Related papers: Frobenius Problem for Semigroups ${\sl S}(d_1,d_2,…

Explicit birational geometry of 3-folds represents a second phase of Mori theory, going beyond the foundational work of the 1980s. This paper is a tutorial and colloquial introduction to the explicit classification of Fano 3-folds (Q-Fano…

We develop a geometric procedure for finding the Ap\'ery set of any numerical semigroup with embedding dimension four. Previous methods of comparable strength worked only for embedding dimension three or under very specific conditions. We…

A common tool in the theory of numerical semigroups is to interpret a desired class of semigroups as the integer lattice points in a rational polyhedron in order to leverage computational and enumerative techniques from polyhedral geometry.…

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…

A generating function of the number of homomorphisms from the fundamental group of a compact oriented or non-orientable surface without boundary into a finite group is obtained in terms of an integral over a real group algebra. We calculate…

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…

The two subjects in the title are related via the specialization of symmetric polynomials at roots of unity. Let $f(z_1,\ldots,z_n)\in\mathbb{Z}[z_1,\ldots,z_n]$ be a symmetric polynomial with integer coefficients and let $\omega$ be a…

We define a Frobenius algebra over fusion categories of the form Rep$(G)\boxtimes$Rep$(G)$ which generalizes the diagonal subgroup of $G\times G$. This allows us to extend field theoretical constructions which depend on the existence of a…

Let $\mathcal{G}$ denote the variety generated by infinite dimensional Grassmann algebras; i.e., the collection of all unitary associative algebras satisfying the identity $[[z_1,z_2],z_3]=0$, where $[z_i,z_j]=z_iz_j-z_jz_i$. Consider the…

A collection of $k$ sets is said to form a $k$-sunflower, or $\Delta$-system, if the intersection of any two sets from the collection is the same, and we call a family of sets $\mathcal{F}$ sunflower-free if it contains no sunflowers.…

Let $p\geq 5$ be a prime and $\ell\neq p$ be a prime not dividing the tame level of a $p$-ordinary Hida family. We prove that the actions of the Frobenius element at $\ell$ on the Galois representations attached to almost all arithmetic…

Given three pairwise coprime positive integers $a_1,a_2,a_3 \in \mathbb{Z}^+$ we show the existence of a relation between the sets of the first elements of the three quotients $\frac{\langle a_i,a_j \rangle}{a_k}$ that can be made for every…

A subalgebra pair of semisimple complex algebras B < A with inclusion matrix M is depth two if MM^t M < nM for some positive integer n and all corresponding entries. If A and B are the group algebras of finite group-subgroup pair H < G, the…

We generalize and prove a hypothesis by V. Arnold on the parity of Frobenius number. For the case of symmetric semigroups with three generators of Frobenius numbers we found an exact formula, which in a sense is the sum of two Sylvester's…

We give explicit formulas for the Kawazumi-Zhang invariant and Faltings delta-invariant of a compact and connected Riemann surface of genus three. The formulas are in terms of two integrals over the associated jacobian, one integral…

We present a unified apoach to the study of separable and Frobenius algebras. The crucial observation is thsat both cases are related to the nonlinear equation $R^{12}R^{23}=R^{23}R^{13}=R^{13}R^{12}$, called the FS-equation. Given a…

We give a complete classification of simple representations of the braid group B_3 with dimension $\leq 5$ over any algebraically closed f ield. In particular, we prove that a simple d-dimensional representation $\rho: B_3 \to GL(V)$ is…

In this article, we study the quotients of numerical semigroups, generated by two coprime positive numbers, named (a,b) over d. We give formulae for the usual invariants of these semigroups, expressed in terms of continued fraction…

Many finite subgroups of SU(3) are commonly used in particle physics. The classification of the finite subgroups of SU(3) began with the work of H.F. Blichfeldt at the beginning of the 20th century. In Blichfeldt's work the two series (C)…

We give two alternate presentations of the Frobenius Heisenberg category, $\mathcal{Heis}_{F,k}$, defined by Savage, when the Frobenius algebra $F=F_1\oplus\dotsb\oplus F_n$ decomposes as a direct sum of Frobenius subalgebras. In these…