµ, are the relaetions from (1) involving only those generators.
∈
Let µ Comp (n). For a 1,...,d , we denote by µ[a] the composition in Comp (n+1) defined by
∈ d ∈{ } d
µ[a] :=µ if b=a, and µ[a] :=µ +1 . (3)
b b 6 a a
Conversely,if µ 1, we define µ Comp (n 1) to be the unique composition such that
a ≥ [a] ∈ d −
(µ )[a] =µ . (4)
[a]
4
We also define the base of µ, denoted by [µ], to be the d-composition defined by
1 if µ 1,
a
[µ] = ≥ for a=1,...,d. (5)
a
0 if µ =0,
(cid:26) a
The composition [µ] belongs to Comp (N) where N is the number of non-zero parts in µ. We denote by
d
Comp0(n) the set of d-compositions of n where all the parts belong to 0,1 ,and we set
d { }
Comp0 := Comp0(n)= [µ] µ Comp (n) . (6)
d d { | ∈ d }
n 1 n 1
[≥ [≥
2.3 The Yokonuma–Hecke algebra
We define the Yokonuma–Hecke algebra Y as the associative C[u 1,v]-algebra (with unit) with a presen-
d,n ±
tation by generators:
g ,g ,...,g ,t ,...,t ,
1 2 n 1 1 n
−
and relations:
g g = g g for all i,j =1,...,n 1 such that i j >1,
i j j i
− | − |
g g g = g g g for all i=1,...,n 2,
i i+1 i i+1 i i+1
−
t t = t t for all i,j =1,...,n,
i j j i
(7)
g t = t g for all i=1,...,n 1 and j =1,...,n,
i j si(j) i −
td = 1 for all j =1,...,n,
j
g2 = u2+ve g for all i=1,...,n 1,
i i i −
where, for all i=1,...,n 1,
− 1
ei := d tsit−i+s1 .
0 s d 1
≤X≤ −
Note that the elements e are idempotents and that we have g e = e g for all i = 1,...,n 1. The
i i i i i
−
elements g are invertible, with
i
gi−1 =u−2gi−u−2vei, for all i=1,...,n−1. (8)
We also set
g :=u 1g , for i 1,...,n 1 .
i − i
∈{ − }
We note that gi2 =1+u−1veigi and also that gi−1 =gi−u−1vei, for i=1,...,n−1.
Let R be a ring and let θ : C[ue 1,v] R be a specialization such that θ(u2)=1 and θ(v) =0 then the
±
→
specializedalgeebraR Y is neaturallyisomorpheicto tehe groupalgebraRG(d,1,n)ofthe complex reflection
θ d,n
groupG(d,1,n)∼=(Z/dZ)≀Sn. Notethatinthecasewhered=1thenY1,nisnothingbuttheIwahori–Hecke
algebra of type A as defined in Subsection 2.1.
n 1
−
Remark 2.2. Thepresentationusedin[3]oftheYokonuma–Heckealgebraisobtained,similarlytoRemark
2.1,byaspecializationθ :C[u 1,v] C[q,q 1]suchthatqisanindeterminate,θ(u2)=1andθ(v)=q q 1.
± − −
→ −
The precise connections between the presentation above and the presentation used in [2, 10, 12, 13, 14, 15]
will be carefully investigated in Section 6 (see also [3, Remark 1]).
△
We set, for w S and s ...s a reduced expression for w,
∈ n i1 ir
gw :=gi1...gir and gw :=u−ℓ(w)gw =gi1...gir . (9)
Again, by Matsumoto’s lemma (see [7, §1.2]), the elements g and g are well-defined. The following
w w
multiplication rules in Y follow directly from the defienitions. For w Se aned i 1,...,n 1 , we have
d,n n
∈ ∈{ − }
e
g if l(ws )>l(w),
g g = wsi i (10)
w i g +u 1vg e if l(ws )<l(w);
(cid:26) wsi − w i i
e
e e
e e5
g if l(s w)>l(w),
g g = siw i (11)
i w g +u 1ve g if l(s w)<l(w).
