We introduce the Frobenius–Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius–Schur theorem including that for semisimple quasi-Hopf algebras, weak Hopf C∗-algebras and association schemes. Our framework also clarifies a mechanism of how the “twisted” theory arises from the ordinary case. As a demonstration, we es...

In this paper, we give a notation on the Singleton bounds for linear codes over a finite commutative quasi-Frobenius ring in the work of Shiromoto [5]. We show that there exists a class of finite commutative quasi-Frobenius rings. The Singleton bounds for linear codes over such rings satisfy d(C) − 1 A ≤ n− log|R| |C|. keywords : Linear codes, General weight, Quasi-Frobenius ring, Singleton bounds

1. p-centers 1 1.1. Notation 1 1.2. Frobenius twist for symplectic varieties 1 1.3. Differential operators 2 1.4. Universal enveloping algebras 2 2. Classical Hamiltonian reduction 2 2.1. GIT in positive characteristic 2 2.2. Frobenius kernel 3 2.3. Hamiltonian actions and reduction 4 3. Frobenius constant quantizations 4 3.1. Definition 4 3.2. Quantum Hamiltonian reduction 5 4. Quantization of...

This paper sets a foundation for the study of linear codes over nite rings. The nite Frobenius rings are singled out as the most appropriate for coding theoretic purposes because two classical theorems of MacWilliams, the extension theorem and the MacWilliams identities, generalize from nite elds to nite Frobenius rings. It is over Frobenius rings that certain key identi cations can be made bet...

In this work we present a new class of numerical semigroups called GSI-semigroups. We see the relations between them and others families give explicitly their set gaps. Moreover, an algorithm to obtain all GSI-semigroups up given Frobenius number is provided realization positive integers as numbers studied.

We investigate numerical semigroups generated by any quadratic sequence with initial term zero and an infinite number of terms. find efficient algorithm for calculating the Apéry set, as well bounds on elements set. also Frobenius genus, asymptotic behavior genus. Finally, we embedding dimension all such semigroups.

The theory of Frobenius groups with complements even order largely reduces to tractable algebraic number theory. If we consider only an upper bound s on the distinct primes dividing their commutator subgroups, then proportion these odd is less than 1/2s. A positive lower also given.

A notion of a Frobenius manifold with a nice real structure was introduced by Hertling. It is called CDV structure (Cecotti-Dubrovin-Vafa structure). In this paper, we introduce a “positivity condition” on CDV structures and show that any Frobenius manifold of rank two with real spectrum can be equipped with a positive CDV structure. We extend naturally the symmetries of Frobenius structures gi...

In this article we describe three constructions of complex variations of Hodge structure, proving the existence of interesting opposite filtrations that generalize a construction of Deligne. We also analyze the relation between deformations of Frobenius modules and certain maximally degenerate variations of Hodge structures. Finally, under a certain generation hypothesis, we show how to constru...