§ Mnemonics For Symmetric Polynomials

§ Some notation for partitions

• Consider a partition $\lambda \equiv (\lambda_1, \lambda_2, \dots \lambda_l)$ of a partition of $N$.
• The $L_0$ norm of the partition will be $1 + 1 + \dots 1$ ( $l$ times), which is equal to $N$. Thus, $|\lambda|_0 = l$.
• So the $L_0$ norm of a partition is the number of parts of the partition.
• The $L_1$ norm of the partition will be $|\lambda_1| + |\lambda_2| + \dots + |\lambda_l|$ which equals $N$.
• So the $L_1$ norm of a partition is the number it is partitoining. Thus, $|\lambda|_1 = N$.

§ Elementary Symmetric Polynomials (integer)

• We need to define $e_k(\vec r)$ for $k \in \mathbb N$, $r \in X^d$ a sequence of variables ( $r$ for "roots").
• These were elementary for Newton/Galois, and so has to do with the structure of roots.
• The value of $e_k(\vec r)$ is the coefficients of the "root polynomial" $(x - \vec r)$, that is:

• Formally, we define $e_k(\vec r)$ to be the product of all terms $(r_a r_b\dots, r_k)$ for distinct numbers $(a, b, \dots, k) \in [1, n]$.

§ Elementary Symmetric Polynomials (partition)

• For a partition $\vec \lambda \equiv (\lambda_1, \lambda_2, \dots, \lambda_l)$, the elementary symmetric polynomial $e_\lambda$is the product of the elementary symmetric polynomial $e_{\lambda_1} \cdot e_{\lambda_2} \dots e_{\lambda_l}$.

§ Monomial Symmetric Polynomials (partition)

• We symmetrize the monomial dictated by the partition. To calculate $m_\lambda(\vec r)$, we compute $\vec r^\lambda \equiv r_1^{\lambda_1} r_2^{\lambda_2} \dots r_l^{\lambda_l}$, and then symmetrize the above monomial.
• For example, $m_{(3, 1, 1)}(r_1, r_2, r_3)$ is given by symmetrizing $r_1^3 r_2^1 r_3^1$. So we must add the terms $r_1 r_2^3 r_3$and $r_1 r_2 r_3^3$.
• Thus, $m_{(3, 1, 1)}(r_1, r_2, r_3) \equiv r_1^3 r_2 r_3 + r_1 r_2^3 r_3 + r_1 r_2 r_3^3$.

§ Power Sum Symmetric Polynomials (number)

• It's all in the name: take a sum of powers.
• Alternatively, take a power and symmetrize it.
• $P_k(\vec r) \equiv r_1^k + r_2^k + \dots + r_n^k$.

§ Power Sum Symmetric Polynomials (partition)

• Extend to partitions by taking product of power sets of numbers.
• $P_\lambda(\vec r) \equiv P_{\lambda_1}(\vec r) + P_{\lambda 2}(\vec r) + \dots + P_{\lambda_l}(\vec r)$.