\(\textbf{Definition: }\)A partition \(\lambda\) is a (finite or infinite) sequence $$\lambda = (\lambda_1, \lambda_2, \dots , \lambda_l,\dots)$$ of non-negative integers such that \(\lambda_1\geq \lambda_2\geq \dots \geq \lambda_l\geq \dots\) and containing only finitely many non-zero terms.
\(\textbf{Definition: }\)The non-zero terms of a partition \(\lambda\) are called the parts of \(\lambda\).
\(\textbf{Example: }\)The parts of \(\lambda = (5,4,1,0,0)\) are \(5,4,1\).
\(\textbf{Definition: }\)The length of a partition \(\lambda\) is the number of its parts. It is denoted by \(l(\lambda)\).
\(\textbf{Example: }\)The length of \(\lambda = (5,4,1,0,0)\) is \(3\), (i.e, \(l(\lambda) = 3\)).
\(\textbf{Definition: }\)The weight of a partition \(\lambda\) is the sum of its parts. It is denoted by \(|\lambda|\): $$|\lambda| = \lambda_1 + \lambda_2 + \dots + \lambda_l + \dots$$ If \(|\lambda|=n\), we say that \(\lambda\) is a partiton of \(n\).
\(\textbf{Definition: }\)Let \(p(x_1, x_2, \dots, x_n)\in\mathbb{Z}[x_1, x_2, \dots, x_n]\) and \(\sigma\in \mathfrak{S}_n\)(i.e. Symmetric group). \(p(x_1, x_2, \dots, x_n)\) is symmetric if $$p(x_1, x_2, \dots, x_n) = p(x_{\sigma(1)}, x_{\sigma(2)}, \dots, x_{\sigma(n)}).$$
\(\textbf{Example: }\)\(p(x_1, x_2) = x_1^4 + 5x_1^3x_2 + 10x_1^2x_2^2 + 5x_1x_2^3 + x_2^4\) is symmetric under action of \(\mathfrak{S}_2\).
Definiton: For each \(\alpha = (\alpha_1, \dots, \alpha_n)\in\mathbb{N}^n\) we denote by \(x^\alpha\) the monomial $$ x^\alpha = x_1^{\alpha_1}x_2^{\alpha_2}\dots x_n^{\alpha_n}. $$ Let \(\lambda\) be any partition of length less than \(n\) (i.e. \(l(\lambda)\leq n\)). The polynomial \(m_\lambda(x_1, \dots, x_n)\) is defined as below: $$ m_\lambda(x_1, \dots, x_n) = \sum_{\alpha\in A} x^\alpha, $$ where \(A=\{\alpha\;|\;\alpha = \sigma\cdot \lambda\text{ for }\sigma\in\mathfrak{S}_n\}\). \(m_\lambda(x_1, \dots, x_n)\) is called the monomial symmetric polynomial.
\(\textbf{Example: }\)Let \(\lambda = (5, 3, 0)\). Since \(A = \{(5,3,0),(3,5,0), (0,3,5), (5,0,3), (0,5,3), (3,0,5)\}\), we can get $$ \begin{align*} m_\lambda(x_1, x_2, x_3) &= x_1^5x_2^3x_3^0 + x_1^5x_2^0x_3^3 + x_1^3x_2^5x_3^0 + x_1^0x_2^5x_3^3 + x_1^3x_2^0x_3^5 + x_1^0x_2^3x_3^5\\ &= x_1^5x_2^3 + x_1^5x_3^3 + x_1^3x_2^5 + x_2^5x_3^3 + x_1^3x_3^5 + x_2^3x_3^5 \end{align*} $$
\(\textbf{Definition: }\)For each integer \(0\leq r\leq n\) the \(r\)th elementary symmetric polynomial \(e_r\) is defined as below: $$ e_0(x_1, x_2, \dots, x_n)=1, $$ $$ e_r(x_1, x_2, \dots, x_n) = \sum_{1\leq i_1 < i_2 < \dots < i_r\leq n} x_{i_1}x_{i_2}\dots x_{i_r}. $$
\(\textbf{Example: }\) $$ e_2(x_1, x_2, x_3, x_4)= x_1x_2+x_1x_3+ x_1x_4+x_2x_3+x_2x_4+x_3x_4. $$
Oberservation: When \(r=n\), we have $$ e_n(x_1, x_2, \dots, x_n) = x_1x_2\dots x_n = m_{(1^n)}(x_1, x_2, \dots, x_n), $$ where \((1^n)\) is a partition with length \(n\) and all parts are equal to \(1\) (e.g. \((1^4) = (1,1,1,1)\)).
