CHAPTER 1
Symmetric Function s
1. Th e rin g o f symmetri c function s
Let a?i,.. . ,xn b e independen t indeterminates . Th e symmetri c grou p S
n
act s
on th e polynomia l rin g Z[#i,... , xn] b y permutin g th e x' s , and w e shall writ e
An = Z[x
u
...,xn]Sn
for th e subrin g o f symmetric polynomial s i n # i , . . . , xn. I f / G An, w e may writ e
r0
where f^ i s th e homogeneou s componen t o f / o f degre e r . Eac h f^ i s itsel f
symmetric, an d s o A
n
i s a graded ring :
r0
whereAJ^ is the additiv e grou p o f homogeneous symmetri c polynomial s o f degree r
in x\, ..., x n. (B y convention , 0 is homogeneous o f every degree. )
If w e no w adjoi n anothe r indeterminat e x
n
+i w e c a n for m A
n +
i =
Z[#i,..., xn+i]Sn+1, an d w e have a surjectiv e homomorphis m (o f graded rings )
(1.1) A
n + 1n
^ A ,
defined b y settin g x
n
+\ 0. Th e mappin g A
n+1
A
n
i s surjectiv e fo r al l r 0 ,
and bijectiv e i f and onl y i f r n.
Often i t i s convenient t o pas s t o th e limit . Le t
Ar = limA ;
n
for eac h r 0 , and le t
A = 0A".
r0
By th e definitio n o f invers e (o r projective ) limits , a n elemen t o f A r i s a sequenc e
{fn)no wher e f
n
G A^ fo r eac h n, an d f
n
i s obtained fro m /
n + 1
b y setting x
n+
i
0. W e ma y therefor e regar d th e f
n
a s th e partia l sum s o f a n infinit e serie s / o f
monomials o f degre e r i n infinitel y man y indeterminate s x\,X2, .... Fo r example ,
if f
n
= x\ - f + x
n
for eac h n , the n / = X^S i
x
i- Thu s th e element s o f A are n o
longer polynomials , an d ar e traditionall y calle d symmetri c functions.
For eac h n ther e i s a surjectiv e homomorphis m A —» A n, obtaine d b y settin g
Xjn = 0 for al l m n. Th e grade d rin g A is the ring of symmetric functions. I f R
is any commutativ e ring , w e shall writ e
A# = A 0Z R, A n,R = A n®zR
I
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