Symmetric functions and combinatorial operators on polynomials

96 Pages·2001·0.599 MB·English

by  Alain Lascoux

SYMMETRIC FUNCTIONS AND
COMBINATORIAL OPERATORS ON
POLYNOMIALS
Alain Las
oux
CNRS, Institut Gaspard Monge, Universit(cid:19)e de Marne-la-Vall(cid:19)ee
77454 Marne-la-Vall(cid:19)ee Cedex, Fran
e
Alain.Las
ouxuniv-mlv.fr
Abstra
t
We present notes about symmetri
fun
tions and their generaliza-
tions.
The theory of symmetri
fun
tions allows to
ompute with fun
-
tions of several variables without seeing the variables. Histori
ally,
the \variables" were roots of a polynomial, and be
aue it was impos-
sible to expli
it them in degree higher than 4, the
lassi
s, starting
from Girard [10℄ and Newton [40℄, gave formulas to express power
sums, monomial fun
tions,
omplete fun
tions in terms of elementary
symmetri
fun
tions ( = the
oeÆ
ients).
We shall not get involved into
hange of bases in the ring Sym
and refer to Ma
donald [37℄ for that topi
.
Weprefertostressthefun
torialaspe
t: symmetri
fun
tionsmust
be
onsidered as operators on the ring of polynomials (in other inde-
terminates), or as operators on ve
tor spa
es or modules.
Thisiswhy we adopt the point of view of (cid:21)-rings, due to Grothen-
i i i
die
k, using (cid:3) , S , (cid:9) , instead of the more
ommon ei; hi; pi. In fa
t,
S
hur fun
tors in pla
e of S
hur fun
tions, are unavoidable as soon as
one takes several alphabets at the same time.
The avantage of (cid:21)-rings, apart from providing
ompa
t notations,
is to show that all properties of symmetri
fun
tions stem from the
Q
Cau
hy kernel i;j1=(1(cid:0)aibj). We should also have re
ourse to the
\plethysm", but our knowledge in this domain istoo s
anty to permit
using it widely.
To expli
it the multipli
ative stru
ture on Sym, one relies mostly
on Pieri formulas.
1
With these tools only, it just remain to translate in terms of sym-
metri
fun
tions su
h
lassi
al topi
s as Eu
lidean division,
ontinued
fra
tions, Pad(cid:19)e approximants, orthogonal polynomials, to see S
hur
fun
tions appear.
Instead of using only the invariants of the symmetri
group, it
is more enlightening to dire
tly use the symmetri
group itself, and

onsider the di(cid:11)erent ways it
an be made to a
t on polynomials. We
put a spe
ial emphasis on the Newton divided di(cid:11)eren
es whi
h have
been negle
ted during three hundred years.
The basis of S
hubert polynomials
an be introdu
ed by general-
izing Newton interpolation to several variables.
Fixing the number of variables, one has a non-symmetri
Cau
hy
kernel, a quadrati
form, and S
hubert polynomials appear now as
the appropriate generalization of S
hur fun
tions, instead of being a
linear basis for interpolation.
For the moment, we do not
onsider extensions to the other
las-
si
al groups, and end these notes with a non-
ommutative tou
h, em-
bedding Sym into the non-
ommutative algebra of Young tableaux.
2
oooo'oo ooo'ooo oooo'oo oooo'oo oooo'oo oooo'oo ooo'ooo oooo'oo ooo'ooo oooo'oo oooo'oo ooo'ooo oooo'oo ooo'ooo oooo'oo oooo'oo oooo'oo oooo'oo ooo'ooo oooo'oo oooo'oo oooo'oo oooo'oo ooo'ooo oooo'oo ooo'ooo oooo'oo oooo'oo
Symmetri
fun
tions
oooo'oo oooo'oo ooo'ooo oooo'oo ooo'ooo oooo'oo oooo'oo oooo'oo oooo'oo ooo'ooo oooo'oo oooo'oo oooo'oo oooo'oo ooo'ooo oooo'oo
We shall handle fun
tions on di(cid:11)erent sets of indeterminates (
alled al-
phabets, though we shall mostly use
ommutative indeterminates).
