Related papers: Frobenius Divisibility and Hopf Centers

We classify finite-dimensional complex Hopf algebras $A$ which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elements $G(A)$ is abelian such that all prime divisors of the order…

We generalize the Frobenius-Schur theorem to $C^*$-categories. From this category-theoretical point of view, we introduce the notions of real, complex and quaternionic representations of Hopf $C^*$-algebras. Based on these definitions, we…

Let $H$ be a finite dimensional pointed rank one Hopf algebra of nilpotent type. We first determine all finite dimensional indecomposable $H$-modules up to isomorphism, and then establish the Clebsch-Gordan formulas for the decompositions…

Inspired by the Capelli identities for group determinants obtained by T\^oru Umeda, we give a basis of the center of the group algebra of any finite group by using Capelli identities for irreducible representations. The Capelli identities…

In 1904, Issai Schur proved the following result. If $G$ is an arbitrary group such that $G/\Z(G)$ is finite, where $\Z(G)$ denotes the center of the group $G$, then the commutator subgroup of $G$ is finite. A partial converse of this…

We study the higher Frobenius-Schur indicators of modules over semisimple Hopf algebras, and relate them to other invariants as the exponent, the order, and the index. We prove various divisibility and integrality results for these…

We study Frobenius-Schur indicators of the regular representations of finite-dimensional semisimple Hopf algebras, especially group-theoretical ones. Those of various Hopf algebras are computed explicitly. In view of our computational…

Let $G$ be a finite non-cyclic $p$-group of order at least $p^3$. If $G$ has an abelian maximal subgroup, or if $G$ has an elementary abelian centre with $C_G(Z(\Phi(G))) \ne \Phi(G)$, then $|G|$ divides $|\text{Aut}(G)|$.

We describe the center of the Hecke algebra of a type attached to a Bernstein block under some hypothesis. When $\bf G$ is a connected reductive group over non-archimedean local field $F$ that splits over a tamely ramified extension of $F$…

We discuss algebraic and representation theoretic structures in braided tensor categories C which obey certain finiteness conditions. Much interesting structure of such a category is encoded in a Hopf algebra H in C. In particular, the Hopf…

S. Montgomery and S. Witherspoon proved that upper and lower semisolvable, semisimple, finite dimensional Hopf algebras are of Froebenius type when their dimensions are not divisible by the characteristic of the base field. In this note we…

If G is a finite group and k is a field, there is a natural construction of a Hopf algebra over k associated to G, the Drinfel'd double D(G). We prove that if G is any finite real reflection group with Drinfel'd double D(G) over an…

Let $G$ be a finite solvable group. Then $G$ always has a useful presentation, which we call a "long presentation". Using a "long presentation" of $G$, we present an inductive method of constructing the irreducible representations of $G$…

In this paper, we study the Green ring and the stable Green ring of a Hopf algebra $H$ by means of bilinear forms. We show that the Green ring of a Hopf algebra of finite representation type is a Frobenius algebra over $\mathbb{Z}$ with a…

We prove Wigner-Eckart theorem for the irreducible tensor operators for arbitrary Hopf algebras, provided that tensor product of their irreducible representation is completely reducible. The proof is based on the properties of the…

Let $\mathcal{C}$ be a finite braided multitensor category. Let $B$ be Majid's automorphism braided group of $\mathcal{C}$, then $B$ is a cocommutative Hopf algebra in $\mathcal{C}$. We show that the center of $\mathcal{C}$ is isomorphic to…

Frobenius' Theorem states that the algebra of quaternions $\mathbb H$ is, besides the fields of real and complex numbers, the only finite-dimensional real division algebra. We first give a short elementary proof of this theorem, then…

Let $G$ be a finite group, and $F$ a field of characteristic $0$ or prime to the order of $G$. In $1952$, Witt and in $1956$, Berman independently proved that the number of inequivalent irreducible $F$-representations of $G$ is equal to the…

The notion of Hopf center and Hopf cocenter of a Hopf algebra is investigated by the extension theory of Hopf algebras. We prove that each of them yields an exact sequence of Hopf algebras. Moreover the exact sequences are shown to satisfy…

In this note, we give short proofs of the well-known results that the exponent of the Schur multiplier $\M$ divides the exponent of $\G$ for finite $\p$-groups of maximal class and potent $\p$-groups. Moreover, we prove the same for a…