\(\textbf{Definition: }\)The generating function for the \(e_r\): $$ E(r) = \prod_{i=1}^\infty (1+x_ir)=\sum_{i=0}^\infty e_i(x_1, x_2, \dots, x_n)r^i. $$
Definition: For each \(r\geq 0\) the \(r\)th complete symmetric function \(h_r\), is the sum of all monomials of total degree \(r\) in the variables \(x_1, x_2, \dots,\) so that $$ h_0(x_1, x_2, \dots, x_n)=0. $$ $$ h_1(x_1, x_2, \dots, x_n) = 1. $$ $$ h_r(x_1, x_2, \dots, x_n) = \sum_{|\lambda|=r} m_\lambda(x_1, x_2, \dots, x_n). $$
\(\textbf{Example: }\) We want to know \(h_2(x_1, x_2, x_3, x_4)\). There are only two partition which has weight \(2\): \((2, 0, 0, 0), (1, 1, 0, 0)\). $$ \begin{align*} h_2(x_1, x_2, x_3, x_4)&= m_{(2, 0, 0, 0)}(x_1, x_2, x_3, x_4) + m_{(1, 1, 0, 0)}(x_1, x_2, x_3, x_4)\\ &= x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_1x_2 + x_1x_3 + \\ &\phantom=x_1x_4 + x_2x_3 + x_2x_4 + x_3x_4. \end{align*} $$
\(\textbf{Definition: }\)The generating function for the \(h_r\): $$ H(r) = \prod_{i=1}^\infty (1-x_ir)^{-1}=\sum_{i=0}^\infty h_i(x_1, x_2, \dots, x_n)r^i. $$
\(\textbf{Definition 17.1}\) Let \( R \) be an integral domain, typically \( \mathbb{Q} \) or \( \mathbb{Z} \). A symmetric polynomial is a polynomial in \( R[x_{1}, \dots, x_{n}] \) that is invariant under permuting the variables.
The homogeneous degree \( d \) symmetric polynomials form a finitely generated, free \( R \)-module \( \Lambda_{d}(R) \).
\(\textbf{Example 17.2}\) If \( n=3 \), then up to scalar multiplication, the only symmetric polynomial of degree one in \( x, y, z \) is \( x+y+z \). In degree two, we have \( x^{2}+y^{2}+z^{2} \) and \( xy+yz+xz \). Every other symmetric polynomial in \( \Lambda_{2}(R) \) is an \( R \)-linear combination of these two because the coefficients of \( x^{2}+xy \) determine all other coefficients. Similarly, \[ \begin{align} & m_{3}(x, y, z) = x^{3}+y^{3}+z^{3}, \\ & m_{21}(x, y, z) = x^{2}y + xy^{2} + x^{2}z + xz^{2} + y^{2}z + yz^{2}, \\ & m_{111}(x, y, z) = xyz \end{align} \] form a basis for \( \Lambda_{3}(R) \).
Note that each symmetric polynomial above is a sum of monomials in a single orbit of \( S_{3} \) (for the natural action), and is indexed by an integer partition whose parts are the exponents of its monomials.
\(\textbf{Definition.}\) In general, for \( \lambda = (\lambda_{1}, \ldots, \lambda_{l}) \), the monomial symmetric polynomial is defined by \[ m_{\lambda}(x_{1}, \dots, x_{n}) = \sum_{\{a_{1}, \ldots, a_{l}\} \subseteq [n]} x_{a_{1}}^{\lambda_{1}} x_{a_{2}}^{\lambda_{2}} \cdots x_{a_{l}}^{\lambda_{l}}. \] There is an annoyance here: these depend on both \( l \) and \( n \). In particular, if \( l > n \), then the sum is empty. To handle arbitrary partitions without worrying about a fixed \( n \), we use a countably infinite set of variables, resulting in a formal power series.
