We introduce and study several affine (=annular in this paper) versions of the classical diagram algebras such as Temperley-Lieb, partition, Brauer, Motzkin, rook Brauer, rook, planar partition, and planar rook algebras. We give generators and relation presentation for them and their associated categories, study their representation theory, and the asymptotic behavior of tensor products of their representations in the monoid case.

The cocycle stability rate of a simplicial complex measures how far cochains with small coboundaries are from actual cocycles. In this paper, we study the 1-dimensional cocycle stability rate with permutation coefficients of random 2-dimensional Linial--Meshulam complexes. Our main contribution is the following: If, in the middle traingle density range, these random complexes asymptotically almost surely have a linear cocycle stability rate, then there exists a non-sofic hyperbolic group. As no non-sofic group, nor any non-residually finite hyperbolic group, are known to exist, this opens a potential approach to both problems simultaneously. Our proof method is inspired by a well known fact about the non local testability of Sipser--Spielman expander codes.

We prove for a $\Theta-$positive representation from a discrete subgroup $\Gamma\subset \mathsf{PSL}(2,\mathbb{R})$, the critical exponent for any $\alpha\in \Theta$ is not greater than one. When $\Gamma$ is geometrically finite, the equality holds if and only if $\Gamma$ is a lattice.

Let $E$ be a subset of the affine plane over a finite field $\mathbb{F}_q$. We bound the size of the subgroup of $SL_2(\mathbb{F}_q)$ that preserves $E$. As a consequence, we show that if $E$ has size $\ll q^\alpha$ and is preserved by $\gg q^\beta$ elements of $SL_2(\mathbb{F}_q)$ with $\beta\geq 3\alpha/2$, then $E$ is contained in a line. This result is sharp in general, and will be proved by using combinatorial arguments and applying a point-line incidence bound in $\mathbb{F}_q^3$ due to Mockenhaupt and Tao (2004).

Let $E$ be a subset in $\mathbb{F}_p^2$ and $S$ be a subset in the special linear group $SL_2(\mathbb{F}_p)$ or the $1$-dimensional Heisenberg linear group $\mathbb{H}_1(\mathbb{F}_p)$. We define $S(E):= \bigcup_{\theta \in S} \theta (E)$. In this paper, we provide optimal conditions on $S$ and $E$ such that the set $S(E)$ covers a positive proportion of all elements in the plane $\mathbb{F}_p$. When the sizes of $S$ and $E$ are small, we prove structural theorems that guarantee that $|S(E)|\gg |E|^{1+\epsilon}$ for some $\epsilon>0$. The main ingredients in our proofs are novel results on algebraic incidence-type structures associated with the groups, in which energy estimates play a crucial role. The higher-dimensional version will also be discussed in this paper.

We show that pure subgroups of infinitely braided Thompson's are bi-orderable. For every finitely generated pure subgroup, we give explicit sets of generators.

It is well known that Rubik's cube has a set of group invariants. These values do not change if any layer was rotated, but they can change in case if some of the cubes were removed from the puzzle, mixed up and returned back. In this paper, we generalize the puzzle to the case of an arbitrary dimension after which we describe all the invariants.

Existing equivariant neural networks require prior knowledge of the symmetry group and discretization for continuous groups. We propose to work with Lie algebras (infinitesimal generators) instead of Lie groups. Our model, the Lie algebra convolutional network (L-conv) can automatically discover symmetries and does not require discretization of the group. We show that L-conv can serve as a building block to construct any group equivariant feedforward architecture. Both CNNs and Graph Convolutional Networks can be expressed as L-conv with appropriate groups. We discover direct connections between L-conv and physics: (1) group invariant loss generalizes field theory (2) Euler-Lagrange equation measures the robustness, and (3) equivariance leads to conservation laws and Noether this http URL connections open up new avenues for designing more general equivariant networks and applying them to important problems in physical sciences

In this note we present explicit canonical forms for all the elements in the two-qubit CNOT-Dihedral group, with minimal numbers of controlled-S (CS) and controlled-X (CX) gates, using the generating set of quantum gates [X, T, CX, CS]. We provide an algorithm to successively construct the n-qubit CNOT-Dihedral group, asserting an optimal number of controlled-X (CX) gates. These results are needed to estimate gate errors via non-Clifford randomized benchmarking and may have further applications to circuit optimization over fault-tolerant gate sets.

Given a simple graph $A$ on a group $G$ and an equivalence relation $B$ on $G$, the $B$ super $A$ graph is defined as a simple graph, whose vertex set is $G$ and two vertices $g$, $h$ are adjacent if either they are in the same equivalence class or there exist $g^{\prime} \in[g]$ and $h^{\prime} \in[h]$ such that $g^{\prime}$ and $h^{\prime}$ are adjacent in $A$. In the literature, the $B$ super $A$ graphs have been investigated by considering $A$ to be either power graph, enhanced power graph, or commuting graph and $B$ to be an equality, order or conjugacy relation. In this paper, we investigate the Sombor spectrums of these $B$ super $A$ graphs for certain non-abelian groups, viz. the dihedral group, generalized quaternion group and the semidihedral group, respectively.

