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A symmetric function is a polynomial which is invariant under
permutations of its indeterminates. The symmetric functions over a
commutative ring with unity, with an arbitrary
number of indeterminates, form an algebra, denoted by Λ.
The symmetric functions of fixed degree n form a submodule of Λ
denoted by Λn. When analysing and computing with elements of
Λ, there are 5 important bases to consider. These are the
bases consisting of the
Schur, Homogeneous, Power Sum,
Elementary and Monomial symmetric functions.
The size of any basis of Λn is equal to the number of partitions
of weight n, and each basis has its elements indexed by those partitions.
A partition is a weakly decreasing sequence of positive integers.
So, for example, [5, 3, 1] is a partition of weight 9, and [3, 1] is
a partition of weight 4.
For more information on partitions
see Section Partitions.
Magma allows computations with symmetric functions
in any of the 5 bases, and any symmetric function can be expressed in
any of the different bases.
The main reference on the theory of symmetric functions used here is the book
of Macdonald [Mac95].
In order to understand the different symmetric function bases, it is necessary
to have an understanding of Young Tableaux.
Firstly, for a given partition let us visualise its shape
making a diagram.
We arrange "boxes" in left-justified rows
corresponding to the entries of the partition.
So for example, the partition [4, 3, 1]
of weight 8 gives
matrix (  & & &
& &
)
This is known as a Ferrers diagram.
Now, to create a Young tableau, we must fill these boxes with integer entries,
but subject to some restrictions. We require that each row of
the tableau must be weakly increasing, and each column must be
strictly increasing. So, for example, a possible tableau of shape [4, 3, 1]
would be
T = matrix ( 1 & 3 & 3 & 4
2 & 4 & 5
4
)
A tableau can be mapped onto a monomial by treating each entry in the
tableau as an indeterminate. So, the above tableau will correspond to the
monomial T |-> xT = x1 x2 x32 x43 x5
With this connection between tableaux and monomials we are now able to define
the bases of an algebra of symmetric functions.
An important point to realise is that when dealing with a symmetric function,
the number of indeterminates of the corresponding polynomial is neither
fixed nor important. In fact a symmetric function has an image as a symmetric
polynomial in an arbitrary number of indeterminates, even an infinite
number. Magma is only equipped to deal with polynomial rings with
finite rank.
Since the partitions of weight n prescribe the symmetric function basis
elements of degree n which generate Λn, (the symmetric polynomials
in n indeterminates), then the number of indeterminates being used
will normally be equal to the degree of the symmetric functions considered.
- Schur Functions
Given a partition λ = [λ1, λ2, ... , λl]
of weight n,
the Schur function sλ is given
by sλ = ∑T xT where the sum
is taken over all Young tableaux with shape λ, whose entries are
all less than or equal to m, (the number of required indeterminates).
- Elementary Functions
For a positive integer k, define the k-th elementary symmetric function,
ek,
as the Schur function of [1k]. Then, for any partition
λ = [λ1, λ2, ..., λl], we define the elementary symmetric function
indexed by λ to be eλ = ∏i eλi.
- Homogeneous Functions
For a positive integer k, define the k-th homogeneous symmetric function,
hk, as the Schur function of [k]. Then, for any partition
λ = [λ1, λ2, ..., λl], we define the homogeneous symmetric function
indexed by λ to be hλ = ∏i hλi.
- Monomial Functions
For a partition λ = [λ1, λ2, ..., λl],
let xλ be given by
x1λ1x2λ2 ... xlλl.
The monomial symmetric function
mλ is the sum over the orbits of xλ
under the action of the symmetric group. Where the number of indeterminates
exceeds the length of the partition, λ can be considered to have
trailing zeros.
- Power Sums
For a positive integer k, define the k-th power sum symmetric function,
pk,
as the monomial function of [k]. Then, for any partition
λ = [λ1, λ2, ..., λl],
we define the power sum symmetric function
indexed by λ to be pλ = ∏i pλi.
For each of these classes of functions, the set of all functions
indexed by partitions
of weight n forms a basis for Λn.
Taking functions indexed by partitions of all weights gives a
basis of Λ.
It is possible to define an inner product on Λ by requiring
that the bases (mλ) and (hλ) be dual to each other:
< mλ, hλ ' > = δλ, λ '
With respect to this inner product, the Schur functions are an
orthonormal basis of Λn and Λ and the power sum
basis is an orthogonal basis.
We give an example illustrating the correspondence between symmetric functions
written with respect to a symmetric function basis and symmetric polynomials.
While the degree of the symmetric function fixes the
degree of the resulting polynomial, the number of indeterminates is
arbitrary.
> Q := Rationals();
> S := SFASchur(Rationals());
> e := S.[2,1];
> P2<[x]> := PolynomialRing(Q, 2);
> P2 ! e;
x[1]^2*x[2] + x[1]*x[2]^2
> P3<[y]> := PolynomialRing(Q, 3);
> P3 ! e;
y[1]^2*y[2] + y[1]^2*y[3] + y[1]*y[2]^2 + 2*y[1]*y[2]*y[3] + y[1]*y[3]^2 +
y[2]^2*y[3] + y[2]*y[3]^2
> P4<[z]> := PolynomialRing(Q, 4);
> P4 ! e;
z[1]^2*z[2] + z[1]^2*z[3] + z[1]^2*z[4] + z[1]*z[2]^2 + 2*z[1]*z[2]*z[3] +
2*z[1]*z[2]*z[4] + z[1]*z[3]^2 + 2*z[1]*z[3]*z[4] + z[1]*z[4]^2 +
z[2]^2*z[3] + z[2]^2*z[4] + z[2]*z[3]^2 + 2*z[2]*z[3]*z[4] +
z[2]*z[4]^2 + z[3]^2*z[4] + z[3]*z[4]^2
>
> P5<[w]> := PolynomialRing(Q, 5);
> P5 ! e;
w[1]^2*w[2] + w[1]^2*w[3] + w[1]^2*w[4] + w[1]^2*w[5] + w[1]*w[2]^2 +
2*w[1]*w[2]*w[3] + 2*w[1]*w[2]*w[4] + 2*w[1]*w[2]*w[5] + w[1]*w[3]^2 +
2*w[1]*w[3]*w[4] + 2*w[1]*w[3]*w[5] + w[1]*w[4]^2 + 2*w[1]*w[4]*w[5] +
w[1]*w[5]^2 + w[2]^2*w[3] + w[2]^2*w[4] + w[2]^2*w[5] + w[2]*w[3]^2 +
2*w[2]*w[3]*w[4] + 2*w[2]*w[3]*w[5] + w[2]*w[4]^2 + 2*w[2]*w[4]*w[5] +
w[2]*w[5]^2 + w[3]^2*w[4] + w[3]^2*w[5] + w[3]*w[4]^2 +
2*w[3]*w[4]*w[5] + w[3]*w[5]^2 + w[4]^2*w[5] + w[4]*w[5]^2
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