Search Results for “ring of symmetric functions”

Source: Ring of symmetric functions

In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number n of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the representation theory of the symmetric group.
The ring of symmetric functions can be given a coproduct and a bilinear form making it into a positive selfadjoint graded Hopf algebra that is both commutative and cocommutative.

Symmetric polynomials

symmetric

The ring of symmetric functions

= Definitions

= Defining individual symmetric functions

can be taken as the definition of an elementary symmetric function if the number of indeterminates is infinite, or as the definition of an elementary symmetric polynomial in any finite number of indeterminates. Symmetric polynomials for the same symmetric function should be compatible with the homomorphisms ρn (decreasing the number of indeterminates is obtained by setting some of them to zero, so that the coefficients of any monomial in the remaining indeterminates is unchanged), and their degree should remain bounded. (An example of a family of symmetric polynomials that fails both conditions is

∏

i
=
1

n

X

i

{\displaystyle \textstyle \prod _{i=1}^{n}X_{i}}

; the family

∏

i
=
1

n

(

X

i

+
1
)

{\displaystyle \textstyle \prod _{i=1}^{n}(X_{i}+1)}

fails only the second condition.) Any symmetric polynomial in n indeterminates can be used to construct a compatible family of symmetric polynomials, using the homomorphisms ρi for i < n to decrease the number of indeterminates, and φi for i ≥ n to increase the number of indeterminates (which amounts to adding all monomials in new indeterminates obtained by symmetry from monomials already present).
The following are fundamental examples of symmetric functions.

The monomial symmetric functions mα. Suppose α = (α1,α2,...) is a sequence of non-negative integers, only finitely many of which are non-zero. Then we can consider the monomial defined by α: Xα = X1α1X2α2X3α3.... Then mα is the symmetric function determined by Xα, i.e. the sum of all monomials obtained from Xα by symmetry. For a formal definition, define β ~ α to mean that the sequence β is a permutation of the sequence α and set

m

α

=

∑

β
∼
α

X

β

.

{\displaystyle m_{\alpha }=\sum \nolimits _{\beta \sim \alpha }X^{\beta }.}

This symmetric function corresponds to the monomial symmetric polynomial mα(X1,...,Xn) for any n large enough to have the monomial Xα. The distinct monomial symmetric functions are parametrized by the integer partitions (each mα has a unique representative monomial Xλ with the parts λi in weakly decreasing order). Since any symmetric function containing any of the monomials of some mα must contain all of them with the same coefficient, each symmetric function can be written as an R-linear combination of monomial symmetric functions, and the distinct monomial symmetric functions therefore form a basis of ΛR as an R-module.
The elementary symmetric functions ek, for any natural number k; one has ek = mα where

X

α

=

∏

i
=
1

k

X

i

{\displaystyle \textstyle X^{\alpha }=\prod _{i=1}^{k}X_{i}}

. As a power series, this is the sum of all distinct products of k distinct indeterminates. This symmetric function corresponds to the elementary symmetric polynomial ek(X1,...,Xn) for any n ≥ k.
The power sum symmetric functions pk, for any positive integer k; one has pk = m(k), the monomial symmetric function for the monomial X1k. This symmetric function corresponds to the power sum symmetric polynomial pk(X1,...,Xn) = X1k + ... + Xnk for any n ≥ 1.
The complete homogeneous symmetric functions hk, for any natural number k; hk is the sum of all monomial symmetric functions mα where α is a partition of k. As a power series, this is the sum of all monomials of degree k, which is what motivates its name. This symmetric function corresponds to the complete homogeneous symmetric polynomial hk(X1,...,Xn) for any n ≥ k.
The Schur functions sλ for any partition λ, which corresponds to the Schur polynomial sλ(X1,...,Xn) for any n large enough to have the monomial Xλ.
There is no power sum symmetric function p0: although it is possible (and in some contexts natural) to define

p

0

(

X

1

,
…
,

X

n

)
=

∑

i
=
1

n

X

i

0

=
n

{\displaystyle \textstyle p_{0}(X_{1},\ldots ,X_{n})=\sum _{i=1}^{n}X_{i}^{0}=n}

as a symmetric polynomial in n variables, these values are not compatible with the morphisms ρn. The "discriminant"

(

∏

i
<
j

(

X

i

−

X

j

)

)

2

{\displaystyle \textstyle (\prod _{i
is another example of an expression giving a symmetric polynomial for all n, but not defining any symmetric function. The expressions defining Schur polynomials as a quotient of alternating polynomials are somewhat similar to that for the discriminant, but the polynomials sλ(X1,...,Xn) turn out to be compatible for varying n, and therefore do define a symmetric function.

