Let K denote a field. A polynomial f(x) ∈ K[x] is said to be decomposable over K if f(x)= g(h(x)) for some polynomials g(x) and h(x)∈K[x] with 1< deg(h) < deg(f ). Otherwise f(x) is called indecomposable. If f(x)= g(xm) with m> 1, then f(x) is said to be trivially decomposable. In this paper, we show that xd+ax+b is indecomposable and that if e denotes the largest proper divisor of d, then xd+a...

For indecomposable representations of a finite group G in characteristic p , the theory of vertices and sources introduced by J.A. Green in 1959 [G1] is a fundamental tool in modular representation theory. The vertex and source of an indecomposable module are (up to conjugation) two invariants of the module and only finitely many modules (up to isomorphism) have the same invariants. Thus a natu...

Motivated by examples in infinite group theory, we classify the finite groups whose subgroups can never be decomposed as direct products.

We give an exact spectral equivalence between the quantum group invariant XXZ chain with arbitrary left boundary term and the same XXZ chain with purely diagonal boundary terms. This equivalence, and a further one with a link pattern Hamiltonian, can be understood as arising from different representations of the one-boundary Temperley-Lieb algebra. For a system of size L these representations a...

Let A be a finite multiset of integers. A second multiset of integers T is said to be an A-tiling of level d if every integer can be expressed in exactly d ways as the sum of an element of A and of an element of T . The set T is indecomposable if it cannot be written as the disjoint union of two proper subsets that are also A-tilings. In this paper we show how to construct indecomposable tiling...

Let G = (V,A) be a directed graph. With any subset X of V is associated the directed subgraph G[X ] = (X,A ∩ (X × X)) of G induced by X . A subset X of V is an interval of G provided that for a, b ∈ X and x ∈ V \ X , (a, x) ∈ A if and only if (b, x) ∈ A, This work was supported, in part, by an NSF EPSCoR grant EPS-0346476 and by a Nebraska Research Initiative (NRI) grant on high performance wir...

In this paper we answer a question of Wayne Lewis by proving that if X is a one-dimensional, hereditarily indecomposable continuum and if HX(X) is finitely generated, then C(X), the hyperspace of subcontinua of X, has dimension 2. Let C(A) be the hyperspace of subcontinua of the continuum X with the topology determined by the Hausdorff metric. A classical theorem of J. L. Kelley [4] asserts tha...