in which cases either f(x) is irreducible or f(x) is the product of two irreducible polynomials of equal degree. If |an| = n > 1, then for some choice of a1, . . . , an−1 ∈ Z and a0 = ±1, we have that f(x) is reducible. I. Schur (in [8]) obtained this result in the special case that an = ±1. Further results along the nature of Theorem 1 are also discussed in [6]. The purpose of this paper is to...

Let H be a Hilbert space. Given a bounded positive definite operator S on H, and a bounded sequence c = {ck}k∈N of non negative real numbers, the pair (S, c) is frame admissible, if there exists a frame {fk}k∈N on H with frame operator S, such that ‖fk‖ 2 = ck, k ∈ N. We relate the existence of such frames with the Schur-Horn theorem of majorization, and give a reformulation of the extended ver...

We apply a theorem of Geronimus to derive some new formulas connecting Schur functions with orthogonal polynomials on the unit circle. The applications include the description of the associated measures and a short proof of Boyd’s result about Schur functions. We also give a simple proof for the above mentioned theorem of Geronimus. 1. Schur functions In what follows we adopt the following nota...

Let V ⊗n be the n–fold tensor product of a vector space V. Following I. Schur we consider the action of the symmetric group Sn on V ⊗n by permuting coordinates. In the ‘super’ (Z2 graded) case V = V0 ⊕ V1, a ± sign is added [BR]. These actions give rise to the corresponding Schur algebras S(Sn, V ). Here S(Sn, V ) is compared with S(An, V ), the Schur algebra corresponding to the alternating su...

In this paper, we construct the q-Schur modules as left principle ideals of the cyclotomic q-Schur algebras, and prove that they are isomorphic to those cell modules defined in [3] and [10] at any level r. Then we prove that these q-Schur modules are free modules and construct their bases. This result gives us new versions of several results about the standard basis and the branching theorem. W...

In this paper a general coset construction for an arbitrary vertex operator algebra is studied and a duality theorem of Schur-Weyl type is obtained.