xii INTRODUCTIO N
(ii) A matrix representatio n p of the genera l linea r grou p GL n(C) i s said t o b e
polynomial i f the matrix coefficients pij(X), X G GLn(C), ar e polynomial function s
of th e entrie s o f th e matri x X. Th e irreducibl e polynomia l representation s p\ o f
GLn(C) ar e indexe d b y th e partition s A = (Ai,...,A
n
) o f lengt h n , an d th e
character o f p\ i s
trace px(X) = s
A
(£i, ,fn)
where £1,... , £
n
ar e th e eigenvalue s o f the matri x X G GLn (C).
The Schu r function s als o arise naturally i n various combinatoria l contexts : no t
only those associated wit h the combinatoric s o f the symmetri c groups , bu t also , fo r
example, i n the enumeration o f plane partitions. A plane partition ma y be though t
of a s a n infinit e matri x n = (^ij)iji i n whic h eac h entr y 7r ^ i s a nonnegativ e
integer, suc h tha t 7r ^ Tn+ij an d 7r ^ 7r^ J+i fo r al l (i , j), an d suc h tha t |7r | =
^2i Tiij is finite (s o that al l but a finite numbe r o f the iX{j ar e zero) . Thu s a plan e
partition i s not s o very different fro m a tableau, a s defined i n Chapter I , §7; and th e
expression of the Schur function s\ a s a sum of monomials indexed by tableaux (loc .
cit.) ca n therefor e b e exploited , a s wa s first realize d b y R . Stanle y [S] , to deriv e
generating function s fo r variou s classe s o f plan e partitions , suc h a s (fo r example )
MacMahon's famou s generatin g functio n fo r al l plane partitions :
$
^H
=
n(i_tn)_n-
7r
n l
The Schu r function s len d themselve s readil y t o generalizatio n i n various direc -
tions, an d som e (bu t b y n o means all ) o f these generalization s ar e briefl y surveye d
in §§8-12 o f Chapte r I . O f these , bot h th e zona l polynomial s (§8 ) an d th e Hall -
Littlewood function s (§10) hav e close connections wit h representatio n theory : the y
may b e interprete d a s th e value s o f zonal spherica l function s o n th e genera l linea r
group GL
n
(F), wher e F i s a loca l field, relativ e t o a maxima l compac t subgrou p
K. I n the cas e of the zona l polynomials, F i s the rea l field and K i s the orthogona l
group; an d i n th e cas e o f th e Hall-Littlewoo d functions , F i s a p-adi c field an d
K = GL n(o), wher e o is the rin g o f integers i n F (an d th e paramete r t i s equal t o
1/q, wher e q is the cardinality o f the residue field of F). Moreover , just a s the Schu r
functions determin e th e characte r table s o f the symmetri c groups , a s we have see n
above, s o the Hall-Littlewoo d function s pla y a crucia l rol e i n th e determinatio n o f
the irreducibl e comple x character s o f th e finite genera l linea r groups , a s wa s first
realized b y J . A . Gree n [Gl] .
Finally, bot h th e zona l polynomials an d th e Hall-Littlewoo d function s ar e sub-
sumed a s limitin g case s o f th e two-paramete r symmetri c function s tha t ar e th e
subject o f §§11and 12. A s remarked a t th e en d o f §12, the structur e o f some of the
formulas suggest s strongl y tha t thes e symmetri c function s shoul d b e regarde d a s
attached to root system s of type A an d that the y should have counterparts attache d
to othe r roo t systems .
This mor e genera l theor y i s th e subjec t o f Chapte r II . Apar t fro m symmetri c
functions, ther e ar e tw o other forerunner s o f this theory , namel y th e theor y o f hy -
pergeometric function s an d Jacob i polynomial s attache d t o root system s develope d
by G . Heckma n an d E . Opda m [HO] , and th e successio n o f constan t ter m conjec -
tures (now , fortunately , al l theorems ) startin g wit h thos e o f F . J . Dyso n [D ] an d
G. E . Andrew s [A] . W e consider thes e input s i n turn .
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