Reducibility without qualification means in this lecture reducibility over the rational field. Questions on such reducibility occupy an intermediate place between questions on reducibility of polynomials over an algebraically closed field and those on primality. I shall refer to these two cases as to the algebraic and the arithmetic one and I shall try to exhibit some of the analogies consideri...

• Generic algebras • Strict Iwahori-Hecke algebras • Representations with Iwahori-fixed vectors • Proof of the Borel-Matsumoto theorem • Irreducibility criteria Using the ideas of [Casselman 1980] descended from the Borel-Matsumoto theorem on admissible representations of p-adic reductive groups containing Iwahori-fixed vectors, it is possible to give an easily verifiable sufficient criterion f...

A graphs G is clique irreducible if every clique in G of size at least two, has an edge which does not lie in any other clique of G and is clique reducible if it is not clique irreducible. A graph G is clique vertex irreducible if every clique in G has a vertex which does not lie in any other clique of G and clique vertex reducible if it is not clique vertex irreducible. The clique vertex irred...

Integrated information theory [1–3] is a mathematical, quantifiable theory of conscious experience. The linchpin of this theory, the φ measure, quantifies a system’s irreducibility to disjoint parts. Purely as a measure of irreducibility, we pinpoint three concerns about φ and propose a revised measure, ψ, which addresses them. Our measure ψ is rigorously grounded in Partial Information Decompo...

Friedland (1981) showed that for a nonnegative square matrix A, the spectral radius r(eA) is a log-convex functional over the real diagonal matrices D. He showed that for fully indecomposable A, log r(eA) is strictly convex over D1,D2 if and only if D1 −D2 6= c I for any c ∈ R. Here the condition of full indecomposability is shown to be replaceable by the weaker condition that A and A>A be irre...

Let T N,χ p,k (x) be the characteristic polynomial of the Hecke operator T p acting on the space of cusp forms S k (N, χ). We describe the factorization of T N,χ p,k (x) mod as k varies, and we explicitly calculate those factorizations for N = 1 and small. These factorizations are used to deduce the irreducibility of certain T 1,1 q,k (x) from the irreducibility of T 1,1 2,k (x).

The irreducibility of a polynomial in a prime modulus implies irreducibility over the rationals. However, the converse is certainly not true. In fact, there are some polynomials that are irreducible over the rationals yet reducible in every prime modulus. We show that the nth cyclotomic polynomial reduces modulo all primes if and only if the discriminant of the nth cyclotomic polynomial is a sq...

Let T N; p;k (x) be the characteristic polynomial of the Hecke operator Tp acting on the space of cusp forms Sk(N; ). We describe the factorization of T N; p;k (x) mod ` as k varies, and we explicitly calculate those factorizations for N = 1 and small `. These factorizations are used to deduce the irreducibility of certain T 1;1 q;k (x) from the irreducibility of T 1;1 2;k (x).

Two results in Differential Algebra, Kolchin’s Irreducibility Theorem, and a result on descent of projective varieties (due to Buium) are proved using methods of “modern” or “Grothendieck style” algebraic geometry.