2
1. SYMMETRI C FUNCTION S
for th e rin g o f symmetri c function s (resp . symmetri c polynomial s i n n indetermi -
nates) wit h coefficient s i n R.
In th e followin g section s o f thi s chapte r w e shal l surve y variou s base s o f th e
ring o f symmetric functions . The y wil l al l b e indexe d b y partitions. A partition i s
a (finit e o r infinite ) sequenc e
A = (Ai , A2, A3,...)
of integers, suc h tha t A i A 2 0 , an d
|A|
=
Y1X* o c
so tha t fro m a certai n poin t onward s (i f th e sequenc e A is infinite ) al l th e A ? are
zero. W e shal l no t distinguis h betwee n tw o suc h sequence s whic h diffe r onl y b y
a strin g o f zero s a t th e end . Thu s (2,1), (2,1,0), an d (2,1, 0, 0,...) ar e al l t o b e
regarded a s the sam e partition .
The nonzer o A ^ are calle d th e parts o f A , and th e numbe r o f parts i s the length
l(X) o f A . I f A has m\ part s equa l t o 1, rri2 part s equa l t o 2 , an d s o on , w e shal l
occasionally writ e A = (l
m
i2
m 2
•••) , althoug h strictl y speakin g w e shoul d writ e
this i n revers e order .
Let V denot e the se t o f all partitions, an d V
n
th e se t o f all partitions o f n (i.e. ,
such tha t |A | = n) . Th e natural (o r dominance) partia l orde r o n V i s define d a s
follows.
(1.2) A \i if and only if |A | = |/x | and A H ( - Ar fi\ -\ h fj,r for all r 1.
It i s a total orde r o n V
n
fo r n 5 , but no t fo r n 6.
With each partition A we associate a diagram, consisting of the points (i,j) £ Z 2
such tha t 1 j X{. W e adop t th e conventio n (a s wit h matrices ) tha t th e first
coordinate i (th e ro w index ) increase s a s on e goe s downwards , an d th e secon d
coordinate j (th e colum n index ) increase s fro m lef t t o right . Ofte n i t i s mor e
convenient t o replac e th e lattic e point s (i,j ) b y squares , an d the n th e diagra m o f
A consists o f A i boxe s i n th e to p row , A 2 boxes i n th e secon d row , an d s o on ; th e
whole arrangemen t o f boxes bein g left-justified .
If w e rea d th e diagra m o f a partitio n A by columns , w e obtai n th e conjugate
partition A' . Thu s A ^ is the numbe r o f boxe s i n th e jt h colum n o f th e diagra m o f
A, and henc e i s equal t o th e numbe r o f part s o f A that ar e j . Equivalently , th e
diagram o f A ' i s obtaine d fro m tha t o f A by reflectio n i n th e mai n diagonal . I t i s
not difficul t t o sho w tha t
A / i ^ [i A' .
2. Monomia l symmetri c function s
Let A = (Ai , A2,. •.) b e a partition. I t determine s a monomia l
T ^ ^ l / r ^ 2 . . .
Jb Jb -I Jby
The monomial symmetric function m\ i s th e su m o f al l distinc t monomial s tha t
can be obtained fro m x
x
b y permutations o f the x's. Fo r example, m^i) = Yl
x1jix
summed ove r al l pairs (i,j) suc h tha t i ^ j .
For each partition A , the image 771^(^1 ?
xn)
otm\ i n An (obtaine d by setting
Xm 0 fo r al l m n) i s zer o i f an d onl y i f /(A ) n . A s A runs throug h th e
partitions o f lengt h n , th e m\(xi, ..., x n) for m a Z-basi s o f A n, an d a s A runs
through al l partitions th e m\ for m a Z-basi s o f A.
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