= A principle relating symmetric polynomials and symmetric functions

=
For any symmetric function P, the corresponding symmetric polynomials in n indeterminates for any natural number n may be designated by P(X1,...,Xn). The second definition of the ring of symmetric functions implies the following fundamental principle:

If P and Q are symmetric functions of degree d, then one has the identity

P
=
Q

{\displaystyle P=Q}

of symmetric functions if and only if one has the identity P(X1,...,Xd) = Q(X1,...,Xd) of symmetric polynomials in d indeterminates. In this case one has in fact P(X1,...,Xn) = Q(X1,...,Xn) for any number n of indeterminates.
This is because one can always reduce the number of variables by substituting zero for some variables, and one can increase the number of variables by applying the homomorphisms φn; the definition of those homomorphisms assures that φn(P(X1,...,Xn)) = P(X1,...,Xn+1) (and similarly for Q) whenever n ≥ d. See a proof of Newton's identities for an effective application of this principle.

Properties of the ring of symmetric functions

= Identities

=
The ring of symmetric functions is a convenient tool for writing identities between symmetric polynomials that are independent of the number of indeterminates: in ΛR there is no such number, yet by the above principle any identity in ΛR automatically gives identities the rings of symmetric polynomials over R in any number of indeterminates. Some fundamental identities are

∑

i
=
0

k

(
−
1

)

i

e

i

h

k
−
i

=
0
=

∑

i
=
0

k

(
−
1

)

i

h

i

e

k
−
i

for all

k
>
0
,

{\displaystyle \sum _{i=0}^{k}(-1)^{i}e_{i}h_{k-i}=0=\sum _{i=0}^{k}(-1)^{i}h_{i}e_{k-i}\quad {\mbox{for all }}k>0,}

which shows a symmetry between elementary and complete homogeneous symmetric functions; these relations are explained under complete homogeneous symmetric polynomial.

k

e

k

=

∑

i
=
1

k

(
−
1

)

i
−
1

p

i

e

k
−
i

for all

k
≥
0
,

{\displaystyle ke_{k}=\sum _{i=1}^{k}(-1)^{i-1}p_{i}e_{k-i}\quad {\mbox{for all }}k\geq 0,}

the Newton identities, which also have a variant for complete homogeneous symmetric functions:

k

h

k

=

∑

i
=
1

k

p

i

h

k
−
i

for all

k
≥
0.

{\displaystyle kh_{k}=\sum _{i=1}^{k}p_{i}h_{k-i}\quad {\mbox{for all }}k\geq 0.}

= Structural properties of ΛR

=
Important properties of ΛR include the following.

The set of monomial symmetric functions parametrized by partitions form a basis of ΛR as a graded R-module, those parametrized by partitions of d being homogeneous of degree d; the same is true for the set of Schur functions (also parametrized by partitions).
ΛR is isomorphic as a graded R-algebra to a polynomial ring R[Y1,Y2, ...] in infinitely many variables, where Yi is given degree i for all i > 0, one isomorphism being the one that sends Yi to ei ∈ ΛR for every i.
There is an involutory automorphism ω of ΛR that interchanges the elementary symmetric functions ei and the complete homogeneous symmetric function hi for all i. It also sends each power sum symmetric function pi to (−1)i−1pi, and it permutes the Schur functions among each other, interchanging sλ and sλt where λt is the transpose partition of λ.
Property 2 is the essence of the fundamental theorem of symmetric polynomials. It immediately implies some other properties:

The subring of ΛR generated by its elements of degree at most n is isomorphic to the ring of symmetric polynomials over R in n variables;
The Hilbert–Poincaré series of ΛR is

∏

i
=
1

∞

1

1
−

t

i

{\displaystyle \textstyle \prod _{i=1}^{\infty }{\frac {1}{1-t^{i}}}}

, the generating function of the integer partitions (this also follows from property 1);
For every n > 0, the R-module formed by the homogeneous part of ΛR of degree n, modulo its intersection with the subring generated by its elements of degree strictly less than n, is free of rank 1, and (the image of) en is a generator of this R-module;
For every family of symmetric functions (fi)i>0 in which fi is homogeneous of degree i and gives a generator of the free R-module of the previous point (for all i), there is an alternative isomorphism of graded R-algebras from R[Y1,Y2, ...] as above to ΛR that sends Yi to fi; in other words, the family (fi)i>0 forms a set of free polynomial generators of ΛR.
This final point applies in particular to the family (hi)i>0 of complete homogeneous symmetric functions.
If R contains the field

Q

{\displaystyle \mathbb {Q} }

of rational numbers, it applies also to the family (pi)i>0 of power sum symmetric functions. This explains why the first n elements of each of these families define sets of symmetric polynomials in n variables that are free polynomial generators of that ring of symmetric polynomials.
The fact that the complete homogeneous symmetric functions form a set of free polynomial generators of ΛR already shows the existence of an automorphism ω sending the elementary symmetric functions to the complete homogeneous ones, as mentioned in property 3. The fact that ω is an involution of ΛR follows from the symmetry between elementary and complete homogeneous symmetric functions expressed by the first set of relations given above.
The ring of symmetric functions ΛZ is the Exp ring of the integers Z. It is also a lambda-ring in a natural fashion; in fact it is the universal lambda-ring in one generator.

