Lectures on Hecke algebras

70 Pages·2003·0.499 MB·English

by  O. Ogievetsky, P. Pyatov

1
Le
ture on He
ke algebras
(cid:3)) 2 (cid:3)(cid:3))
O. Ogievetsky & P. Pyatov
(cid:3))
Center of Theoreti
al Physi
s, Luminy, 13288 Marseille,
Fran
e
(cid:3)(cid:3))
Bogoliubov Laboratory of Theoreti
al Physi
s, JINR,
141980 Dubna, Mos
ow region, Russia
&
Max-Plan
k-Institut fu(cid:127)r Mathematik, Vivatsgasse 7,
53111 Bonn, Germany
1
Based on le
tures presented at the International S
hool "Symmetries and Integrable
Systems", Dubna, 8-11 June, 1999.
2
On leave of absen
e from P. N. Lebedev Physi
al Institute, Theoreti
al Department,
Leninsky pr. 53, 117924Mos
ow, Russia
1
Contents
1 Introdu
tion 3
2 Algebrai
ba
kground 4
3 Coxeter groups and He
ke algebras 10
3.1 De(cid:12)nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Semisimpli
ity . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Symmetri
groups and A-type He
ke algebras 16
5 Young diagrams and tableaux 20
5.1 Partitions, diagrams, tableaux and Young graph . . . . . . . . 20
5.2 Hook formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6 Ju
ys-Murphy elements 26
7 Primitive orthogonal idempotents 27
8 Central idempotents 33
8.1 Chara
teristi
polynomials . . . . . . . . . . . . . . . . . . . . 33
8.2 One-dimensional and Burau representations . . . . . . . . . . 35
8.3 Young symmetrizers . . . . . . . . . . . . . . . . . . . . . . . 36
9 Matrix stru
ture 38
10 Appli
ation: Temperley{Lieb algebras 46
11 Representations 48
11.1 Constru
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
11.2 AÆne He
ke algebra . . . . . . . . . . . . . . . . . . . . . . . 53
12 Examples 60
2
1 Introdu
tion
This le
ture is an introdu
tive review of the stru
ture and representation
theory of He
ke algebras. To be more pre
ise we shall dis
uss the An-type
He
ke algebras whi
h have found lastly a lot of appli
ations in the theory
of integrable systems and whi
h are related to the representation theory of
the An series of quantum groups (see, e.g., [ChP℄). We have no intention to
give an exhaustive a

ount of the subje
t here, therefore some of important
topi
s (like the R-matrix representations and the tra
es over He
ke algebras
or the theory of Kazhdan-Lusztig polynomials) as well as most of the proofs
will be omitted in our le
ture. Instead we shall try to present the subje
t in
a
lear and memorizable way appealing to analogies with a familiar example
of matrixalgebra and developing simple mnemoni
s. Forthis purpose we use
a re
ently developed approa
h due to R. Dipper, G. James, G. Murphy and
others. The
entral role in this approa
h is played by a spe
ial
ommutative
set of theso-
alledJu
ys-Murphy elements whi
h are used for
onstru
tionof
a fullset of primitiveidempotents (Young tableaux) and
entral idempotents
(Young diagrams) in the He
ke algebra and, then, for developing the repre-
sentation theory and for an expli
it des
ription of a
orresponden
e between
the He
ke algebra and the matrix algebras.
Before goingon with the presentation let us emphasize an importantfa
t:
(cid:2)
the An-type He
ke algebra Hn+1(q) in
ase if its parameter q 2 C is not a
root of unity is isomorphi
to the group algebra of the symmetri
group
C [Sn+1℄. So the He
ke algebra just provides another language for des
ription
of the permutation symmetry and this language appears to be very useful in
models where the permutation symmetry is realized in a nontrivial way, i.e.,
not by the standard permutation matri
es.
