mλ = x12x2+ x22x1+ x21x3+ x32x1+ x22x3+ x23x2, eλ = (x1x2+ x1x3+ x2x3)(x1+ x2+ x3), pλ = (x21+ x22+ x32)(x1+ x2+ x3), hλ = eλ+ pλ
Note V.17: It can be convenient to consider N arbitrary large or even infinite – the identities amongst symmetric functions still hold.
Each of these classes of symmetric functions allows to generate all:
Theorem V.18: Suppose that N ≥ n and f a homogeneous symmetric function of degree n in x1, . . . , xN. Then
a) f = ∑λ⊢ncλmλfor suitable coefficients cλ. b) f = ∑λ⊢ncλeλfor suitable coefficients cλ. c) f = ∑λ⊢ncλhλfor suitable coefficients cλ. d) f = ∑λ⊢ncλpλfor suitable coefficients cλ.
In each case the coefficients cλ are unique, furthermore in cases a,b,c), if f has integral coefficients, then all cλare integral.
Corollary V.19: Any symmetric function f (x1, . . . , xN) can be written as a poly-nomial g(z1, . . . , zN), where z stands for one of the symbols e,h, p.
Note V.20: For algebraists we remark that one can show that the ring of symmetric polynomials in N variables has transcendence degree N and that the {ei}, the {hi} and the {pi} each form a transcendence basis of this ring.
Proof:[of Theorem V.18, part a)] If f = ∑αcαxαis homogeneous of degree n, then
f =∑λ⊢ncλmλ. ◻
Corollary V.21: The set {mλ}λ⊢nis a vector space basis of Λn, thus dim(Λn) = p(n).
We shall see below that each of the other sets, {eλ}, {hλ}, {pλ}, also forms a basis.
V.6 Base Changes
The proof we shall give for the remaining parts of Theorem V.18 will be based on linear algebra: We shall consider the coefficients of expressing one set of symmetric
functions in terms of another set, and obtain a combinatorial interpretation of these coefficients. From this we shall deduce that the matrix of coefficients is invertible (and possibly has an integral inverse). This implies that the other sets also form a basis and thus the remaining parts of the theorem.
As we will argue with matrices, it involves an arrangement of basis vectors, we thus need to define an ordering on partitions:
Definition V.22: Let λ = (λ1, λ2, . . . , λk) and Let µ = (µ1, µ2, . . . , µl) be parti-tions of n. We assume WLOG that k = l by allowing cells of size 0.
We say that µ ≤ λ in the natural order, (also called dominance order or ma-jorization order) if µ /= λ and for any i ≥ 1 we have that
µ1+ µ2+ ⋯ + µi≤ λ1+ λ2+ ⋯ + λi.
Following Theorem III.3, this partial order has a total order as a linear exten-sion, one such total order is given in exercise ??.
Definition V.23: Let A = (ai j) a matrix. We consider row sums ri = ∑jai jand column sums cj= ∑iai jand call row(A) = (r1, r2, . . .) the row sum vector, respec-tively col(A) = (c1, c2, . . .) the column sum vector.
Since the mλform a basis there are integral coefficients Mλ µ, such that eλ= ∑
µ⊢n
Mλ µmµ. In fact Mλ µis simply the coefficient of xµin eλ.
Lemma V.24: Mλ µ equals the number of (0, 1)-matrices (that is, matrices whose entries are either 0 or 1) A = (ai j) satisfying row(A) = λ and col(A) = µ.
Proof: Let λ = (λ1, λ2, . . .). Any term of eλis a product of a term of eλ1, eλ2, etc. and any term of eλi is the product of i different variables. We shall describe any term using a matrix. Let
X =
⎛⎜
⎝
x1 x2 x3 ⋯ x1 x2 x3 ⋯
⋮ ⋮
⎞⎟
⎠
Then each monomial xαof eλ is the product of λ1entries from the first row, λ2
entries from the second row and so on. We describe the selection of the factors by a (0, 1)-matrix A with 1 indicating the selected factors.
The products contributing to a particular monomial xαcorrespond exactly to the matrices with row sums λ and column sums α. The statement follows. ◻
We denote by M = (Mλ µ) the matrix of these coefficients.
As the transpose of a matrix exchanges rows and columns we get Corollary V.25: The matrix M is symmetric, that is Mλ µ= Mµ λ.
V.6. BASE CHANGES 81 We now establish the fact that the matrix M is, for a suitable arrangement of the partitions, triangular with diagonal entries 1. This shows6that det M = 1, that is M is invertible over Z. This in turn implies Theorem V.18, b).
Proposition V.26: Let λ, µ ⊢ n. Then Mλ µ = 0, unless µ ≤ λ∗. Also Mλ λ∗ = 1.
Thus, if the λ are arranged in (a linear extension of) the natural ordering, and the µ according to ordering of its duals, then M is upper triangular with diagonal 1.