A symmetri
fun
tion of an alphabet A is a fun
tion of the letters whi
h
is invariant under permutation of the letters of A.
The simpler symmetri
fun
tions are best de(cid:12)ned through generating
fun
tions. We shall not use the
lassi
al notations for symmetri
fun
tions
(as they
an be found in Ma
donald's book), be
ause it will be
ome
lear in
the
ourse of these le
tures that we need to
onsider symmetri
fun
tions as
fun
tors, and
onne
t them with operations on ve
tor spa
es and representa-
tions. It is a small burden imposed on the reader, but the
ompa
t notations
that we propose greatlysimpli(cid:12)esmanipulationsof symmetri
fun
tions. No-
J
ti
e that exponents are used for produ
ts, and that S is di(cid:11)erent from SJ,
ex
ept if J is of length one (i.e. is an integer).
We need operations on alphabets, the (cid:12)rst one being the addition, that
is the disjoint union that we shall denote by a `+`-sign :
(cid:16) (cid:17)
A = fag ; B = fbg 7! A +B := fag[fbg
More operations will be introdu
ed in the se
ond se
tion.
(cid:15) Generating Fun
tions of symmetri
fun
tions
Taking an extra indeterminate z, one has three series
1
Y Y XX
1
i i
(cid:21)z(A) := (1+za) ; (cid:27)z(A) := ; (cid:9)z(A) := z a =i (1:1)
1(cid:0)za
a2A a2A i=1 a2A
i
the expansion of whi
h gives the elementary symmetri
fun
tions (cid:3) (A) the
i

omplete fun
tions S (A), and the power sums i(A) :
1
X X X
i i i i i
(cid:21)z(A) = z (cid:3) (A) ; (cid:27)z(A) = z S (A) ; (cid:9)z(A) = z i(A)=i : (1:2)
i=1
P
i
Sin
e log(1=(1(cid:0)a) = a =i, one has
i>0
(cid:27)z(A) = exp((cid:9)z(A)) ; (cid:9)z(A) = log((cid:27)z(A)) (1:3)
Addition of alphabets implies produ
t of generating series
(cid:21)z(A +B) = (cid:21)z(A)(cid:21)z(B) ; (cid:27)z(A +B) = (cid:27)z(A)(cid:27)z(B) : (1:4)
3
However, sin
e one
an invert formal series beginning by 1, or take any
power of them, one
an extend (1.1) by
Q
b2B(1(cid:0)zb)
(cid:27)z(A (cid:0)B) := Q ; (cid:27)z(
A) = ((cid:27)z(A) ;
2 C (1:5)
(1(cid:0)za)
a2A
In other words, formalseries beginning by 1 should be treated as generat-
ing series of
omplete or elementary symmetri
fun
tions of some alphabet,
that one
an manipulate without knowing its elements (as one
an use a
polynomial without being able to fa
torize it!).
Having written A + B for the disjoint union of alphabets for
e us to

onsider a (cid:12)nite alphabet as the sum of its sub-alphabets of
ardinality 1, i.e
P
to identify A and a.
a2A
Given a (cid:12)nite alphabet A, let Sym(A) be the ring of symmetri
polyno-
mials in A. As a ve
tor spa
e, it has (multipli
ative) bases
8
I i1 i2
<(cid:3) (A) := (cid:3) (A)(cid:3) (A)(cid:1)(cid:1)(cid:1)
I i1 i2
S (A) := S (A)S (A)(cid:1)(cid:1)(cid:1) ;
: I i1 i2
(cid:9) (A) := (cid:9) (A)(cid:9) (A)(cid:1)(cid:1)(cid:1)
sum over all k, all partitions I = [i1;::: ;ik℄, ik (cid:20)
ard(A).