(cid:26) siw − i w i
By [10] and Remark 2.2, Yd,neisea free Ce[u±1,v]-module with basis
e e
tk1...tkng w S , k ,...,k Z/dZ (12)
{ 1 n w | ∈ n 1 n ∈ }
and the rank of Y is dnn!. The algebra Y naturally embeds in the algebra Y in an obvious way.
d,n ed,n−1 d,n
Remark 2.3. For the isomorphism theorem, we will mainly use the elements g and g instead of g and
i w i
g . Concerning this part, we could have given the presentation of Y in terms of the generators g and
w d,n i
thusremoveoneofthe variablesuorv. However,the generatorsg willbe usedsystematicallystartingfrom
i e e
Section 5 for applications to links theory.
e
△
2.4 A decomposition of Y
d,n
We consider the commutative subalgebra := t ,...,t of Y . This algebra is naturally isomorphic to
n 1 n d,n
the group algebra of (Z/dZ)n over C[u 1A,v], anhd we willialways implicitly make this identification in the
±
following.
A complex character χ of the group (Z/dZ)n is characterized by the choice of χ(t ) ξ ,...,ξ for
j 1 d
each j =1,...,n. We denote by Irr( ) the set of complex characters of (Z/dZ)n, extend∈ed{to . }
n n
A A
Definition 2.4. For each χ Irr( ), we denote by E the primitive idempotent of associated to χ,
n χ n
∈ A A
that is, characterized by χ(E )=0 if χ =χ and χ(E )=1.
′ χ ′ χ
6
The idempotent E is explicitly written in terms of the generators as follows:
χ
1
Eχ = d χ(ti)st−i s . (13)
1 i n 0 s d 1
≤Y≤ ≤X≤ −
 
By definition, we have, for all χ Irr( ) and i=1,...,n,
n
∈ A
t E =E t =χ(t )E . (14)
i χ χ i i χ
The symmetricgroupS actsby permutationson (Z/dZ)n andinturnactson Irr( ). The actionis given
n n
A
by the formula:
w(χ) ti =χ(tw−1(i)), for all i=1,...,n, w Sn and χ Irr( n).
∈ ∈ A
In the algebra Y , du(cid:0)e(cid:1)to the relation g t =t g for i=1,...,n and w S , we have
d,n w i w(i) w n
∈
g E =E g and g E =E g . (15)
w χ w(χ) w w χ w(χ) w
Let χ Irr( ). For a = 1,...,d, denote by µ the cardinal of the set j 1,...,n χ(t ) = ξ .
n a e e j a
∈ A { ∈ { } | }
Then the sequence (µ ,...,µ ) is a d-composition of n which is denoted by
1 d
Comp(χ):=(µ ,...,µ ) Comp (n) .
1 d ∈ d
Let µ Comp (n). Then we denote by
∈ d
(µ):= χ Irr( ) Comp(χ)=µ
n
O { ∈ A | }
the orbit of the element χ Irr( ) under the action of the symmetric group and
n
∈ A
n!
m :=♯ = .
µ µ
O µ !µ !...µ !
1 2 d
6
Definition 2.5. Let µ Comp (n). We set
∈ d
E := E = E .
µ χ χ
ComXp(χ)=µ χ∈XO(µ)
Due to the commutation relation (15), the elements E , with µ Comp (n), are central in Y . More-
µ ∈ d d,n
over, as the set E χ Irr( ) is a complete set of orthogonal idempotents, it follows at once that the
χ n
{ | ∈ A }
set E µ Comp (n) forms a complete set of central orthogonal idempotents in Y . In particular, we
{ µ | ∈ d } d,n
have the following decomposition of Y into a direct sum of two-sided ideals:
d,n
Y = E Y . (16)
d,n µ d,n
µ∈CMompd(n)
2.5 Another basis for Y
d,n
We here give another basis for Y using the idempotents we just defined. As the subalgebra of Y is
d,n n d,n
isomorphicto the groupalgebraof(Z/dZ)n overC[u 1,v], the set E χ Irr( ) is a C[uA1,v]-basisof
± χ n ±
{ | ∈ A }
An, as well as the set {tk11...tknn | k1,...,kn ∈Z/dZ}. So from the knowledge of the C[u±1,v]-basis (12) of
Y , we also have that the set
d,n
E g χ Irr( ), w S (17)
χ w n n
{ | ∈ A ∈ }
is a C[u 1,v]-basis of Y . Moreover,this basis is compatible with the decomposition (16) of Y since, for
± d,n d,n
µ Comp (n), we have E g E Y ifeand only if Comp(χ)=µ. In other words, the set
∈ d χ w ∈ µ d,n
E g χ Irr( ) with Comp(χ)=µ, w S
χ w n n
e{ | ∈ A ∈ }
is a C[u 1,v]-basis of E Y .
