About the Book
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 53. Chapters: Antimatroid, Association scheme, Bender-Knuth involution, Bose-Mesner algebra, Buekenhout geometry, Building (mathematics), Combinatorial commutative algebra, Combinatorial species, Coxeter complex, Dominance order, Dyson conjecture, Eulerian poset, Finite ring, Garnir relations, Graded poset, H-vector, Hall-Littlewood polynomials, Hessenberg variety, Incidence algebra, Kazhdan-Lusztig polynomial, Kruskal-Katona theorem, Lattice word, Littelmann path model, Littlewood-Richardson rule, LLT polynomial, Macdonald polynomials, N! conjecture, Newton's identities, Picture (mathematics), Quasi-polynomial, Quasisymmetric function, Restricted representation, Ring of symmetric functions, Robinson-Schensted correspondence, Robinson-Schensted-Knuth correspondence, Schubert variety, Simplicial sphere, Stanley's reciprocity theorem, Stanley-Reisner ring, Viennot's geometric construction. Excerpt: In mathematics, Newton's identities, also known as the Newton-Girard formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P (counted with their multiplicity) in terms of the coefficients of P, without actually finding those roots. These identities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard. They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity. Let x1, ..., xn be variables, denote for k 1 by pk(x1, ..., xn) the k-th power sum: and for k 0 denote by ek(x1, ..., xn) the elementary symmetric polynomial that is the sum of all distinct products of k distinct variables, so in particular Then Newton's identities can be stated as valid for all k 1. Concretely, one gets for the first few values of k: The form and validity of these equations do not depend on the number n of variables (although the point where the left-hand side becomes 0 does, namely after the n-th identity), which makes it possible to state them as identities in the ring of symmetric functions. In that ring one has and so on; here the left-hand sides never become zero. These equations allow to recursively express the ei in terms of the pk; to be able to do the inverse, one may rewrite them as Now view the xi as parameters rather than as variables, and consider the monic polynomial in t with roots x1, ..., xn: where the coefficients are given by the elementary symmetric polynomials in the roots: ak = ek(x1, ..., xn). Now consider the power sums of the roots Then according to Newton's identities these can be expressed recursively in terms of the coefficients of the polynomial using When the polynomial above