Proof: Suppose that Mλ µ /= 0, so by Lemma V.24 there is a (0, 1)-matrix A with row(A) = λ and col(A) = µ. Let A∗be the matrix with row(A∗) = λ and the 1’s left justified, that is A∗i j= 1 if and only if 1 ≤ j ≤ λi. By definition of the dual, we have that col(A∗) = λ∗. On the other hand, for any j the number of 1’s in the first j columns of A is not less that the number of 1’s in the first j columns of A, so in the natural order of partitions we have that
λ∗= col(A∗) ≥ col(A) = µ.
To see that Mλ λ∗ = 1 we observe that A∗is the only matrix with row(A∗) = λ and
col(A∗) = λ∗. ◻
Note V.27: This approach of proving that the eλform a basis — expressing the eλ in terms of the mµ, giving a combinatorial interpretation of the coefficients in this expression, and deducing from there that the coefficient matrix must be invertible
— could also be applied to the other two bases, hλand pλ. We shall instead give other proofs, in part because they illustrate further aspects of symmetric functions.
For the reader who feels that we have moved far away from enumerating com-binatorial structures, we close this section with the following observation (whose proof is left as exercise ??):
Suppose we have n balls in total, of which λiballs are labeled with the number i. We also have boxes 1, 2,. . . . Then Mλ µis the number of ways to place the balls into the boxes such that box j contains exactly µjballs, but no box contains more than one ball with the same label.
Complete Homogeneous Symmetric Functions
For the complete homogeneous symmetric functions hλwe get the following ana-log to Lemma V.24:
Lemma V.28: Define Nλ µas the coefficient of xµin hλ, that is hλ= ∑
µ⊢n
Nλ µmµ.
Then Nλ µequals the number of N0-matrices A = (ai j) satisfying row(A) = λ and col(A) = µ.
6A consequence of the Cayley-Hamilton theorem is that the inverse of a matrix A is a polynomial in A with denominators being divisors of det(A).
Proof: Exercise ??
We now establish a duality between the e-polynomials and the h-polynomials.
Definition V.29: We define a map ω∶ Λ → Λ by setting
ω∶ en→ hn
eλ→ hλ
and extending linearly.
As the {eλ}λ⊢nfor a basis of Λnthis defines ω uniquely and shows that ω is a linear map. Since the eiare also algebraically independent (a fact we shall take as given), ω is also an algebra endomorphism, that is it preserves products.
Proposition V.30: The endomorphism ω is an involution, that is ω2is the identity.
In particular ω(hn) = enand ω(hλ) = eλ.
We note that Theorem V.18, c) is an immediate consequence of this proposition.
Proof: We go in the ring Λ[[t]] of formal power series over Λ and set
E(t) ∶= ∑
n≥0
entn,
H(t) ∶= ∑
n≥0
hntn.
We have that
E(t) = ∏N
r=1
(1 + xrt),
H(t) = ∏
r
(1 + xrt + x2rt2+ ⋯) =∏N
r=1
(1 − xrt)−1,
as can be seen by expanding the products on the right hand side. This implies that H(t)E(−t) = 1. Comparing coefficients of tnon both sides yields
0 =∑n
i=0
(−1)ieihn−i, n ≥ 1.
We now apply ω, use ω(ei) = hi, and reindex and thus get 0 =∑n
i=0
(−1)ihiω(hn−i) and 0 =∑n
i=0
(−1)ihn−iω(hi)
We consider this as a linear system of equations with the ω(hi) as variables. The coefficient matrix of this system is lower triangular with diagonal entries h0 = 1, thus has full rank. The solution therefore must be unique, but we know already
V.6. BASE CHANGES 83
that the eiform a solution, proving the theorem. ◻
We finally come to the power sum functions pλ. Define P(t) ∶=∑
n≥1
pntn−1
Proposition V.31: ddt H(t) = P(t)H(t) and d
dt E(t) = P(−t)E(t).
The result for H follows by logarithmic differentiation. The argument for E is
ana-log. ◻
By considering the coefficients of the power series, we can write this result in the form nhn = ∑nr=1prhn−r. This allows us to express the hi in terms of the pi, albeit at the cost of introducing denominators. For example, h2= 21(p21+ p2).
Therefore the pλ generate as vector space and we get Theorem V.18 d); albeit not necessarily with integral coefficients.
We close this section with the mention of another basis of symmetric polyno-mials which is of significant practical relevance, but more complicated to define.
Given a partition λ = (λ1, λ2, . . .), define
By the properties of the determinant, snum is invariant under all even permuta-tions of the variables, but is not symmetric. It thus must be divisible by the Vander-monde determinant ∏i< j(xi− xj), and the quotient
sλ(x1, . . . , xn) = snumλ(x1, . . . , xn)
∏i< j(xi− xj)
is symmetric. We call the set of sλthe Schur polynomials. They form another basis of Λn, but we will not prove this here.
Amongst the many interesting properties of Schur functions, one can show, for example, that in the expression sλ = ∑µKλ µmµ, the coefficient Kλ µ(called a Kostka Number) is given by the number of semistandard tableaux (that is, we allow duplicate entry and enforce strict increase in columns, but allow equality or increase in rows) of shape λ. Also the base change matrix from {pλ} to {sλ} is the character table of the symmetric group Sn.