± µ d,n
e
Now we will label the elements of Irr( ) in a useful way for the following. This is done as follows. We
n
A
first consider a distinguished element in each orbit (µ). Let µ Comp (n). We denote
O ∈ d
χµ Irr( ) ,
1 ∈ An
the character given by
χµ(t ) = ... = χµ(t ) = ξ ,
1 1 1 µ1 1
χµ(t ) = ... = χµ(t ) = ξ ,
 1 µ.1+1 . . . 1 µ.1+µ2 . 2. (18)
 .. .. .. .. .. .. ..
χµ(t ) = ... = χµ(t ) = ξ .
Notethatthestabilizerofχµ1 1undµe1+r·t·h·+eµad−ct1i+o1nofSn istheYoung1suµbdgroupSµ.dIneachleftcosetinSn/Sµ,
we take a representative of minimal length (such a representative is unique, see [7, §2.1]). We denote by
π ,...,π
{ 1,µ mµ,µ}
this set of distinguished left coset representatives of S /Sµ with the convention that π = 1 (recall that
n 1,µ
m :=♯ (µ)). Then, if we set for all k =1,...,m :
µ µ
O
χµ :=π (χµ), (19)
k k,µ 1
we have by construction that
(µ)= χµ,...,χµ .
O { 1 mµ}
To sum up, we have the following C[u 1,v]-basis of Y :
± d,n
{Eχµkgw | w ∈Sn, k=1,...,mµ, µ∈Compd(n)} , (20)
where, for each µ ∈ Compd(n), ethe subset {Eχµkgw | w ∈ Sn, k = 1,...,mµ} is a C[u±1,v]-basis of the
two-sided ideal E Y .
µ d,n
e
7
3 The isomorphism theorem
The aim of this part is to study an isomorphism between Y and Mat ( µ), and then to
analyse the inclusion Y Y from this point of view.d,Tnheoremµ3∈.1Cobmeplodw(n)is inmfaµctHa special case of
d,n 1 d,n
− ⊂ L
a result by G. Lusztig (see [19, §34]). Nevertheless we give full details of the proof as the explicit formulas
will then be used throughout the rest of the paper.
3.1 The statement
Let µ Comp (n). We recall that E Y is a two-sided ideal of Y and is also a unital subalgebra with
∈ d µ d,n d,n
unit E . We define a linear map
µ
Φ :Mat ( µ) E Y ,
µ mµ H → µ d,n
by setting, for any matrix consisting of basis elements T of µ (that is, with w Sµ),
wi,j H i,j ∈
Φµ (Twi,j)1≤i,j≤mµ = e Eχµi gπi,µwi,jπj−,µ1Eχµj . (21)
(cid:0) (cid:1) 1≤iX,j≤mµ
e e
We also define a linear map
Ψ :E Y Mat ( µ) ,
µ µ d,n → mµ H
as follows. Let k 1,...,m and w S , and let j 1,...,m be uniquely defined (given k) by the
µ n µ
relation w(χµj)=χ∈µk{. Note tha}t we thus∈have πk−,1µwπj,µ∈∈{Sµ. We th}en set
Ψµ(Eχµkgw)=Tπk−,1µwπj,µMk,j , (22)
where we recall that Mk,j denotes the elementaery maetrix with 1 in position (k,j).
