Related papers: Generating functions for ranks of pre-modular cate…

We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the…

In this work, we consider the generating function of Kim's q-Euler polynomials and introduce new generalization of q-Genocchi polynomials and numbers of higher order. Also, we give surprising identities for studying in Analytic Numbers…

We study the bimodule structure of the quantum function algebra at roots of 1 and prove that it admits an increasing filtration with factors isomorphic to the tensor products of the dual of Weyl modules $V_\lambda^* \otimes V_{- \omega_0…

We introduce group corings, and study functors between categories of comodules over group corings, and the relationship to graded modules over graded rings. Galois group corings are defined, and a Structure Theorem for the $G$-comodules…

Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the…

In this paper, we present a formulation of the moduli problem for rank-2 algebras over general base rings in functorial terms, providing presentations as presheaf quotients of affine schemes by group scheme actions.

We consider the computational problem of determining the unit group of a finite ring, by which we mean the computation of a finite presentation together with an algorithm to express units as words in the generators. We show that the problem…

We calculate generating functions for the Poincare polynomials of moduli spaces of pointed curves of genus zero and of Configuration Spaces of Fulton and MacPherson. We also prove that contributions of multiple coverings of curves in a…

Rank functors on a quiver $Q$ are certain additive functors from the category of representations of $Q$ to the category of finite-dimensional vector spaces. Composing with the dimension function on vector spaces gives a rank function on…

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…

We pursue a classification of low-rank super-modular categories parallel to that of modular categories. We classify all super-modular categories up to rank=$6$, and spin modular categories up to rank=$11$. In particular, we show that, up to…

We give a diagrammatic presentation of the category of $\textbf{U}_q(\mathfrak{sl}_2)$-tilting modules $\mathfrak{T}$ for $q$ being a root of unity and introduce a grading on $\mathfrak{T}$. This grading is a "root of unity phenomenon" and…

We define a zeta-function of a pre-triangulated dg-category and investigate its relationship with the motivic zeta-function in the geometric case.

Representations of Quantum Groups U_q (g_n), g_n any semi simple Lie algebra of rank n, are constructed from arbitrary representations of rank n-1 quantum groups for q a root of unity. Representations which have the maximal dimension and…

By work of Bringmann, Ono, and Rhoades it is known that the generating function of the $M_2$-rank of partitions without repeated odd parts is the so-called holomorphic part of a certain harmonic Maass form. Here we improve the standing of…

We first prove an analogue of Lagrange theorem for global dimensions of fusion categories, then we give a complete classifications of pre-modular fusion categories of integer global dimensions less than or equal to $10$.

We survey the existing parts of a classification of finite groups generated by orthogonal transformations in a finite-dimensional Euclidean space whose fixed point subspace has codimension one or two and extend it to a complete…

Categories over a field $k$ can be graded by different groups in a connected way; we consider morphisms between these gradings in order to define the fundamental grading group. We prove that this group is isomorphic to the fundamental group…

We study several structure aspects of functor categories from a small additive category to a module category, in particular the category F(A,K) of functors from finitely generated free modules over a commutative ring A to vector spaces over…

We characterize which graph invariants are partition functions of a spin model over the complex numbers, in terms of the rank growth of associated `connection matrices'.