Related papers: A Note on the Group-theoretic Approach to Fast Mat…

Groups of almost upper triangular infinite matrices with entries indexed by integers are studied. It is shown that, when the matrices are over a finite field, these groups admit a nondiscrete totally disconnected, locally compact group…

In this paper, we introduce novel fast matrix inversion algorithms that leverage triangular decomposition and recurrent formalism, incorporating Strassen's fast matrix multiplication. Our research places particular emphasis on triangular…

Irrespective of whether n is prime, prime power with exponent >1, or composite, the group U_n of units of Z_n can sometimes be obtained as the direct product of cyclic groups generated by x, x+k and x+2k, for x, k in Z_n. Indeed, for many…

A sequence of integers $ \{ s_n \}_{n \in \mathbb{N}} $ is called a T-sequence if there exists a Hausdorff group topology on $ \mathbb{Z} $ such that $ \{ s_n \}_{n \in \mathbb{N}} $ converges to zero. For every finite set of primes $ S $…

A group action H on X is called "telescopic" if for any finitely presented group G, there exists a subgroup H' in H such that G is isomorphic to the fundamental group of X/H'. We construct examples of telescopic actions on some CAT[-1]…

The $G$-representation variety $R_G(\Sigma_g)$ parametrizes the representations of the fundamental groups of surfaces $\pi_1(\Sigma_g)$ into an algebraic group $G$. Taking $G$ to be the groups of $n \times n$ upper triangular or unipotent…

Let $G$ be a finite nilpotent group and $n\in \{3,4, 5\}$. Consider $S_n\times G$ as a subgroup of $S_n\times S_{|G|}\subset S_{n|G|}$, where $G$ embeds into the second factor of $S_n\times S_{|G|}$ via the regular representation. Over any…

In this work, we prove limitations on the known methods for designing matrix multiplication algorithms. Alman and Vassilevska Williams recently defined the Universal Method, which substantially generalizes all the known approaches including…

Transformer-based entropy models have gained prominence in recent years due to their superior ability to capture long-range dependencies in probability distribution estimation compared to convolution-based methods. However, previous…

Let K be a field of positive characteristic p, let R be either a group algebra K[G] or a restricted enveloping algebra u(L), and let I be the augmentation ideal of R. We first characterize those R for which I satisfies a polynomial identity…

In the classical transformer attention scheme, we are given three $n \times d$ size matrices $Q, K, V$ (the query, key, and value tokens), and the goal is to compute a new $n \times d$ size matrix $D^{-1} \exp(QK^\top) V$ where $D =…

We define tensors, corresponding to cubic polynomials, which have the same exponent $\omega$ as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor $sM_n$ defined on an $n\times n$ matrix…

Random matrices in the large N expansion and the so-called double scaling limit can be used as toy models for quantum gravity: 2D quantum gravity coupled to conformal matter. This has generated a tremendous expansion of random matrix…

A new procedure is presented for computing the matrix cosine and sine simultaneously by means of Taylor polynomial approximations. These are factorized so as to reduce the number of matrix products involved. Two versions are developed to be…

In a recent paper, an algorithm has been presented for determining implications between a particular kind of category theoretic property represented by matrices -- the so called `matrix properties'. In this paper we extend this algorithm to…

We resolve three interrelated problems on \emph{reduced Kronecker coefficients} $\overline{g}(\alpha,\beta,\gamma)$. First, we disprove the \emph{saturation property} which states that $\overline{g}(N\alpha,N\beta,N\gamma)>0$ implies…

For the rational group algebra $\mathbb{Q}G$ of a finite group $G$, we provide an effective method to compute a complete set of matrix units and, in particular, primitive orthogonal idempotents in a simple component of $\mathbb{Q}G$, which…

Given a group $G$, we write $g^G$ for the conjugacy class of $G$ containing the element $g$. A theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the commutator subgroup…

We have developed polynomial-time algorithms to generate terms of the cogrowth series for groups $\mathbb{Z}\wr \mathbb{Z},$ the lamplighter group, $(\mathbb{Z}\wr \mathbb{Z})\wr \mathbb{Z}$ and the Navas-Brin group $B.$ We have also given…

We show how to represent a class of expressions involving discrete sums over partitions as matrix models. We apply this technique to the partition functions of 2* theories, i.e. Seiberg-Witten theories with the massive hypermultiplet in the…