Now we can state the isomorphism theorem. Recall the decomposition (16) of Y .
d,n
Theorem 3.1. Let µ Comp (n). The linear map Φ is an isomorphism of C[u 1,v]-algebra with inverse
∈ d µ ±
map Ψ . In turn,
µ
Φ := Φ : Mat ( µ) Y
n µ mµ H → d,n
µ∈CMompd(n) µ∈CMompd(n)
is an isomorphism with inverse map:
Ψ := Ψ : Y Mat ( µ).
n µ d,n → mµ H
µ∈CMompd(n) µ∈CMompd(n)
3.2 Preliminary results
We first prove a series of useful lemmas.
Lemma 3.2. Let µ Comp (n) and i 1,...,m . We consider a reduced expression s ...s of π .
∈ d ∈ { µ} i1 ik i,µ
Then for all l 1,...,k , we have:
∈{ }
eilEsil+1...sik(χµ1) =Esil+1...sik(χµ1)eil =0
Proof. By definition, π is the (unique) element of S with minimal length satisfying π (χµ)=χµ. As a
i,µ n i,µ 1 i
consequence, we have for all l =1,...,k:
s s ...s (χµ)=s ...s (χµ)
il · il+1 ik 1 6 il+1 ik 1
which is equivalent to
s ...s (χµ)(t )=s ...s (χµ)(t ) .
il+1 ik 1 il 6 il+1 ik 1 il+1
8
Thus by (14), we have
tilEsil+1...sik(χµ1) 6=til+1Esil+1...sik(χµ1).
This discussion shows that
tilt−il+11Esil+1...sik(χµ1) =ξjEsil+1...sik(χµ1)
for a d-root of unity ξ =1. We conclude that
j
6
Esil+1...sik(χµ1)eil =eilEsil+1...sik(χµ1) =( ξjs)Esil+1...sik(χµ1) =0 ,
0 s d 1
≤X≤ −
where we note, for the first equality, that e commutes with any E .
il χ
Lemma 3.3. For all µ Comp (n), 1 i,j m and w S , we have:
∈ d ≤ ≤ µ ∈ n
(i) Eχµ1 gπ−i1,µgwgπj,µEχµ1 =Eχµ1 gπi−,µ1wπj,µEχµ1 ;
(ii) Eχµi geπi,µgewgeπ−j1,µEχµj =Eχµi geπi,µwπj−,µ1Eχµj .
Proof. Let us denote a reduced expression of π by s ...s . We have :
e e e e i,µ i1 ik
Eχµ1 gπ−i1,µ gw = Eχµ1 gi−k1...gi−11 gw
e e = gi−k1.e..gi−21 Eesi2.e..sik(χµ1) gi−11 gw .
Recall that for all j =1,...,n, we have gj−1 =gej−u−e1vej. Thus, witherepeeated applications of Lemma 3.2
together with the multiplication rule (11), we deduce that:
Eχµ1 gπ−i1,µ gwe = gei−k1...gi−21 Esi2...sik(χµ1) gsi1w
e e = gei−k1...gei−31 Esi3...sik(χµ1) gei−21 gsi1w
= ...
e e e e
= Eχµ1 gπi−,µ1w .
Now let us denote by s ...s a reduced expression of π , we have:
j1 jl e j,µ
Eχµ1 gπ−i1,µ gw gπj,µ Eχµ1 = Eχµ1 gπi−,µ1w gπj,µ Eχµ1
e e e = Eχµ1 geπi−,µ1w gej1...gjl Eχµ1
= Eχµ1 geπi−,µ1w gej1 Esje2...sjl(χµ1) gi2...gil .
