Let V be a finite-dimensional representation of a finite group G of order n over a field k. Then by a classical theorem of Hermann Weyl [5] on polarizations one obtains a nice generating set for the invariant ring (SymV ) consisting of er(f), r = 1, 2, . . . , n, f ∈ V ∗ where er(f) denotes the r-th elementary symmetric polynomial in σ.f, σ ∈ G. Also, for any point x ∈ P(V ) there is a G-invari...

Structure constants for the multiplication of Schubert polynomials by Schur symmetric polynomials are known to be related to the enumeration of chains in a new partial order on S∞, the universal k-Bruhat order. Here we present a monoid M for this order. We show that M is analogous to the nil-Coxeter monoid for the weak order on S∞. For this, we develop the theory of reduced sequences for M. We ...

This thesis is concerned with the development and application of categorical tools in the study of the cluster algebras of S. Fomin and A. Zelevinsky. C. Amiot’s generalized cluster category is a triangulated category which has been used, in the case where it is Hom-finite, to categorify a certain class of cluster algebras, using cluster characters in the sense of Y. Palu. In this thesis, we ge...

Newman polynomials are those with all coefficients in {0, 1}. We consider here the problem of finding Newman polynomials P such that all the coefficients of P 2 are so small as possible for deg P and P (1) given. A set A ⊂ [1, N ] is called a B2[g] sequence if every integer n has at most g distinct representations as n = a1 + a2 with a1, a2 ∈ A and a1 ≤ a2. Gang Yu [4] introduced a new idea to ...

We introduce and study the notion of read-k projections of the determinant: a polynomial f ∈ F[x1, . . . , xn] is called a read-k projection of determinant if f = det(M), where entries of matrix M are either field elements or variables such that each variable appears at most k times in M . A monomial set S is said to be expressible as read-k projection of determinant if there is a read-k projec...

The distribution for the number of searches needed to find k of n lost objects is expressed in terms of a refinement of the q-Eulerian polynomials, for which formulae are developed involving homogeneous symmetric polynomials. In the case when k � n and the find probability remains constant, relatively simple and efficient formulas are obtained. From our main theorem, we further (1) deduce the i...

We relate the prepotentials F0 and effective superpotentials Weff of N = 1 supersymmetric U(N) gauge theories with different tree-level superpotentials W (Φ) and Ŵ (Φ) for an adjoint chiral multiplet. For W (x) = Ŵ (h(x)) we show how F0 and Weff on certain submanifolds of the moduli space can be expressed in terms of F̂0 and Ŵeff . In particular, if W (x) is a Zk symmetric polynomial of degree 2...

We construct the first nontrivial examples of Calabi-Yau monopoles. Our main interest on these, comes from Donaldson and Segal’s suggestion [3] that it may be possible to define an invariant of certain noncompact Calabi-Yau manifolds from these gauge theoretical equations. We focus on the Stenzel metric on the cotangent bundle of the 3-sphere T ∗S3 and study monopoles under a symmetry assumptio...

Introduction-The question whether an isomorphism test for two graphs may be found, which is polynomial in the number of vertices, , n stands open for quite a while now. The purpose of the present article is to answer this question affirmatively by presenting an algorithm which decides whether two graphs are isomorphic or not and showing that the number of n independent elementary operations nee...