We analyze the structure of the algebra K〈x〉n of symmetric polynomials in non-commuting variables in so far as it relates to K[x]n , its commutative counterpart. Using the “place-action” of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of K〈x〉n analogous to the classical theorems of Chevalley, Sh...

The family of general Jacobi polynomials P (α,β) n where α, β ∈ C can be characterised by complex (nonhermitian) orthogonality relations (cf. [15]). The special subclass of Jacobi polynomials P (α,β) n where α, β ∈ R are classical and the real orthogonality, quasi-orthogonality as well as related properties, such as the behaviour of the n real zeros, have been well studied. There is another spe...

This paper presents some applications using several properties of three important symmetric polynomials: elementary symmetric polynomials, complete symmetric polynomials and the power sum symmetric polynomials. The applications includes a simple proof of El-Mikkawy conjecture in [M.E.A. El-Mikkawy, Appl. Math. Comput. 146 (2003) 759–769] and a very easy proof of the Newton–Girard formula. In ad...

We give some results towards the conjecture that X(2, 2`− 1) are the only nonlinear balanced elementary symmetric polynomials over GF (2), where t and ` are any positive integers and X(d, n) = ∑ 1≤i1<i2<···<id≤n xi1xi2 · · ·xid .

We prove that there does not exist a non-square quadratic polynomial with integer coefficients and an axis of symmetry which takes square values for N consecutive integers for N = 7 or N ≥ 9. At the opposite, if N ≤ 6 or N = 8 there are infinitely many.

We considerN = 1 supersymmetric U(N) gauge theories with Zk symmetric treelevel superpotentials W (Φ) for an adjoint chiral multiplet. We show that (for 2N/k ∈ Z) this Zk symmetry survives in the quantum effective theory as a corresponding symmetry of the effective superpotential Weff(Si) under permutations of the Si. For W (x) = Ŵ (ξ), ξ = x , this allows us to express the prepotential F0 and ...

In this paper we use a bicharacter construction to define an HDquantum vertex algebra structure corresponding to the quantum vertex operators describing classes of symmetric polynomials.

The Lie module of the group algebra FSn of the symmetric group is known to be not projective if and only if the characteristic p of F divides n. We show that in this case its nonprojective summands belong to the principal block of FSn. Let V be a vector space of dimension m over F , and let L(V ) be the n-th homogeneous part of the free Lie algebra on V ; this is a polynomial representation of ...

A large class of Boolean functions, as well as almost all symmetr ic Boolean functions, are shown to have no two-level completely robust path-delay-fault testable ( R P D F T ) realization b y combinational circuits. Exact and asymptotic formulae are derived fo r the number of symmetric Boolean functions which have two-level completely R P D F T realization. To achieve completely R P D F T real...