We shall see that in the approa
h to He
ke algebras based on Ju
ys-
Murphy elements, the main obje
ts of the theory of symmetri
groups, like
Young diagrams and tableaux, or restri
tions of representations of the sym-
metri
group Sn to its subgroup Sn(cid:0)1, appear naturally within the algebra
itself.
Note also that the approa
h using the Ju
ys-Murphy elements whi
h was
originally developed for the An-type He
ke algebras has proved its useful-
ness in appli
ation to the He
ke algebras of other types and the Birman-
Murakami-Wenzl algebras (for some history and referen
es see [R℄).
3
The le
ture is aimed for graduate and postgraduate students; the reading
assumes some basi
knowledge of linear algebra and
ommutative algebra.
The le
ture is organized as follows. In the next Se
tion we re
all some
basi
fa
ts about the (cid:12)nite-dimensional semi-simple algebras. In Se
tion 3
we give de(cid:12)nitions of Coxeter groups, their Coxeter graphs and He
ke alge-
bras. A Subse
tion with a proof of several formulations of semisimpli
ity of
He
ke algebras is in
luded. Se
tion 4
ontains additional information on the
An-type He
ke algebras. An introdu
tion to the theory of representations
of symmetri
groups is
ontained in Se
tion 5, together with a proof of the
hook formula whi
h gives the dimensions of irredu
ible representations. In
Se
tion 6 the set of the Ju
ys-Murphy elements for the An-type He
ke al-
gebas is introdu
ed. In Se
tion 7 eigenvalues of the Ju
ys-Murphy elements
in di(cid:11)erent representations are
al
ulated and the primitive idempotents in
the He
ke algebras are
onstru
ted. Se
tion 8 deals with the eigenvalues of

entral idempotents in di(cid:11)erent representations. There is a Subse
tion de-
s
ribing a He
ke algebra analogue of the tautologi
al representation of the
symmetri
groups by permutation matri
es and 1-dimensional representa-
tions of the He
ke algebra. Another Subse
tion
ontains generalities about
a di(cid:11)erent system of primitive idempotents,
alled Young symmetrizers, for
thesymmetri
group andtheHe
ke algebra. An isomorphismfromthe He
ke
algebra to the dire
t produ
t of matrix algebras is expli
itly
onstru
ted in
Se
tion 9. This isequivalent tothe des
riptionof the He
ke algebrarepresen-
tations. Weapplythematrixstru
ture tothe
onstru
tionofTemperley-Lieb
algebras in Se
tion 10. In Se
tion 11 we re
onsider the representation theory
of the He
ke algebra from a slightly di(cid:11)erent point of view. An appli
ation
to the
onstru
tion of the Burau representation is given there. Se
tion 12
illustrates results of Se
tions 9 and 11 on simple examples.
In the text, the symbol 2 marks the end of a proof.
2 Algebrai
ba
kground
In thisSe
tion we shallremindsomebasi
resultsfromthe theory ofsemisim-
ple (cid:12)nite dimensional algebras. Proofs and a more systemati
introdu
tion
into the subje
t
an be found in [DK℄.
LetA(K)bea (cid:12)nite-dimensionalunitalasso
iativealgebraovera(cid:12)eld K.
We shall
onsider a regular A-bimoduleAreg. As a linearspa
e Areg
oin
ides
4
with A so that one
an identify elements of Areg and A. The stru
ture of
the left or right A-a
tion on Areg is given by multipli
ation from the left or
right hand side in A, i.e., 8 a 2 A ; b 2 Areg the left and right a
tion of a
on b is given, respe
tively, by
a B b := ab ; bC a := ba : (2.1)
The representation of the algebra A given by its left (as well as right)
a
tion on Areg is faithful, that is, the image of A in the algebra End(Areg) of
3
the endomorphisms of Areg is isomorphi
to A itself .
Clearly, investigationofan algebraisequivalenttothestudy ofits regular
representation. The left and right module stru
tures of the regular bimodule

arry the sameinformation,so we shalldis
uss onlythe leftmodulestru
ture
of Areg in what follows.