I
The basis (cid:3) is due to Newton. Matri
es of
hange of bases are fairly
well known and
an be
omputed by having re
ourse to some
ombinatorial
obje
ts su
h as Young tableaux, matri
es with (cid:12)xed row and
olumn sums,
et
.
J
The sum of allelementsinthe orbitsof a monomiala under thea
tionof
thesymmetri
groupS(A) isof
ourseasymmetri
fun
tion,
alledmonomial
fun
tion and we shall denote it (cid:9)J(A), (J partition), rather than mJ.
It has been sin
e long realized that one should use alphabets of in(cid:12)nite

ardinality, and thus
onsider a universal ring Sym from whi
h one gets by
spe
ialization the rings Sym(A), for spe
i(cid:12)
alphabets A of (cid:12)nite
ardinality
(more generally, we shall use spe
ializations su
h that letters are no more
algebrai
ally independent.
(cid:15) Matrix generating fun
tions
Let z stands now for the in(cid:12)nite matrix with diagonal j (cid:0) i = 1 (cid:12)lled
with 1's, all other entries being 0's. Now (cid:27)z(A) is an Toeplitz matrix (i.e. a
matrix with
onstant values in ea
h diagonal) that we shall denote by S(A);
similarly (cid:21)z(A) is a matrix denoted L(A ) :
(cid:16) (cid:17) (cid:16) (cid:17)
j(cid:0)i j(cid:0)i
S(A) = S (A) & L(A ) = (cid:3) (A) (1:6)
i;j(cid:21)0 i;j(cid:21)0
4
Addition or subtra
tion of alphabets still
orrespond to produ
t of ma-
tri
es
(cid:6)1 (cid:6)1
S(A (cid:6)B) = S(A)S(B) & L(A (cid:6)B) = L(A )L(B) : (1:7)
Theadvantageofmatri
es,
omparedtoformalseries, isthattheyo(cid:11)erus
theirminors,thatwe shallindexby(in
reasing)partitions,ormoregenerally,
by ve
tors with
omponents in Z. More pre
isely, given I = (i1;::: ;in) 2
n n
Z , J = (j1;::: ;jn) 2 Z one de(cid:12)nes the skew S
hur fun
tion SJ=I(A) to
be the minor of S(A) taken on rows i1 + 1;i2 + 2;::: ;in + n and
olumns
j1 + 1;::: ;jn + n (the minor is 0 if one of these numbers is < 0). When
n
I = 0 , the minor is
alled a S
hur fun
tion and one writes SJ(A) instead
of SJ=0n(A).
In other words,
(cid:12) (cid:12)
SJ=I(A) = (cid:12)Sjk(cid:0)ih+k(cid:0)h(A)(cid:12)1(cid:20)h;k(cid:20)n : (1:8)
It is
onvenient to also use determinants in elementary symmetri
fun
-
tions :
(cid:12) (cid:12)
(cid:3)J=I(A) = (cid:12)(cid:3)jk(cid:0)ih+k(cid:0)h(A)(cid:12)1(cid:20)h;k(cid:20)n : (1:9)
i i i
Of
ourse, one must not forget that (cid:3) (A) = ((cid:0)1) S ((cid:0)A), i 2 Z, and
thus the (cid:3)J=I(A) are also skew S
hur fun
tions, but indexed by \
olumn
lengths".
Weshallalsoneed re
tangularsub-matri
esofS(A) thatwe shall
ontinue
to index the same way: SJ=I(A) is the sub-matrix of S(A) taken on rows
i1 +1;i2 +2;:::, and
olumns j1 +1;j2 +2;:::.
Binet-Cau
hy theorem for minors of the produ
t of two matri
es implies,
taking S(A +B), the following expansion on skew-S
hur fun
tions
X
SJ=I(A +B) = SJ=K(A)SK=I(B) ; (1:10)
K
sum over all partitions ( only those K : I (cid:18) K (cid:18) J give a non-zero
ontri-
bution).