= Generating functions

=
The first definition of ΛR as a subring of

R
[
[

X

1

,

X

2

,
.
.
.
]
]

{\displaystyle R[[X_{1},X_{2},...]]}

allows the generating functions of several sequences of symmetric functions to be elegantly expressed. Contrary to the relations mentioned earlier, which are internal to ΛR, these expressions involve operations taking place in R[[X1,X2,...;t]] but outside its subring ΛR[[t]], so they are meaningful only if symmetric functions are viewed as formal power series in indeterminates Xi. We shall write "(X)" after the symmetric functions to stress this interpretation.
The generating function for the elementary symmetric functions is

E
(
t
)
=

∑

k
≥
0

e

k

(
X
)

t

k

=

∏

i
=
1

∞

(
1
+

X

i

t
)
.

{\displaystyle E(t)=\sum _{k\geq 0}e_{k}(X)t^{k}=\prod _{i=1}^{\infty }(1+X_{i}t).}

Similarly one has for complete homogeneous symmetric functions

H
(
t
)
=

∑

k
≥
0

h

k

(
X
)

t

k

=

∏

i
=
1

∞

(

∑

k
≥
0

(

X

i

t

)

k

)

=

∏

i
=
1

∞

1

1
−

X

i

t

.

{\displaystyle H(t)=\sum _{k\geq 0}h_{k}(X)t^{k}=\prod _{i=1}^{\infty }\left(\sum _{k\geq 0}(X_{i}t)^{k}\right)=\prod _{i=1}^{\infty }{\frac {1}{1-X_{i}t}}.}

The obvious fact that

E
(
−
t
)
H
(
t
)
=
1
=
E
(
t
)
H
(
−
t
)

{\displaystyle E(-t)H(t)=1=E(t)H(-t)}

explains the symmetry between elementary and complete homogeneous symmetric functions.
The generating function for the power sum symmetric functions can be expressed as

P
(
t
)
=

∑

k
>
0

p

k

(
X
)

t

k

=

∑

k
>
0

∑

i
=
1

∞

(

X

i

t

)

k

=

∑

i
=
1

∞

X

i

t

1
−

X

i

t

=

t

E
′

(
−
t
)

E
(
−
t
)

=

t

H
′

(
t
)

H
(
t
)

{\displaystyle P(t)=\sum _{k>0}p_{k}(X)t^{k}=\sum _{k>0}\sum _{i=1}^{\infty }(X_{i}t)^{k}=\sum _{i=1}^{\infty }{\frac {X_{i}t}{1-X_{i}t}}={\frac {tE'(-t)}{E(-t)}}={\frac {tH'(t)}{H(t)}}}

((Macdonald, 1979) defines P(t) as Σk>0 pk(X)tk−1, and its expressions therefore lack a factor t with respect to those given here). The two final expressions, involving the formal derivatives of the generating functions E(t) and H(t), imply Newton's identities and their variants for the complete homogeneous symmetric functions. These expressions are sometimes written as

P
(
t
)
=
−
t

d

d
t

log
⁡
(
E
(
−
t
)
)
=
t

d

d
t

log
⁡
(
H
(
t
)
)
,

{\displaystyle P(t)=-t{\frac {d}{dt}}\log(E(-t))=t{\frac {d}{dt}}\log(H(t)),}

which amounts to the same, but requires that R contain the rational numbers, so that the logarithm of power series with constant term 1 is defined (by

log
⁡
(
1
−
t
S
)
=
−

∑

i
>
0

1
i

(
t
S

)

i

{\displaystyle \textstyle \log(1-tS)=-\sum _{i>0}{\frac {1}{i}}(tS)^{i}}

).

Specializations

Let

Λ

{\displaystyle \Lambda }

be the ring of symmetric functions and

R

{\displaystyle R}

a commutative algebra with unit element. An algebra homomorphism

φ
:
Λ
→
R
,

f
↦
f
(
φ
)

{\displaystyle \varphi :\Lambda \to R,\quad f\mapsto f(\varphi )}

is called a specialization.
Example:

Given some real numbers

a

1

,
…
,

a

k

{\displaystyle a_{1},\dots ,a_{k}}

and

f
(

x

1

,

x

2

,
…
,
)
∈
Λ

{\displaystyle f(x_{1},x_{2},\dots ,)\in \Lambda }

, then the substitution

x

1

=

a

1

,
…
,

x

k

=

a

k

{\displaystyle x_{1}=a_{1},\dots ,x_{k}=a_{k}}

and

x

j

=
0
,
∀
j
>
k

{\displaystyle x_{j}=0,\forall j>k}

is a specialization.
Let

f
∈
Λ

{\displaystyle f\in \Lambda }

, then

ps
⁡
(
f
)
:=
f
(
1
,
q
,

q

2

,

q

3

,
…
)

{\displaystyle \operatorname {ps} (f):=f(1,q,q^{2},q^{3},\dots )}

is called principal specialization.

References

Macdonald, I. G. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 1979. viii+180 pp. ISBN 0-19-853530-9 MR553598
Macdonald, I. G. Symmetric functions and Hall polynomials. Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. ISBN 0-19-853489-2 MR1354144
Stanley, Richard P. Enumerative Combinatorics, Vol. 2, Cambridge University Press, 1999. ISBN 0-521-56069-1 (hardback) ISBN 0-521-78987-7 (paperback).