Suppose Areg de
omposes into a dire
t sum of left A-modules Mi, i =
1;:::;s;
Ms
Areg = Mi : (2.2)
i=1
The subspa
es Mi (cid:26) A are left ideals of A.
The
orresponding de
omposition of the unit element of Areg ((cid:17) of A) is
Xs
1 = ei where ei 2 Mi : (2.3)
i=1
Consider the a
tion of the element ei 2 A on the identity
Xs
ei (cid:17) ei B 1 = ei B ej :
j=1
Sin
e ei B ej (cid:17) eiej 2 Mj it follows that
eiej = Æijei : (2.4)
s
So, the elements feigi=1 form the set of mutually orthogonal idempotents in
P
s
A. A de
omposition of unity, 1 = i=1ei into a sum of mutually orthogonal
idempotents ei is
alled a resolution of unity.
3
Toseethis,itisenoughto
onsiderthea
tionofelementsofAontheidentityelement
12Areg.
5
The module Mi is inde
omposable if and only if the
orresponding idem-
potent ei
annot be further resolved into a sum of nontrivial mutually or-
thogonal idempotents. In this
ase one
alls ei the primitive idempotent.
The arguments used in the derivation of the formula (2.4) remain valid if
one repla
es the idempotent ei by an arbitrary element vi 2 Mi. Hen
e, one
has viej = Æijvi and therefore Mi = Aei.
We have seen that to a de
omposition (2.2) one asso
iates a resolution
P
s
of unity. In the other dire
tion, to a resolution of unity 1 = i=1ei one
asso
iates a de
omposition (2.2), where Mi = Aei.
To sum up, there is a one-to-one
orresponden
e between the de
om-
positions of the regular module Areg into a dire
t sum of submodules and
the resolutions of the unit element of the algebra A into a sum of mutually
orthogonal idempotents.
Inde
omposable submodules in a de
omposition of Areg into a dire
t
sum
orrespond to primitive idempotents.
The inde
omposable submodules of a de
omposition of the module Areg
intoadire
tsumofmodulesare
alledprin
ipalmodules. Thesetofprin
ipal
modules is de(cid:12)ned uniquely:
P P
s t
Let 1 = i=1ei = j=1e~j be two resolutions of the unity element.
(cid:0)1
Then s = t and there exists an element
2 A su
h that e~i =
e(cid:25)(i)
for
some permutation (cid:25).
Correspondingly, the de
omposition of Areg into inde
omposable sub-
modules is unique in a sense that any two su
h de
ompositions
ontain
the same number of
omponents whi
h under appropriate identi(cid:12)
ation
are pairwise isomorphi
.
The de
omposition of the left regular module Areg (or, equivalently, of
the algebra A)
orresponding to a given resolution (2.3), (2.4) of the unit
looks like
Ms
A = Aei : (2.5)
i=1
It is
alled the left Peir
e de
omposition of the algebra A.
In the same way one
an de(cid:12)ne the right Peir
e de
omposition A =
L
s
i=1eiA. Applying then both left and right Peir
e de
ompositions simul-
6
taneously one obtains the (two-sided) Peir
e de
omposition of the algebra
A
Ms
A = eiAej : (2.6)
i;j=1
The
omponents Aij := eiAej of the Peir
e de
omposition being in gen-
eral neither left nor right ideals in A ((cid:17) left or right submodules in Areg)
provide a very
onvenient matrix interpretation for the elements of A. In-
deed, let a and b be any elements of A and let
Xs Xs
a = aij ; b = bij ; where aij := eiaej 2 Aij ; bij := eibej 2 Aij
i;j=1 i;j=1
be their Peir
e de
omposition. Then, the
omponents of the Peir
e de
om-
position of their sum and produ
t look, respe
tively, as
Xs
(a+b)ij = aij +bij ; (ab)ij = aikbkj ;
k=1
s
so that if one represents elements a, b of A by the matri
es jjaijjji;j=1,
s
jjbijjji;j=1 of their Peir
e
omponents, the addition and multipli
ation in A
will be reprodu
ed by the usual matrix addition and multipli
ation.