Ja
obi's theorem on minors of the inverse of a matrix implies, denoting
(cid:24)
by J the
onjugate partition (obtained by transposing the Ferrers' diagram
of J):
jJ=Ij
(cid:3)J=I(A) = SJ(cid:24)=I(cid:24)(A) = ((cid:0)1) SJ=I((cid:0)A) (1:11)
One needs to enlarge the de(cid:12)nition of a S
hur fun
tion, to be able to play
with di(cid:11)erent alphabets at the same time.
5
Given n, given two sets of alphabets fA1;A2;::: ;Ang, fB1;B2;::: ;Bng,
n
and I;J 2 N , we de(cid:12)ne the multi-S
hur fun
tion
(cid:12) (cid:12)
SJ=I(A1 (cid:0)B1;::: ;An (cid:0)Bn) := (cid:12)Sjk(cid:0)ih+k(cid:0)h(Ak (cid:0)Bk)(cid:12)1(cid:20)h;k(cid:20)n : (1:12)
In the
ase where the alphabets are repeated, we indi
ate by a semi
olon
p q
the
orresponding blo
k separation : given H 2 Z , K 2 Z , then SH;K(A (cid:0)
B;C (cid:0)D) stands for the multi-S
hur fun
tion with index the
on
atenation
of H and K, and alphabets A1 = (cid:1)(cid:1)(cid:1) = Ap = A, B1 = (cid:1)(cid:1)(cid:1) = Bp = B,
Ap+1 = (cid:1)(cid:1)(cid:1) = Ap+q = C, Bp+1 = (cid:1)(cid:1)(cid:1) = Bp+q = D.
These fun
tions are now suÆ
iently general to allow easy indu
tions,
thanks to the following transformation lemma.
Lemma 1 Let SJ(A1 (cid:0) B1;::: ;An (cid:0) Bn) be a multi-S
hur fun
tion, and
D0;D1;::: ;Dn(cid:0)1 be a family of (cid:12)nite alphabets su
h that
ard(Di) (cid:20) i,
0 (cid:20) i (cid:20) n(cid:0)1. Then SJ(A1 (cid:0)B1;::: ;An (cid:0)Bn) is equal to the determinant
(cid:12) (cid:12)
(cid:12)Sjk(cid:0)ih+k(cid:0)h(Ak (cid:0)Bk (cid:0)Dn(cid:0)h)(cid:12)1(cid:20)h;k(cid:20)n
In other words, one does not
hange the value of a multi-S
hur fun
tion
SJ by repla
ing in row h the di(cid:11)eren
e A (cid:0) B by A (cid:0) B (cid:0) Dn(cid:0)h. Indeed,
thanks to the expansion (1.10) :
Sj(A(cid:0)B(cid:0)Dh) = Sj(A(cid:0)B)+S1((cid:0)Dh)Sj(cid:0)1(A(cid:0)B)+(cid:1)(cid:1)(cid:1)+Sh((cid:0)Dh)Sj(cid:0)h(A(cid:0)B) ;
the sum terminating be
ause the Sk((cid:0)Dh) are null for k > h, we see that the
determinant has been transformed by multipli
ation by a triangular matrix
with 1's in the diagonal, and therefore has kept its value. QED
For example, taking D0 = ;, D1 = fxg, D2 = fy;zg, one has
2 3
Si(A1 (cid:0)y (cid:0)z) Sj+1(A2 (cid:0)y (cid:0)z) Sh+2(A3 (cid:0)y (cid:0)z)
4 Si(cid:0)1(A1 (cid:0)x) Sj(A2 (cid:0)x) Sh+1(A3 (cid:0)x) 5 =
Si(cid:0)2(A1) Sj(cid:0)1(A2) Sh(A3)
2 3 2 3
1 (cid:0)y (cid:0)z yz Si(A1) Sj+1(A2) Sh+2(A3)
= 40 1 (cid:0)x5(cid:1)4Si(cid:0)1(A1) Sj(A2) Sh+1(A3)5
0 0 1 Si(cid:0)2(A1) Sj(cid:0)1(A2) Sh(A3)
and the determinant of the left matrix is equal to Sijh(A1;A2;A3).
