The F-signature is a numerical invariant defined by the number of free direct summands in Frobenius push-forward, and it measures singularities positive characteristic. It can be generalized focusing on nonfree summands. In this paper, we provide several methods to compute (generalized) Hibi ring which special class toric rings. particular, show that computed counting elements symmetric group s...

We introduce a theory of modules over representation small category taking values in entwining structures semiperfect coalgebra. This takes forward the aim developing categories entwined to same extent as that module well philosophy Mitchell working with rings several objects. The representations are motivated by work Estrada and Virili, who developed preadditive categories, which were then stu...

A splinter is a notion of singularity that has seen numerous recent applications, especially in connection with the direct summand theorem, mixed characteristic minimal model program, Cohen–Macaulayness absolute integral closures and cohomology vanishing theorems. Nevertheless, many basic questions about these singularities remain elusive. One outstanding problem whether property spreads from p...

The Poincaré duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical “characteristic element;” in the classical case this is the Euler class, and in the quantum case this is a deformation of the classical Euler class which we call the “quantum Euler class.” We pro...

G. Navarro raised the question whether the ordinary character table X(G) of a finite group G determines the Sylow numbers of G. In this note we show that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are determined by their ordinary character table. If G and H are finite groups with i...