One also extra
ts useful information about the stru
ture of an algebra

onsidering
entral resolutions of its unit element, i.e. the resolutions (2.3),
(2.4) with
omponents ei belonging to the
enter of the algebra. In this
ase
one has
eiAej = 0 ; 8 i 6= j; eiAei = eiA = Aei ;
and so the left, right and two-sided Peir
e de
ompositions be
ome identi
al
and their
omponents are (two-sided) ideals in A. The ideal Aii := eiAei is
an algebra with a unity element ei. If an ideal Aii is inde
omposable (as an
ideal), it is
alled a blo
k of the algebra A.
Thus, there is a one-to-one
orresponden
e between the
entral resolu-
tionsofthe unit element ofthe algebraAandthe de
ompositionsof Ainto
s
a dire
t sum of its ideals A = (cid:8)i=1Aii, or, equivalently, de
ompositions of
A into a dire
t produ
t of the subalgebras A ' A11 (cid:2):::(cid:2)Ass.
The de
omposition of A into blo
ks is unique in the following sense: if
0 0
A = A11 (cid:8):::(cid:8)Ass = A11 (cid:8):::(cid:8)Att are two su
h de
ompositions then
t = s and Aii = A(cid:25)(i)(cid:25)(i) (i = 1;:::;s) for some permutation (cid:25).
7
We shall now illustrate and re(cid:12)ne the general results des
ribed above on
a basi
example of the matrix algebra Mn(K).
Re
all that Mn(K) is the algebra of n(cid:2)n matri
es with
omponents in
2
the (cid:12)eld K. A linear basis in Mn(K) is given by a set of n matrix units
jjeijjjkm := ÆikÆjm. The multipli
ation rules in this basis look like
eij ekm = Æjk eim : (2.7)
The diagonalmatrixunitseii, i = 1;:::;n, aremutuallyorthogonalprimitive
idempotents de(cid:12)ning suitable resolution of the identity matrix I
Xn
I = eii : (2.8)
i=1
The
orresponding left Peir
e de
omposition of the regular module is
n
Mn(K)reg = (cid:8)i=1 Vi, where Vi := Mn(K)eii are simple modules. Note, the
simpli
ity property of Vi strengthens the inde
omposability
ondition from
the general theory.
The basis in Vi is given by the elements eki, k = 1;:::;n, so that dimVi =
n and one
an think of Vi as the i-th
olumn of the matrix. In fa
t, the
algebra Mn(K) has only one nontrivial simple module V and allthe modules
Vi are isomorphi
to V. Any Mn(K)-module is isomorphi
to a dire
t sum
of several
opies of the simple module V, i.e., it is semisimple. In parti
ular,
Mn(K)reg ' nV.
The
omponents of the (two-sided) Peir
e de
omposition | eiiMn(K)ejj
| are multiples of the matrix units
1
eiiMn(K)ejj = Keij ' K :
The unit matrix does not admit
entral resolutions in the matrix algebra,
andmoreover, thealgebraMn(K)doesnothave nontrivial(two-sided)ideals,
i.e., itissimple. The
enter Z(Mn(K)) of Mn(K) is spanned by the multiples
of the unit matrix, Z(Mn(K)) = KI ' K.
In fa
t, the family of matrix algebras Mn(K), n = 1;2;:::;
ompletes the
list of (cid:12)nite-dimensional simple K-algebras in
ase when K is algebrai
ally

losed. Next by generality is the family of semisimple algebras, i.e., the
algebras whose right regular module
an be de
omposed into a dire
t sum
of simple modules. The general stru
ture results for this
ase are re(cid:12)ned in
8
the Wedderburn-Artin theorem. Here we shall formulate it for a parti
ular
situation when A is the algebra over an algebrai
ally
losed (cid:12)eld K.
Wedderburn-Artin theorem. Let K be an algebrai
ally
losed (cid:12)eld.