We show that the Graph Isomorphism (GI) problem and the related problems of String Isomorphism (under group action) (SI) and Coset Intersection (CI) can be solved in quasipolynomial ($\exp((\log n)^{O(1)})$) time. The best previous bound for GI was $\exp(O(\sqrt{n\log n}))$, where $n$ is the number of vertices (Luks, 1983); for the other two problems, the bound was similar, $\exp(\tilde{O}(\sqrt{n}))$, where $n$ is the size of the permutation domain (Babai, 1983). The algorithm builds on Luks's SI framework and attacks the barrier configurations for Luks's algorithm by group theoretic "local certificates" and combinatorial canonical partitioning techniques. We show that in a well-defined sense, Johnson graphs are the only obstructions to effective canonical partitioning. Luks's barrier situation is characterized by a homomorphism {\phi} that maps a given permutation group $G$ onto $S_k$ or $A_k$, the symmetric or alternating group of degree $k$, where $k$ is not too small. We say that an element $x$ in the permutation domain on which $G$ acts is affected by {\phi} if the {\phi}-image of the stabilizer of $x$ does not contain $A_k$. The affected/unaffected dichotomy underlies the core "local certificates" routine and is the central divide-and-conquer tool of the algorithm.

As an analogue of the topological boundary of discrete groups $\Gamma$, we define the noncommutative topological boundary of tracial von Neumann algebras $(M, \tau)$ and apply it to generalize the main results of [AHO23], showing that for a trace-preserving action $\Gamma \curvearrowright (A, \tau_A)$ on an amenable tracial von Neumann algebra, any $\Gamma$-invariant amenable intermediate subalgebra between $A$ and $\Gamma \ltimes A$ is necessarily a subalgebra of $\mathrm{Rad}(\Gamma) \ltimes A$. By taking $(A, \tau_A) = L^\infty(X, \nu_X)$ for a free pmp action $\Gamma \curvearrowright (X, \nu_X)$, we obtain a similar result for the invariant subequivalence relations of $\mathcal{R}_{\Gamma \curvearrowright X}$.

We prove a formula for the ${\mathbb S}_n$-equivariant Euler characteristic of the moduli space of graphs $\mathcal{MG}_{g,n}$. Moreover, we prove that the rational ${\mathbb S}_n$-invariant cohomology of $\mathcal{MG}_{g,n}$ stabilizes for large $n$. That means, if $n \geq g \geq 2$, then there are isomorphisms $H^k(\mathcal{MG}_{g,n};\mathbb{Q})^{{\mathbb S}_n} \rightarrow H^k(\mathcal{MG}_{g,n+1};\mathbb{Q})^{{\mathbb S}_{n+1}}$for all$k$.

The classification problem of $P$- and $Q$-polynomial association schemes has been one of the central problems in algebraic combinatorics. Generalizing the concept of $P$- and $Q$-polynomial association schemes to multivariate cases, namely to consider higher rank $P$- and $Q$-polynomial association schemes, has been tried by some authors, but it seems that so far there were neither very well-established definition nor results. Very recently, Bernard, Cramp\'{e}, d'Andecy, Vinet, and Zaimi [arXiv:2212.10824], defined bivariate $P$-polynomial association schemes, as well as bivariate $Q$-polynomial association schemes. In this paper, we study these concepts and propose a new modified definition concerning a general monomial order, which is more general and more natural and also easy to handle. We prove that there are many interesting families of examples of multivariate $P$- and/or $Q$-polynomial association schemes.

A conjecture regarding the structure of expander graphs is discussed.

We study time-inhomogeneous random walks on finite groups in the case where each random walk step need not be supported on a generating set of the group. When the supports of the random walk steps satisfy a natural condition involving normal subgroups of quotients of the group, we show that the random walk converges to the uniform distribution on the group and give bounds for the convergence rate using spectral properties of the random walk steps. As an application, we prove a universality theorem for cokernels of random integer matrices allowing some dependence between entries.

We study the distribution of the Galois group of a random $q$-additive polynomial over a rational function field: For $q$ a power of a prime $p$, let $f=X^{q^n}+a_{n-1}X^{q^{n-1}}+\ldots+a_1X^q+a_0X$ be a random polynomial chosen uniformly from the set of $q$-additive polynomials of degree $n$ and height $d$, that is, the coefficients are independent uniform polynomials of degree ${\rm deg}\, a_i\leq d$. The Galois group $G_f$ is a random subgroup of ${\rm GL}_n(q)$. Our main result shows that$G_f$is almost surely large as$d,q$ are fixed and $n\to \infty$. For example, we give necessary and sufficient conditions so that ${\rm SL}_n(q)\leq G_f$ asymptotically almost surely. Our proof uses the classification of maximal subgroups of ${\rm GL}_n(q)$. We also consider the limits: $q,n$ fixed, $d\to \infty$ and $d,n$ fixed, $q\to \infty$, which are more elementary.

We devise an explicit method for computing combinatorial formulae for Hadamard products of certain rational generating functions. The latter arise naturally when studying so-called ask zeta functions of direct sums of modules of matrices or class- and orbit-counting zeta functions of direct products of nilpotent groups. Our method relies on shuffle compatibility of coloured permutation statistics and coloured quasisymmetric functions, extending recent work of Gessel and Zhuang.

Let $k_0$ be a field of characteristic $0$ with algebraic closure $k$. Let $G$ be a connected reductive $k$-group, and let $Y$ be a spherical variety over $k$ (a spherical homogeneous space or a spherical embedding). Let $G_0$ be a $k_0$-model ($k_0$-form) of $G$. We give necessary and sufficient conditions for the existence of a $G_0$-equivariant $k_0$-model of $Y$.

Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general lineal group. By means of the Euler-Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimizing paths in the group are shown to have a velocity with constant singular values and multiplicity. In several special cases, these geodesic paths are computed explicitly. In particular the Riemannian geodesics, corresponding to the case p=2, are characterized as the product of two one-parameter groups. It is also shown that geodesics are one-parameter groups if and only if the initial velocity is a normal matrix. These results are further extended to the context of compact operators with p-summable spectrum, where a differential equation for the spectral projections of the velocity vector of an extremal path is obtained.