§ Mnemonics For Symmetric Polynomials
§ Some notation for partitions
- Consider a partition of a partition of .
- The norm of the partition will be ( times), which is equal to . Thus, .
- So the norm of a partition is the number of parts of the partition.
- The norm of the partition will be which equals .
- So the norm of a partition is the number it is partitoining. Thus, .
§ Elementary Symmetric Polynomials (integer)
- We need to define for , a sequence of variables ( for "roots").
- These were elementary for Newton/Galois, and so has to do with the structure of roots.
- The value of is the coefficients of the "root polynomial" , that is:
- Formally, we define to be the product of all terms for distinct numbers .
§ Elementary Symmetric Polynomials (partition)
- For a partition , the elementary symmetric polynomial is the product of the elementary symmetric polynomial .
§ Monomial Symmetric Polynomials (partition)
- We symmetrize the monomial dictated by the partition. To calculate , we compute , and then symmetrize the above monomial.
- For example, is given by symmetrizing . So we must add the terms and .
- Thus, .
§ Power Sum Symmetric Polynomials (number)
- It's all in the name: take a sum of powers.
- Alternatively, take a power and symmetrize it.
§ Power Sum Symmetric Polynomials (partition)
- Extend to partitions by taking product of power sets of numbers.