As above, with repeated applications of Lemma 3.2 together with the multiplication rule (10), we obtain:
e e e e
Eχµ1 gπ−i1,µ gw gπj,µ Eχµ1 = Eχµ1 gπi−,µ1wsj1 Esj2...sjl(χµ1) gj2...gjl
e e e = Eχµ1 geπi−,µ1wsj1 gj2 Esj3...sjl(χeµ1) gj3e...gjl
= ...
e e e e
= Eχµ1 gπi−,µ1wπj,µ Eχµ1 ,
which proves item (i). Let us now prove item (ii). We have by (19) and (15) :
e
Eχµi gπi,µ gw gπ−j1,µ Eχµj = gπi,µEχµ1 gw Eχµ1 gπ−j1,µ
e e e = geπi,µEχµ1 geπi−,µ1πi,µweπj−,µ1πj,µ Eχµ1 gπ−j1,µ
= egπi,µEχµ1 egπ−i1,µ gπi,µwπj−,µ1 gπj,µ Eeχµ1 gπ−j1,µ ,
where the last equality comes from item (i). Tehe proof ies conecluded useing that geπi,µEχµ1 gπ−i1,µ = Eχµi and
gπj,µEχµ1 gπ−j1,µ =Eχµj.
e e
e e
9
Lemma 3.4. Let µ Comp (n). The map
∈ d
φµ : µ EχµYd,nEχµ,
H → 1 1
defined on the generators by
i Iµ, φµ(Ti)=EχµgiEχµ,
∀ ∈ 1 1
extends to an homomorphism of algebras.
Proof. Recall that the subspace EχµYd,nEχµ is a unital subalgebra of Yd,n with unit Eχµ.
We first note that if i I then1s (χµ)=1 χµ. We thus have 1
∈ µ i 1 1
giEχµ =Eχµgi
1 1
andthisrelationeasilyimpliesthattheelementsEχµ1giEχµ1 withi∈Iµ satisfythebraidrelations. Itremains
to check the “quadratic relation”. We have
(EχµgiEχµ)2 = Eχµgi2Eχµ
1 1 1 1
= u2Eχµ +vEχµeigiEχµ
1 1 1
= u2Eχµ +vEχµgiEχµ
1 1 1
The last equality comes from the fact that for i∈Iµ we have tiEχµ1 =ti+1Eχµ1, and thus
eiEχµ =Eχµei =Eχµ.
1 1 1
Thus all the defining relations of µ = are satisfied so that φ can be extended to a
H Hµ1 ⊗···⊗ Hµd µ
homomorphism of algebras.
Remark 3.5. One can actually show that the morphism φ is an isomorphism. Indeed the lemma implies
µ
that φµ is given on the standard basis of µ by φµ(Tw) = EχµgwEχµ, w Sµ. Moreover, if w Sµ then
w(χµ1) = χµ1 and therefore φµ(Tw) = Eχµ1Hgw. So it remains to1check1that∈{Eχµ1gw | w ∈ Sµ} is∈a basis of
EχµYd,nEχµ. The linear independence is immediate, while the spanning property follows from the following
1 1
calculation, for a basis element Eχνigw of Yd,n and ν ∈Compd(n),
Eχµgw if µ=ν, i=1 and w Sµ;
Eχµ1 ·Eχνigw·Eχµ1 =Eχµ1EχνiEw(χµ1)gw =( 0 1 otherwise. ∈
In the following,we will not use the fact that φ is actually anisomorphism(actually, it is a consequence of
µ
Theorem 3.1).
△
3.3 Proof of Theorem 3.1
Let µ Comp (n).
∈ d
1. WefirstshowthatΦ isamorphism. Beforethis,wenotethatbyLemma3.3(ii),forall1 i,j m
µ µ
and w Sµ, we have: ≤ ≤
∈ Eχµi gπi,µwπj−,µ1 Eχµj = Eχµi gπi,µ gw gπ−j1,µ Eχµj
e = gπi,µeEχµ1 egwEeχµ1 gπ−j1,µ (23)
= g φ (T ) g 1 .
eπi,µ µ ew π−j,µe
Now, let i,j,k,l 1,...,m and w,w Sµ. We have:
∈{ µ} ′ ∈ e e e
Φµ TwMi,j Φµ Tw′Mk,l =Eχµi gπi,µwπj−,µ1 Eχµj Eχµk gπk,µw′πl−,µ1 Eχµl .
(cid:0) (cid:1) (cid:0) (cid:1)
e e e e
10