Taking (cid:0)A1;(cid:0)A2;(cid:0)A3 instead of A1;A2;A3, and getting rid of signs be-

ause of \isobarity", one gets by the same token
2 3
(cid:3)i(A1 +y +z) (cid:3)j+1(A2 +y +z) (cid:3)h+2(A3 +y +z)
(cid:3)i;j;h(A1;A2;A3) = 4 (cid:3)i(cid:0)1(A1 +x) (cid:3)j(A2 +x) (cid:3)h+1(A3 +x) 5
(cid:3)i(cid:0)2(A1) (cid:3)j(cid:0)1(A2) (cid:3)h(A3)
6
In the pre
eding lemma,we needed only
onse
utive elements of a
olumn
to be
omplete fun
tions of the same di(cid:11)eren
e of alphabets, of
onse
utive
degrees. A similartransformation
an be performed in rows, when alphabets
are repeated in some
onse
utive
olumns, for partitions having repeated
parts.
Lemma 2 Let j;n be two integers, D0;::: ;Dn(cid:0)1 be a family of alphabets
su
h that
ard(Di) (cid:20) i, 0 (cid:20) i (cid:20) n (cid:0) 1, and let A, B be two arbitrary
alphabets. Let S(cid:5);jn;Æ(|; A(cid:0)B; (cid:127)) be a multi-S
hur fun
tion of whi
h we
have spe
i(cid:12)ed only n
olumns.
Then it is equal to the multi-S
hur fun
tion
S(cid:5);j;:::;j;Æ(|; A(cid:0)B(cid:0)D0;A(cid:0)B(cid:0)D1;::: ; A(cid:0)B(cid:0)Dn(cid:0)1; (cid:127)) :
The notations are a little too heavy for hand-
omputations. Indeed, one
usually play with either indi
es of fun
tions, or with alphabets, but not with
both at the time. For example, in the pre
eding
ases, one transforms only
alphabets, the indi
es of the
omplete fun
tions do not
hange. It seems
appropriate to use \umbral" notations and write alphabets on the border of
the determinant: an entry k in position (i;j) will be interpreted as Sk(A (cid:6)B)
if A is written in
olumn j and (cid:6)B in row i. Thus the pre
eding determinant
will be written (cid:12)
i j +1 h+2 (cid:12) (cid:0)y (cid:0)z
(cid:12)
i(cid:0)1 j h+1 (cid:12) (cid:0)x
(cid:12)
i(cid:0)2 j (cid:0)1 h (cid:12) (cid:1)
A1 A2 A3
The above lemma implies many fa
torization properties, e.g. for r (cid:21) 0,
r
SJ(A(cid:0)B(cid:0)x)x = SJ;r(A(cid:0)B;x) (1:13)
sin
e taking D1 = D2 = (cid:1)(cid:1)(cid:1) = fxg fa
torizes the determinant SJ;r(A(cid:0)B;x).
r
More generally, for an alphabet D of
ardinal (cid:20) r and J 2 N , one has
SI(A(cid:0)B(cid:0)D)SJ(D) = SI;J(A(cid:0)B ; D) : (1:14)
Monomial themselves
an be written as multi-S
hur fun
tions. Given
a totally ordered alphabet A = fa1;a2;:::g, denote, for any n, An :=
!
fa1;::: ;ang. Then, for any J = [j1;::: ;jn℄, denoting J := [jn;::: ;j1℄,
one has
J j1 jn
a := a1 (cid:1)(cid:1)(cid:1)an = SJ!(An;::: ;A2;A1)
Indeed, subtra
t the (cid:13)ag 0;A1;A2;::: in the su

essive rows, starting
from the bottom one. One sees the monomial appearing in the diagonal, the
7
upperpart ofthematrixvanishingbe
ause itis
onstitutedofSk((cid:0)(Aj(cid:0)Ai))
for k > (j (cid:0)i), j (cid:21) i.