Let \( R \) be an integral domain and \( x = \{x_{1}, x_{2}, \dots\} \) a countably infinite set of commuting indeterminates. A monomial is a product \( x^{\alpha} = \prod_{i=1}^{\infty} x_{i}^{\alpha_{i}} \), where \( \alpha_{i} \in \mathbb{N} \) for all \( i \) and all but finitely many \( \alpha_{i} \) are zero. The sequence \( \alpha \) is called the exponent vector of the monomial. Listing the nonzero entries of \( \alpha \) in decreasing order gives a partition \( \lambda(\alpha) \). A formal power series is an expression \[ F = \sum_{\alpha} c_{\alpha} x^{\alpha} \] with \( c_{\alpha} \in R \). Also, \( [x^{\alpha}] F = c_{\alpha} \). The set \( R[[x]] \) of all formal power series is an abelian group under addition. In fact, it is an \( R \)-module, namely the direct product of infinitely many copies of \( R \), one for each exponent vector. It is also a ring, with multiplication \[ \left(\sum_{\alpha} c_{\alpha} x^{\alpha}\right) \left(\sum_{\beta} d_{\beta} x^{\beta}\right) = \sum_{\gamma} \left(\sum_{\alpha+\beta=\gamma} c_{\alpha} d_{\beta}\right) x^{\gamma}, \] where each sum has finitely many nonzero terms.
\(\textbf{Note:}\) We are generally unconcerned with whether (or even where) a formal power series converges in the analytic sense. All that matters is that any operation we define produces well-defined powers and coefficients given by a finite computation in the base ring. For example, even though \( \sum_{n \geq 0} x^{n} \) has well-behaved coefficients, squaring this element would not be well-defined. Familiar functions from analysis, like \(\exp\) and \(\log\), can be regarded as formal power series via their Taylor series. But they are studied using combinatorial methods. For example, \( 1/(1-x) \) is equal to the power series \( 1 + x + x^{2} + \dots \) not by calculating derivatives of \( 1/(1-x) \), but by observing that \( (1 - x)(1 + x + x^{2} + \dots) = 1 \) in \( \mathbb{Z}[[x]] \). Combinatorialists often use differential operators, but we treat them formally as linear transformations that map monomials to monomials. We can now properly define symmetric functions as elements of \( R[[x]] \).
\(\textbf{Definition 17.3}\) Let \( \lambda \vdash n \). The monomial symmetric function \( m_{\lambda} \) is the power series \[ m_{\lambda} = \sum_{\alpha : \lambda(\alpha) = \lambda} x^{\alpha} \] where \( \alpha \) runs over all possible exponent vectors. For example, \[ \begin{aligned} m_{3} &= x_{1}^{3} + x_{2}^{3} + x_{3}^{3} + \cdots = \sum_{i=1}^{\infty} x_{i}^{3}, \\ m_{21} &= x_{1}^{2}x_{2} + x_{1}x_{2}^{2} + x_{1}^{2}x_{3} + x_{1}x_{3}^{2} + \cdots + x_{2}^{2}x_{3} + x_{2}x_{3}^{2} + \cdots = \sum_{i \neq j} x_{i}^{2}x_{j}, \\ m_{111} &= x_{1}x_{2}x_{3} + x_{1}x_{2}x_{4} + \cdots = \sum_{1 \leq i < j < k} x_{i}x_{j}x_{k}. \end{aligned} \]
\(\textbf{Definition 17.4}\) The ring of symmetric functions over \( R \) is \[ \Lambda = \Lambda_{R}(x) = \bigoplus_{d \geq 0} \Lambda_{d}, \] where \( \Lambda_{d}(x) = \{ \text{degree } d \text{ homogeneous symmetric functions in indeterminates } x \text{ with coefficients in } R \} \). Each \( \Lambda_{d} \) is a finitely generated, free \( R \)-module with basis \( \{m_{\lambda} \mid \lambda \vdash d\} \), and their direct sum \( \Lambda \) is a graded \( R \)-algebra.
If \( S_{\infty} \) is the group of permutations of \( \{x_{1}, x_{2}, x_{3}, \dots\} \) with finitely many fixed points, then \( \Lambda \) is the ring of formal power series that have bounded degree and are invariant under the action of \( S_{\infty} \). The monomial symmetric functions \( m_{\lambda} \) provide a natural basis for \( \Lambda \) from an algebraic perspective, as each \( m_{\lambda} \) represents an \( S_{\infty} \)-orbit. However, there are many other bases, and understanding symmetric functions requires understanding these bases and their interactions. Typically, we take \( R = \mathbb{C} \).
\(\textbf{Definition 17.5}\) A basis \( B \) of \( \Lambda \) is an \(\textit{integral basis}\) if the symmetric functions with integer coefficients are precisely the integer linear combinations of elements of \( B \). The set \( \{m_{\lambda}\} \) is an integral basis. This condition is stronger than being a vector space basis, as integer bases are not preserved by scaling.