Introduction
The ai m o f thi s Introductio n i s t o provid e som e backgroun d an d perspectiv e
to th e subjec t o f thes e lecture s fo r th e benefi t o f reader s wh o ar e no t alread y
well-versed i n thes e matter s an d t o explai n it s connection s wit h othe r branche s
of mathematics , suc h a s combinatoric s an d representatio n theory . Reference s i n
square bracket s ar e to the bibliograph y a t th e en d o f this Introduction , not t o tha t
at th e en d o f the book .
Symmetric function s
The theory o f symmetric function s i s one of the mos t classica l parts o f algebra ,
going bac k t o th e 16th an d 17th centurie s an d th e attempt s o f mathematician s
of tha t epoc h t o solv e polynomia l equation s o f degre e highe r tha n th e second .
The coefficient s o f a polynomia l equatio n i n on e unknow n are , u p t o sign , jus t
the elementar y symmetri c function s o f th e root s o f tha t equation , an d henc e an y
symmetric polynomia l i n th e root s ca n b e expresse d uniquel y a s a polynomia l i n
the coefficients . Fo r example , th e equation s
n
nen = y^(-l)
r~1pren_r,
r = l
which serv e t o expres s th e powe r sum s p
r
i n term s o f th e elementar y symmetri c
functions e r, wer e firs t foun d b y Isaa c Newto n i n th e 17th century . Th e firs t par t
of Chapte r I (§§1-6) provide s a rapi d survey , fro m a moder n viewpoint , o f thi s
classical material .
Schur function s an d thei r generalization s
Of th e variou s familie s o f symmetri c functions , th e mos t significan t ar e un -
doubtedly th e Schu r functions , becaus e o f thei r intimat e relationshi p wit h th e ir -
reducible character s o f bot h th e symmetri c group s an d th e genera l linea r groups ,
and fo r thei r combinatoria l applications .
Let u s conside r th e representation-theoreti c aspec t first.
(i) Th e irreducibl e (complex ) character s \ X ° f t n e symmetri c grou p S
n
ar e
indexed b y th e partition s A of n. S o also ar e th e conjugac y classe s i n 5
n
, becaus e
the conjugac y clas s o f a permutatio n i s determine d b y it s cycle-type , whic h i s a
partition o f n. Th e valu e x i ° f th e characte r x
A
a t a n elemen t o f cycle-typ e \i i s
then give n b y th e scala r produc t
Equivalently, \a *
s ^n e
coefficien t o f s\ i n th e expressio n o f p^ a s a su m o f Schu r
functions.
xi
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