Polynomials and Representations V

It was clear from the earlier articles that n (number of variables x_1, \ldots, x_n) plays a minimal role in the combinatorics of the symmetric polynomials. Hence, removing the parameter n turns out to be quite convenient; the process gives us the formal ring of symmetric functions.

Concrete Definition of Ring Λ

Consider the set of all monomials x^a := x_1^{a_1} \ldots x_m^{a_m}, where a = (a_1, a_2, \ldots) is a vector of non-negative integers such that only finitely many terms are non-zero. Thus x_1 x_2 and x_1 x_2^2 x_3^3 are monomials but x_1 x_2 x_3 \ldots is not. Now for d = 0,1,2,\ldots, consider the additive group C_d of all sums

\displaystyle\sum_{\sum a_i = d} c_a x^a, \qquad c_a \in \mathbb{Z} \text{ for each } a.

Though each a has only finitely many non-zero terms, the collection of all c_a‘s may have infinitely many non-zero terms. E.g. x_1 + 2x_3 + 3x_5 + 4x_7 + \ldots is a perfectly valid element of C_1.

Now let \Lambda^{(d)} be the subgroup of all \sum_a c_a x^a \in C_d such that c_a = c_{\sigma(a)} for any permutation \sigma of \{1,2,\ldots\}. Note that each \Lambda^{(d)} is a finite free abelian group with basis elements given by:

\begin{aligned} \Lambda^{(0)} &\longrightarrow \left\{1\right\},\\ \Lambda^{(1)} &\longrightarrow \left\{\sum_{i\ge 1} x_i\right\}, \\ \Lambda^{(2)} &\longrightarrow \left\{\sum_{i\ge 1} x_i^2, \sum_{1\le i<j} x_i x_j\right\},\\ \Lambda^{(3)} &\longrightarrow \left\{\sum_{i\ge 1} x_i^3, \sum_{\substack{i,j\ge 1 \\i\ne j}} x_i^2 x_j, \sum_{1\le i< j< k} x_i x_j x_k\right\}. \end{aligned}

Let: \Lambda := \oplus_{d=0}^\infty \Lambda^{(d)}; this gives us a homogeneous graded ring, called the formal ring of symmetric functions. E.g. in this ring, we have:

\left(\sum_{i=1}^\infty x_i\right)^2 = \left(\sum_{i=1}^\infty x_i^2\right) + 2\left( \sum_{1\le i < j} x_i x_j\right).

For each n, the map \Lambda^{(d)} \to \Lambda_n^{(d)} taking x_{n+1}, x_{n+2}, \ldots \mapsto 0 gives a graded ring homomorphism \Lambda \to \Lambda_n.

Abstract Definition of Λ

Here is an alternate definition: let \Lambda_n \to \Lambda_{n-1} be the map (graded ring homomorphism) taking x_n \mapsto 0. Thus we get a sequence:

\ldots \longrightarrow \Lambda_{n+1} \longrightarrow \Lambda_n \longrightarrow \ldots \longrightarrow \Lambda_1 \longrightarrow 0.

Now \Lambda is defined to be the inverse limit of this sequence, which gives us a graded ring and a set of graded maps \Lambda \to \Lambda_n, one for each n.

Note that if n\ge d, the maps \ldots \to \Lambda_{n+1}^{(d)} \to \Lambda_{n}^{(d)} are all isomorphisms; hence \Lambda^{(d)} \to \Lambda_n^{(d)} is also an isomorphism.

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Symmetric Polynomials in Λ

We define the following in \Lambda, also called the elementary symmetric polynomialcomplete symmetric polynomial and the power sum symmetric polynomial.

\displaystyle\begin{aligned} e_0 = 1,\qquad e_k &= \sum_{1 \le i_1 < \ldots < i_k} x_{i_1} x_{i_2} \ldots x_{i_k},\\ h_0 = 1,\qquad h_k &= \sum_{1 \le i_1 \le \ldots \le i_k} x_{i_1} x_{i_2} \ldots x_{i_k},\\ \qquad p_k &= \sum_{i\ge 1} x_i^k.\end{aligned}

For a partition \lambda, we also define e_\lambda := e_{\lambda_1} \ldots e_{\lambda_l} etc, as before. Finally, the monomial symmetric polynomial m_\lambda is the sum of all x_{i_1}^{a_1} x_{i_2}^{a_2} \ldots x_{i_l}^{a_l}, over all 1 \le i_1 < \ldots < i_l and (a_1, \ldots a_l) such that when sorted in decreasing order, (a_i) becomes (\lambda_i). Note that \deg(m_\lambda) = |\lambda|.

When projected via \Lambda \to \Lambda_n, the above polynomials becomes their counterpart in \Lambda_n. For example

\displaystyle \begin{aligned} m_{21} = \sum_{i\ne j} x_i^2 x_j \in \Lambda \quad &\mapsto \quad x_1^2(x_2 + x_3) + x_2^2 ( x_1 + x_3)+ x_3(x_1 + x_2) \in \Lambda_3,\\ e_3 = \sum_{1\le i < j< k} x_i x_j x_k \in \Lambda \quad&\mapsto\quad x_1 x_2 x_3 \in \Lambda_3.\end{aligned}

Rewriting Earlier Results

We thus have the following:

\displaystyle \begin{aligned} e_\lambda &= \sum_{\mu \vdash d} N_{\lambda\mu} m_\mu,\\ h_\lambda &= \sum_{\mu\vdash d} M_{\lambda\mu} m_\mu, \end{aligned}

where d = |\lambda|. Indeed, the above relations hold in \Lambda_n^{(d)} for any n and d; when n\ge d, the isomorphism \Lambda^{(d)} \cong \Lambda_n^{(d)} shows that the same relations hold in \Lambda. Similarly, the identities from earlier can be copied over verbatim by projecting to \Lambda_n^{(k)} for n\ge k.

\displaystyle\begin{aligned} e_0 h_k - e_1 h_{k-1} + \ldots + (-1)^k e_k h_0 &=0,\\ e_{k-1}p_1 - e_{k-2}p_2 + \ldots + (-1)^{k-1}e_0 p_k &= ke_k, \\ h_{k-1}p_1 + h_{k-2}p_2 + \ldots + h_0 p_k &= kh_k.\end{aligned}.

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Duality in Λ

Recall that we have an involution \omega:\Lambda_n \to \Lambda_n for each n; this map does not commute with the projection \Lambda_{n+1} \to \Lambda_{n+1}. For an easy example, say n=2, the map \omega : \Lambda_3 \to \Lambda_3 takes e_3 \mapsto h_3. However, upon projecting to \Lambda_2, the element e_3 \mapsto 0 while h_3 \mapsto \ne 0.

This can be overcome by taking n\ge d. Since \omega is graded, we thus have the following:

commutative_diagram_formal_ring_d

The vertical lines are isomorphisms and the diagram commutes.

Definition. For each d\ge 0, we define the involution \omega : \Lambda^{(d)} \to \Lambda^{(d)} to be that induced from \omega: \Lambda_n^{(d)}\to \Lambda_n^{(d)} for any n\ge d.

Note that in Λ, we have:

\omega(e_k) = h_k, \quad\omega(h_k) = e_k, \quad\omega(p_k) = (-1)^{k-1}p_k

for all k\ge 1 since these hold in \Lambda_n^{(k)} for any n\ge k.

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