What is now proved was once only imagined.
William Blak e
Introduction
The symmetri c grou p 6
n
i s th e grou p o f permutation s o n 1,2, ...,n . Th e
ordinary irreducible representations of &n ar e very well understood, with everythin g
from thei r degree s an d characte r formulae , t o explicit matri x representation s bein g
known fo r man y years . I n contrast, ver y littl e detaile d informatio n i s known abou t
the modula r representation s o f th e symmetri c groups ; i n fact , i n spit e o f a grea t
deal o f effort , no t tha t muc h progres s ha s bee n mad e sinc e Jame s [84 ] classifie d
and constructe d th e irreducibl e modula r representation s o f th e symmetri c group s
in 1976.
The ai m o f thi s boo k i s t o giv e a self-containe d introductio n t o th e modula r
representation theor y o f the Iwahori-Heck e algebra s o f the symmetri c groups ; thi s
includes the modular representatio n theor y of 6
n
a s a special case. I n studying th e
Iwahori-Hecke algebra s i t i s profitable t o wide n th e scop e o f ou r investigation s t o
include the g-Schu r algebras . Th e study o f these algebra s was pioneered b y Dippe r
and Jame s i n a serie s o f landmark paper s [37,39-41]. Her e w e recast thi s theory ,
taking accoun t o f recen t advances , wit h a primar y goa l o f classifyin g th e block s
and th e simpl e module s o f both algebras . W e have written thes e note s s o as to b e
accessible t o th e advance d graduat e studen t an d als o to b e usefu l t o researcher s i n
the field .
Apart fro m bein g interestin g i n an d o f themselves , th e mai n motivatio n fo r
studying the Iwahori-Heck e algebra s is that the y provide a bridge between th e rep-
resentation theor y o f th e symmetri c an d genera l linea r groups ; thes e connection s
are eve n mor e transparen t wit h th e g-Schu r algebras . I n th e classica l cas e (tha t
is, q = 1), th e Schu r algebra s wer e introduce d b y Schu r [158] wh o use d them ,
together wit h the representation theor y o f the symmetri c groups , to classify th e or -
dinary irreducibl e polynomia l representations o f GLn(C); se e also [74] . Dippe r an d
James' motivatio n fo r introducin g th e g-Schu r algebr a wa s t o stud y th e modula r
representation theor y of GL n(q) ove r fields o f characteristic no t dividin g q (that is ,
in non-definin g characteristic) ; the y showe d tha t th e g-Schu r algebra s completel y
determine th e decompositio n matri x o f GL n{q) i n this cas e [40] .
Our motivatio n fo r studyin g th e g-Schu r algebr a i s mor e modes t i n tha t w e
view the g-Schu r algebr a a s a tool fo r studyin g th e Iwahori-Heck e algebra . A s we
will see, the g-Schur algebra s have a rich and beautiful combinatoria l representatio n
theory whic h i s closely allie d wit h tha t o f the Iwahori-Heck e algebras . Indeed , ful l
knowledge o f th e representatio n theor y o f on e clas s o f thes e algebra s i s equivalen t
to ful l knowledg e o f the other . Further , i t i s often th e cas e tha t th e easies t wa y t o
prove a resul t abou t on e o f thes e algebra s i s to firs t prov e a n analogou s resul t fo r
the other . Throughou t th e 'classical ' theor y fo r th e symmetri c an d genera l linea r
groups ca n b e obtaine d b y settin g q = 1.
ix
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