CHAPTE R 1
Symmetric function s
1.1. Alphabet s
We shal l trea t function s o n differen t set s o f indeterminate s (calle d alphabets,
though w e shal l mostl y us e commutativ e indeterminate s fo r th e moment) .
A symmetri c functio n o f a n alphabe t A i s a functio n o f th e letter s whic h i s
invariant unde r permutatio n o f th e letter s o f A .
The simple r symmetri c function s ar e bes t define d throug h generatin g functions .
We shal l no t us e th e classica l notation s fo r symmetri c function s (a s foun d i n Mac -
donald's boo k [136]), because , a s wil l becom e clea r i n th e cours e o f thes e lectures ,
we nee d t o conside r th e symmetri c function s a s functors, an d connec t the m wit h
operations o n vecto r space s an d representations . Th e compac t notation s tha t w e
propose greatl y simplif y manipulation s o f symmetri c functions . Notic e tha t expo -
nents ar e use d fo r products , an d tha t S J i s differen t fro m Sj, excep t whe n J i s o f
length on e (i.e . i s a n integer) .
J = \j
u
j2i ...]= A J - A J1 A^'2 & S J = S jlSj2 & ^ J = 9^j2...
are differen t fro m Sj, ij)j etc .
In cas e o f lengt h 1, w e shal l indifferentl y writ e indice s o r exponent s fo r th e
same function s :
SJ = Sj , A j = A
3
, V j = ty
3
.
We shal l nee d operation s o n alphabets , th e firs t bein g addition, tha t is , th e
disjoint union , whic h w e denot e b y a 'V-sig n :
(A - {a} , B = {b}) ^ A + B : = {a} U {b}
Other operation s wil l b e introduce d i n Chapte r 2 .
1.2. Partition s
A weakl y increasin g sequenc e o f strictl y positiv e number s / = [ii, i
2
, .., ie] i s
called a partition of the number n of length £(I) H, wher e n \I\ :— i\ + + ie.
One als o use s weakl y decreasin g sequence s instea d o f increasin g ones , an d refe r t o
them a s decreasing partitions, bu t t o handl e minor s o f matrices , ou r conventio n i s
preferable.
A partitio n / ha s a graphica l representatio n du e t o Ferrers , calle d it s diagram:
it i s a diagra m o f lef t packe d squar e boxes , wit h i\, i
2
,..., it th e numbe r o f boxe s
in th e successiv e rows . Readin g th e numbe r o f boxe s i n th e successiv e columns , on e
obtains anothe r partitio n I~ whic h i s calle d th e conjugate partition. Conjugatin g
l
http://dx.doi.org/10.1090/cbms/099/01
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