Skew G-codes

We describe skew [Formula: see text]-codes, which are codes that are the ideals in a skew group ring, where the ring is a finite commutative Frobenius ring and [Formula: see text] is an arbitrary finite group. These codes generalize many of the well-known classes of codes such as cyclic, quasicyclic, constacyclic codes, skew cyclic, skew quasicyclic and skew constacyclic codes. Additionally, using the skew [Formula: see text]-matrices, we can generalize almost all the known constructions in the literature for self-dual codes.

2d TQFTs and baby universes

In this work, we extend the 2d topological gravity model of [1] to have as its bulk action any open/closed TQFT obeying Atiyah’s axioms. The holographic duals of these topological gravity models are ensembles of 1d topological theories with random dimension. Specifically, we find that the TQFT Hilbert space splits into sectors, between which correlators of boundary observables factorize, and that the corresponding sectors of the boundary theory have dimensions independently chosen from different Poisson distributions. As a special case, we study in detail the gravity model built from the bulk action of 2d Dijkgraaf-Witten theory, with or without end-of-the-world branes, and for arbitrary finite group G. The dual of this Dijkgraaf-Witten gravity model can be interpreted as a 1d topological theory whose Hilbert space is a random representation of G and whose aforementioned sectors are labeled by the irreducible representations of G.These holographic interpretations of our gravity models require projecting out negative-norm states from the baby universe Hilbert space, which in [1] was achieved by the (only seemingly) ad hoc solution of adding a nonlocal boundary term to the bulk action. In order to place their solution in the completely local framework of a TQFT with defects, we couple the boundaries of the gravity model to an auxiliary 2d TQFT in a non-gravitational (i.e. fixed topology) region. In this framework, the difficulty of negative-norm states can be remedied in a local way by the introduction of a defect line between the gravitational and non-gravitational regions. The gravity model is then holographically dual to an ensemble of boundary conditions in an open/closed TQFT without gravity.

Composite matrices from group rings, composite G-codes and constructions of self-dual codes

In this work, we define composite matrices which are derived from group rings. We extend the idea of G-codes to composite G-codes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters $$\gamma =7,8$$ γ = 7 , 8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other construction

Bounds for Matchings in Nonabelian Groups

We give upper bounds for triples of subsets of a finite group such that the triples of elements that multiply to $1$ form a perfect matching. Our bounds are the first to give exponential savings in powers of an arbitrary finite group. Previously, Blasiak, Church, Cohn, Grochow, Naslund, Sawin, and Umans (2017) gave similar bounds in abelian groups of bounded exponent, and Petrov (2016) gave exponential bounds in certain $p$-groups.

ASYMPTOTIC BOUNDS FOR THE SIZE OF Hom(A, GLn(q))

Fix an arbitrary finite group A of order a, and let X(n, q) denote the set of homomorphisms from A to the finite general linear group GLn(q). The size of X(n, q) is a polynomial in q. In this note, it is shown that generically this polynomial has degree n2(1 – a−1) − εr and leading coefficient mr, where εr and mr are constants depending only on r := n mod a. We also present an algorithm for explicitly determining these constants.

Character correspondences for symmetric groups and wreath products

The Alperin–McKay conjecture relates irreducible characters of a block of an arbitrary finite group to those of its

Single-query quantum algorithms for symmetric problems

Given a unitary representation of a finite group on a finite-dimensional Hilbert space, we show how to find a state whose translates under the group are distinguishable with the highest probability. We apply this to several quantum oracle problems, including the GROUP MULTIPLICATION problem, in which the product of an ordered n-tuple of group elements is to be determined by querying elements of the tuple. For any finite group G, we give an algorithm to find the product of two elements of G with a single quantum query with probability 2/|G|. This generalizes Deutsch’s Algorithm from Z2 to an arbitrary finite group. We further prove that this algorithm is optimal. We also introduce the HIDDEN CONJUGATING ELEMENT PROBLEM, in which the oracle acts by conjugating by an unknown element of the group. We show that for many groups, including dihedral and symmetric groups, the unknown element can be determined with probability 1 using a single quantum query.

CHARACTERIZATION OF SOLVABLE GROUPS AND SOLVABLE RADICAL

We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for Lie algebras. Some open problems are discussed as well.

Spectrum of group cohomology and support varieties

We give a brief introduction to two fundamental papers by Daniel Quillen appearing in the Annals, 1971. These papers established the foundations of equivariant cohomology and gave a qualitative description of the cohomology of an arbitrary finite group. We briefly describe some of the influence of these seminal papers in the study of cohomology and representations of finite groups, restricted Lie algebras, and related structures.

Local identification of spherical buildings and finite simple groups of Lie type

We give a short proof of the uniqueness of finite spherical buildings of rank at least 3 in terms of the structure of the rank 2 residues and use this result to prove a result making it possible to identify an arbitrary finite group of Lie type from knowledge of its “parabolic structure” alone. Our proof also involves a connection between loops, “Latin chamber systems” and buildings.