Giventwo(cid:12)nitealphabets,thefollowingfa
torizationandvanishingprop-
erties impli
itly appear in many
lassi
al 19-th
entury texts about elimina-
tion theory (modern referen
e is Berele-Regev ?).
p (cid:11)
Proposition 3 Let A, B, be of
ardinalities (cid:11);(cid:12), p 2 N, I 2 N , J 2 N .
Then
Y
Si1;:::;ip;(cid:12)+j1;:::;(cid:12)+j(cid:11)(A (cid:0)B) = SI((cid:0)B)SJ(A) (a(cid:0)b) :
a2A;b2B
(cid:11)+1
Let J be a partition, J (cid:19) ((cid:12) +1) . Then SJ(A (cid:0)B) = 0.
Proof. Subtra
t A in the (cid:12)rst p rows. One gets the fa
torization
SI((cid:0)B)S(cid:12)+j1;:::;(cid:12)+j(cid:11)(A (cid:0)B) :
Now, using the partition K
onjugate to [(cid:12) + j1;::: ;(cid:12) + j(cid:11)℄, one gets the
fa
torization of SK(B (cid:0)A) into a S
hur fun
tion of A and S(cid:11)(cid:12)(B (cid:0)A). This
Q
last fun
tion
anbe seen equaltothe resultant (b(cid:0)a) by subtra
ting
a2A;b2B
the (cid:13)ag 0;B1;B2;:::.
The
ase J too big
an be treated by adding the same letters to A and
Q
B, so that one is redu
ed to the pre
eding
ase. But now the fa
tor (a(cid:0)b)
vanishes.
Given a (cid:12)nite alphabet A (that one will totally order: A = fa1;::: ;ang),
Cau
hy and Ja
obi separately de(cid:12)ned the S
hur fun
tion SJ(A) using the
(in(cid:12)nite) Vandermonde matrix
(cid:2) (cid:3)
j
V(A) = ai 1(cid:20)i(cid:20)n;j(cid:21)0
Q
and the Vandermonde (cid:1)(A) := i>j(ai (cid:0)aj).
n
Proposition 4 Let J 2 N . Then SJ(A)(cid:1)(A ) is equal to the minor of index
n
(0 ;J) of the Vandermonde matrix V(A).
Proof.LetSJ(A) denotes the sub-matrixofS(A)taken on
olumnsj1+1;j2+
2;::: ;jn+n. Consider the produ
t V(A)S((cid:0)A)SJ(A). It
an be fa
torized
in two manners, using (1.7):
V(A)S((cid:0)A)SJ(A) = [Sj(ai (cid:0)A)℄1 (cid:20) i (cid:20) n;j (cid:21) 0SJ(A) = V(A)S(0) :
However, the Sj(ai (cid:0)A) are null for j (cid:21) n, be
ause they are the elementary
fun
tions (up to sign) of alphabets of
ardinality n(cid:0)1. On the other hand,
8
Sj(0) = 0, if j 6= 0. In both
ases, we have obtained matri
es su
h that only
one minor of order n is di(cid:11)erent from 0. QED
(cid:15) Cau
hy formula
The most important formula in the theory of symmetri
fun
tions is the
following expansion, due to Cau
hy.
LetA;B be (cid:12)nitealphabets(thatwe shallsuppose ofthe same
ardinality
n). Then
Y Y X
K(A;B) := 1= (1(cid:0)ab) = SJ(A)SJ(B) ;
a2A b2B J
n
sum over all partitions J 2 N .
One will (cid:12)nd a proof of it using symmetrizing operators, in the exer
ises.
