This paper explores the interplay between the Frobenius functor and Serre’s vanishing conjecture over a Noetherian local ring R of prime characteristic p. We show that the Frobenius functor induces a diagonalizable map on certain Q-vector spaces, which are tensor products of Q with quotients of Grothendieck groups. This allows us to decompose an element (representing a bounded complex of finite...

We give further results for Perron-Frobenius theory on the numericalrange of real matrices and some other results generalized from nonnegative matricesto real matrices. We indicate two techniques for establishing the main theorem ofPerron and Frobenius on the numerical range. In the rst method, we use acorresponding version of Wielandt's lemma. The second technique involves graphtheory.

Motivated by the theory of Painlev\'e equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra vector fields can be deformed to time-dependent integrable spanning same distribution as original algebra. The results applied quasi-St\"ackel systems.

Let X be a smooth projective curve of genus g ≥ 2 over an algebraically closed field k of characteristic p > 0. Let MX be the moduli space of semistable rank-2 vector bundles over X with trivial determinant. The relative Frobenius map F : X → X1 induces by pull-back a rational map V : MX1 99K MX . In this paper we show the following results. (1) For any line bundle L over X , the rank-p vector ...

Compact closed categories have found applications in modeling quantum information protocols by Abramsky-Coecke. They also provide semantics for Lambek’s pregroup algebras, applied to formalizing the grammatical structure of natural language, and are implicit in a distributional model of word meaning based on vector spaces. Specifically, in previous work Coecke-Clark-Sadrzadeh used the product c...

In §1 we use COO-vector methods, essentially Frobenius reciprocity, to derive the Howe-Richardson multiplicity formula for compact nilmanifolds. In §2 we use Frobenius reciprocity to generalize and considerably simplify a reduction procedure developed by Howe for solvable groups to general extensions of nilpotent Lie groups. In §3 we give an application of the previous results to obtain a reduc...

The purpose of these notes is to provide an introduction to the Steenrod algebra in an algebraic manner avoiding any use of cohomology operations. The Steenrod algebra is presented as a subalgebra of the algebra of endomorphisms of a functor. The functor in question assigns to a vector space over a Galois field the algebra of polynomial functions on that vector space: the subalgebra of the endo...

Dedicated to William Kahan and Beresford Parlett on the occasion of their 60th birthdays Let A be an m n matrix, b be an m-vector, and x̃ be a purported solution to the problem of minimizing kb Axk2. We consider the following open problem: find the smallest perturbation E of A such that the vector x̃ exactly minimizes kb (A+E)xk2. This problem is completely solved whenE is measured in the Frobeni...

For N and k positive integers, let M(N, k)C denote the C-vector space of cuspidal modular forms of level N and weight k. This vector space is equipped with the usual Hecke operators Tn, n ≥ 1. If we need to consider several levels or weights at the same time, we will denote this Tn by T N n , or T N,k n . If p is a prime number dividing N , our Tp is also known under the name Up. One of our mai...

A new way to prove the Frobenius conjecture on the dimensions of real algebras without zero divisors is given in the present paper. Firstly, the proof of nonexistence of real algebras without zero divisors in all dimensions except 1,2,4 and 8 was given in [1]. It was based on the simplicial cohomology operation technique. Later on the methods of K-theory cohomology operations gave one a possibi...