Related papers: Frobenius splitting of certain rings of invariants
We construct blocks of finite groups with arbitrarily large Morita Frobenius numbers, an invariant which determines the size of the minimal field of definition of the associated basic algebra. This answers a question of Benson and Kessar.…
Frobenius groups are an object of fundamental importance in finite group theory. As such, several generalizations of these groups have been considered. Some examples include: A Frobenius--Wielandt group is a triple $(G,H,L)$ where $H/L$ is…
In this note I extend two theorems of Sommese regarding abelian varieties to arbitrary characteristic; that an abelian variety cannot be an ample divisor in a smooth projective variety and that a cone over an abelian variety of dimension at…
The values of the normalized homogeneous weight are determined for arbitrary finite Frobenius rings and expressed in a form that is independent from a generating character and the M\"obius function on the ring. The weight naturally induces…
When the quantum parameter $q^{\frac{1}{2}}$ is a root of unity of odd order and the punctured bordered surface has nonempty boundary, we prove the fraction ring of the stated skein algebra (that is the localization over all nonzero…
All indecomposable finite-dimensional representations of the homogeneous Galilei group which when restricted to the rotation subgroup are decomposed to spin 0, 1/2 and 1 representations are constructed and classified. These representations…
The quantum Frobenius map and it splitting are shown to descend to corresponding maps for generalized $q$-Schur algebras at a root of unity. We also define analogs of $q$-Schur algebras for any affine algebra, and prove the corresponding…
In the late 1980s, Friedlander and Parshall studied the representations of a family of algebras which were obtained as deformations of the distribution algebra of the first Frobenius kernel of an algebraic group. The representation theory…
We consider a generalization of representations of quivers that can be derived from the ordinary representations of quivers by considering a product of arbitrary classical groups instead of a product of the general linear groups and by…
We introduce the notion of the Frobenius--Witt cotangent complex, which can be considered as a derived variant of the module of Frobenius--Witt differentials defined by T. Saito. This new object also can be seen as an arithmetic variant of…
The representation ring of an affine algebraic group scheme can be endowed with the structure of a (special) $\lambda$-ring. We show that the same is true for the ring of symmetric representations, i.e. for the Grothendieck-Witt ring of the…
Frobenius' Theorem states that the only finite-dimensional real division algebras are the algebra of real numbers $\mathbb R$, the algebra of complex numbers $\mathbb C$, and the algebra of quaternions $\mathbb H$. We present a short proof…
We show that the Frobenius manifold associated to the pair of a cusp singularity and it's canonical primitive form is isomorphic to the one constructed from the Gromov--Witten theory for an orbifold projective line with at most three…
We generalise some well known properties of irreducible characters of finite groups to zonal spherical functions of Gelfand pairs. This leads to a Frobenius formula for Gelfand pairs. For a given Gelfand pair, the structure coefficients of…
Let $(Q,\sigma)$ be a symmetric quiver, where $Q=(Q_0,Q_1)$ is a finite quiver without oriented cycles and $\sigma$ is a contravariant involution on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form $<,>$ on…
We introduce continuous Frobenius categories. These are topological categories which are constructed using representations of the circle over a discrete valuation ring. We show that they are Krull-Schmidt with one indecomposable object for…
We find the number of compositions over finite abelian groups under two types of restrictions: (i) each part belongs to a given subset and (ii) small runs of consecutive parts must have given properties. Waring's problem over finite fields…
The Frobenius-Perron theory of an endofunctor of a $\Bbbk$-linear category (recently introduced in [CG]) provides new invariants for abelian and triangulated categories. Here we study Frobenius-Perron type invariants for derived categories…
Throughout this paper $A$ is a commutative non-associative algebra over a field $\mathbb{F}$ of characteristic not $2.$ In addition $A$ posses a Frobenius form. We obtain detailed information about the multiplication in $A$ given two axes…
Frobenius observed that the number of times an element of a finite group is obtained as a commutator is given by a specific combination of the irreducible characters of the group. More generally, for any word w the number of times an…