Any semisimple K-algebra A is isomorphi
to an algebra of the type
Mn1(K)(cid:2)Mn2(K)(cid:2):::(cid:2)Mns(K)
for some integer s and an unordered set of integers fn1;:::;nsg. Under
(i) ni
this isomorphism the diagonal matrix units fejjgj=1 (resp., the identity
(i) Pni (i)
matri
es I = j=1ejj) from the subalgebra Mni(K)
orrespond to the
primitive idempotents (resp.,
entral idempotents) in A. The set of data
fn1;:::;nsg de(cid:12)ning A up to an isomorphism is
alled the numeri
data
of A.
Any A-module V is semisimple and it
an be uniquely presented as
L
s (i) (i)
V = i=1kiV ; where V are simple Mni(K)-modules. In parti
ular,
(i)
simple A-modules
orrespond to V , i = 1;:::;s, and the regular module
L
s (i)
de
omposes as Areg = i=1niV .
The numeri
data of A
an be identi(cid:12)ed with a set of dimensions of all
(1) (2) (s) (i)
the simple A-modules V ;V ;:::;V : ni = dimV .
We
on
lude the Se
tion with a useful
riterium of semisimpli
ity for
algebras over a (cid:12)eld of
hara
teristi
0.
Let A be an algebra over a (cid:12)eld K,
har(K) = 0. Consider the left
regularmoduleAreg andlet(cid:26)a betheoperator
orrespondingtoanelement
a 2 A in the representation in the spa
e Areg, (cid:26)a(b) = ab. The bilinear
form (a;b) := Tr((cid:26)a(cid:26)b) is
alled the tra
e form on A.
Then the algebra A is semisimple i(cid:11) the tra
e form is nondegenerate.
In other words, let fe(cid:11)g be a basis of A and D(cid:11)(cid:12) = Tr((cid:26)e(cid:11)(cid:26)e(cid:12)) be the
matrix of the tra
e form in the basis fe(cid:11)g. The determinant D = detD(cid:11)(cid:12)
is
alled the dis
riminant of A in the basis fe(cid:11)g.
Then A is semisimple i(cid:11) D 6= 0.
9
3 Coxeter groups and He
ke algebras
In this Se
tion we give de(cid:12)nitions of Coxeter groups and He
ke algebras.
Subse
tion 3.2 deals with semisipli
ity questions for He
ke algebras.
A detailed presentation of the theory of Coxeter groups
an be found in
[B, H℄. A di(cid:11)erent de(cid:12)nition, based on double
osets, of the He
ke algebra
is given in [B℄, Ch.IV, x 2, Ex.22 (see also [K℄). A
onne
tion between
the two de(cid:12)nitions was established by Iwahori [Iw℄ and for this reason the
He
ke algebras whi
h we will dis
uss here are
alled often the Iwahori-He
ke
algebras.
3.1 De(cid:12)nitions
The Coxeter group is generated by the set of elements fsigi2I subje
t to
following
onditions.
{ Ea
h of the generators is a re(cid:13)e
tion:
2
(si) = 1 ; 8 i 2 I : (3.1)
{ Consider pairs i;j 2 I for whi
h the element sisj is of (cid:12)nite order, i.e.,
there exists an integer mij (cid:21) 2, su
h that
mij
(sisj) = 1 : (3.2)
Then, the system (3.1), (3.2) is a presentation of the Coxeter group by
generators and de(cid:12)ning relations (that is, there are no other relations
on the set of generators fsig).
The se
ond
ondition
an be given a form
sisjsi ::: = sjsisj ::: (3.3)
| {z } | {z }
mij fa
tors mij fa
tors
whi
h in view of (3.1) is equivalent to (3.2).
As follows from the de(cid:12)nition, the Coxeter group is given provided one
(cid:12)xes itsCoxeter matrixM = kmijki;j2I |thesymmetri
matrixwithinteger
entries satisfying
onditions mii = 1, 8 i 2 I; mij (cid:21) 2, 8 i 6= j 2 I (here
10