xii INTRODUCTIO N
The secon d chapte r develop s Graha m an d Lehrer' s [72 ] theory o f cellular alge-
bras. Thes e ar e a class of algebras whic h com e equipped wit h a distinguished basi s
which i s particularly well-adapte d t o th e representatio n theor y o f the algebra . Fo r
example, given a cellular basis one can immediately write down a collection of mod-
ules which contain s al l o f the simpl e module s o f the algebra . Cellula r algebra s ar e
one of the unifyin g thread s runnin g throug h thes e notes a s we construct th e simpl e
modules o f th e Iwahori-Heck e algebra s an d th e g-Schu r algebra s b y firs t showin g
that thes e algebra s ar e cellular .
The thir d chapte r embark s upo n th e stud y o f the representatio n theor y o f JF.
Following Murph y [150] w e sho w tha t J(? has a natura l basi s indexe d b y pair s o f
standard tableaux ; importantly , Murphy' s basi s i s cellular . A s a consequence , fo r
each partition A we obtain a Specht modul e S x; thi s modul e i s a ^-analogue o f th e
usual Spech t modul e o f @
n
. B y th e theor y o f chapte r 2 , ther e i s a n intrinsicall y
defined bilinea r form on each Specht module ; furthermore, S
x
modul o the radical of
its form i s either zer o or absolutely irreducible. Th e last two sections of this chapte r
are a detailed stud y o f the Spech t modules , culminatin g i n the classificatio n o f th e
simple ^-modules . Al l of this theor y closel y parallel s th e modula r representatio n
of the symmetri c group .
Apart fro m bein g cellular , Murphy' s basi s ha s anothe r marvelou s propert y i n
that i t i s possible t o 'lift ' thi s basi s t o giv e a cellular basi s fo r th e g-Schu r algebra ;
this basi s i s indexe d b y pair s o f semistandar d tableaux , an d eac h basi s elemen t
is essentiall y a su m o f Murph y basi s element s ( a semistandar d tableau x ca n b e
thought o f a s a n orbi t o f standar d tableau x an d th e sum s ar e ove r thes e orbits) .
In thi s wa y w e obtai n a ver y clea n an d ver y elegan t constructio n o f th e simpl e
modules o f th e g-Schu r algebras . Fo r free , w e discove r tha t th e g-Schu r algebra s
are quasi-hereditary . Al l of these result s ar e prove d i n chapte r 4 .
Chapter 5 i s devote d t o classifyin g th e block s o f th e g-Schu r algebras ; a s a
corollary thi s yield s th e block s of Jif. I n orde r t o classif y th e block s we first prov e
an analogu e o f the Jantze n su m formul a [98 ] fo r th e Wey l module s o f the g-Schu r
algebras. Th e Jantze n su m formul a i s a stron g resul t whic h give s informatio n
about th e compositio n factor s o f Wey l module s an d Spech t modules ; i t i s prove d
by computing the determinants o f the Gra m matrices of the Weyl modules (tha t w e
can comput e thes e determinant s i n genera l i s in itsel f surprising) . Thi s calculatio n
requires a heavy dos e o f combinatorics; i t i s by far th e mos t technica l par t o f thes e
notes.
The fina l chapte r i s a survey o f some recen t an d importan t result s an d conjec -
tures i n the field ; her e we abandon ou r claim s of being self-contained . Th e chapte r
begins wit h a reasonabl y thoroug h accoun t o f th e LL T algorith m whic h compute s
the decompositio n matrice s o f th e Iwahori-Heck e algebra s define d ove r th e com -
plex field. W e then discuss adjustment matrices , the Kleshchev-Brundan branchin g
rules, th e theor y o f Dippe r an d Jame s [40 ] connectin g th e g-Schu r algebra s an d
the finit e genera l linea r groups , rule s fo r computin g decompositio n matrice s an d
the Ariki-Koik e algebra s an d cyclotomi c g-Schu r algebras .
In addition , th e boo k contain s thre e appendices . Th e firs t o f these provide s a
quick treatment o f the assumed representatio n theor y of finite dimensiona l algebra s
over a field . Thi s appendi x i s intende d a s a prime r fo r thos e ne w t o th e subject ;
although no t necessary , previou s exposur e t o th e ordinar y representatio n theor y
of finit e group s woul d b e advantageous . Th e secon d appendi x contain s table s o f
the crystallize d decompositio n matrice s an d adjustmen t matrice s fo r n 10 (se e
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