For the moment, let us re
all that this identity is intimately
onne
ted to
Binet-Cau
hy theorem, applied to Cau
hy's determinant j1=(1(cid:0)ab)j. This
tr
determinant fa
torizes into the two Vandermonde matri
es V(A)V (B) and
the minors of these matri
es are exa
tly S
hur fun
tions multiplied by the
Vandermondes (cid:1)(A) and (cid:1)(B) respe
tively. Now, the determinant itself is
equal to the sum (`((cid:27)) denoting the length of a permutation (cid:27))
X
1
`((cid:27))
((cid:0)1) (cid:16) (cid:17)(cid:27) :
(cid:27)2S(A) (1(cid:0)a1b1)(cid:1)(cid:1)(cid:1)(1(cid:0)anbn)
Q
Extra
ting the denominator 1=(1(cid:0)ab), one has to
ompute the sum
X (cid:16) (cid:17)(cid:27)
`((cid:27)) n(cid:0)1 n(cid:0)1
((cid:0)1) S (1+b1a1 (cid:0)b1A)(cid:1)(cid:1)(cid:1)S (1+bnan (cid:0)bnA)
(cid:27)2S(A)
This sum is divisible by (cid:1)(A)(cid:1)(B), be
ause it vanishes when two of the
a's, or two of the b's
oin
ide. Be
ause of degree, the quotient is a
onstant
and one has to
he
k that it is equal to 1.
The last step in the above demonstration miss the
ru
ial fa
t that what
is involved is Ja
obi symmetrizer
X
1
`((cid:27)) (cid:27)
C[a1;::: ;an℄ 3 f 7! ((cid:0)1) f ;
(cid:1)(A)
(cid:27)2S(A)
sum over all permutations (cid:27). Ja
obi's symmetrizer provides a
onne
tion
with the theory of
hara
ters (and extends to Weyl's
hara
ter formula for
the
lassi
al groups). We postpone this point of view to another se
tion.
(cid:15) S
alar Produ
t
9
There are other de
ompositions of K(A;B) as a sum of produ
ts of sym-
metri
fun
tions in A and in B. However, there is only one of the type
P
P(A)P(B) over Z: up to signs, the P's are all the S
hur fun
tions in-
dexed by partitions in N. Thus K(A;B), that we shall
all Cau
hy kernel,
determines the S
hur fun
tions. One
an interpret di(cid:11)erently the kernel, as
de(cid:12)ning a s
alar produ
t on the spa
e of symmetri
fun
tions, the S
hur
fun
tions
onstituting the only orthogonal basis. Now, any expansion of the
type
X
K(A;B) = PJ(A)QJ(B)
de(cid:12)ne a pair of adjoint bases fPJg, fQJg, with respe
t to the
anoni
al
s
alar produ
t (; ) indu
ed by K(A;B), i.e. the s
alar produ
t su
h that
(SJ;SJ) = 1, for all partition J.
There are some subtleties for what
on
erns s
alar produ
ts when taking
(cid:12)nite alphabets, and in the rest of the se
tion, we shall take only in(cid:12)nite
alphabets.
The expansion
Y Y Y(cid:16)X (cid:17) X
1
i i I
K(A;B) = ( ) = b S (A) = (cid:9)I(B)S (A)
1(cid:0)ab
b a b I
I
shows that the basis adjoint to S , I partition, is the monomial basis (cid:9)I.
The expansion
Y(cid:16)X (cid:17) X
i i I I
K(A;B) = a b = (cid:9) (A)(cid:9) (B)=zI ;
a;b I
m1 m2 m3
where zI = m1!1 m2!2 m3!3 (cid:1)(cid:1)(cid:1) (using the exponential notation for
m1 m2 m3
I = [1 ;2 ;3 ;:::℄), shows that the basis of produ
ts of power sums
is orthogonal, with s
alar produ
t ((cid:9)I;(cid:9)I) = zI.
The name kernel is justi(cid:12)ed by the following property, whi
h is just an-
other way of stating that K(A;B) de(cid:12)nes a s
alar produ
t:
Lemma 5 Let f be a symmetri
fun
tion and (; ) be the s
alar produ
t on
symmetri
fun
tions in A. Then
(K(A;B); f(A)) = f(B) :
(cid:15) Di(cid:11)erential
al
ulus
Having a s
alar produ
t, one
an now obtain operators adjoint to some
simple ones. We did not use the multipli
ative stru
ture of Sym up to now.
Any symmetri
fun
tion S
an be thought as the operator \multipli
ation
by S".
10