4. COMPLET E SYMMETRI C FUNCTION S
3
3. Elementar y symmetri c function s
When A = (l
r
), m\ i s the rt h elementary symmetric function e r:
i\---ir
(For r = 0, we define e o to b e 1.) Th e e
r
hav e th e generatin g functio n
(3.1) E(t) =
Y,ertr
= Y[(l + Xit)
r0 il
where t i s another indeterminate .
For eac h partitio n A = (Ai , A2,...) le t
ex = e Xlex2
Then the ex form another Z-basis of A as A runs through all partitions. Equivalently ,
we hav e
(3.2) A = Z[ei , e2,... ], and ei , e2,. •. ar e algebraically independent over Z .
PROOF. Conside r th e produc t ev , wher e A ' is the conjugat e o f A . Whe n mul -
tiplied ou t i t i s a sum o f monomials o f the for m
\Xi1Xi2 - - )\Xj1Xj2 ') x ,
say, wher e i\ Z2 i\ , ji 32 " ' j \ '
2
,
a n
d s o on. I f w e now ente r th e
numbers ii , ^2? •» *A' m orde r dow n the first colum n o f the diagra m o f A, then th e
numbers 31,32, ,j\'2 i n order dow n the secon d column , an d s o on, i t i s clear tha t
for eac h r 1 all th e number s r s o entere d i n th e diagra m o f A must occu r i n
the to p r rows . Henc e a\ + + a
r
Ai + -f A
r
fo r eac h r 1, and i t follow s
that e\i i s of the for m
(3.3) e\* =^x + ^2 aA ^mM
with nonnegativ e intege r coefficient s ax^. Th e triangula r for m o f thes e equation s
shows tha t the y ca n b e solve d t o giv e mx a s a linea r combinatio n o f the e
M
.
4. Complet e symmetri c function s
For eac h r 1, the rt h complete symmetric function h
r
i s
hr = ^
m
A,
|A|=r
the su m o f al l monomial s o f degre e r . (For r 0, w e defin e ho t o b e 1.) Th e
generating functio n o f the h
r
i s
(4.1) if(t)=^M r =n( i - a : **)~ 1'
r0 i\
as on e see s b y expandin g eac h facto r ( 1 Xit)~
l
a s a geometri c serie s an d the n
multiplying thes e serie s together. Fro m (3.1) an d (4.1) i t follow s tha t
H(t)E(-t) = 1,
Previous Page Next Page