Related papers: Row and column detection complexities of character…

Integrality properties of partial sums over irreducible representations, along columns of character tables of finite groups, were recently derived using combinatorial topological string theories (CTST). These CTST were based on…

The character table of the symmetric group $S_n$, of permutations of $n$ objects, is of fundamental interest in theoretical physics, combinatorics as well as computational complexity theory. We investigate the implications of an identity,…

In the rapidly evolving field of quantum computing, quantifying circuit complexity remains a critical challenge. This paper introduces Character Complexity, a novel measure that bridges Group-theoretic concepts with practical quantum…

In this paper we consider finite groups G satisfying the following condition: G has two columns in its character table which differ by exactly one entry. It turns out that such groups exist and they are exactly the finite groups with a…

Let $N$ be a normal subgroup of a finite group $G$. From a result due to Brauer, it can be derived that the character table of $G$ contains square submatrices which are induced by the $G$-conjugacy classes of elements in $N$ and the…

We describe an easy way how to find supercharacter theories for a finite group, if its character table is known. Namely, we show how an arbitrary partition of the conjugacy classes or of the irreducible characters can be refined to the…

Let F be a finite field with q elements, let A be a finite dimensional F-algebra and let J=J(A) be the Jacobson radical of A. Then G=1+J is a p-group, where p is the characteristic of F. We refer to G as an F-algebra group. A subgroup H of…

Algebraic characterizations of the computational aspects of functions defined over the real numbers provide very effective tool to understand what computability and complexity over the reals, and generally over continuous spaces, mean. This…

This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…

Motivated by recent work of Peluse and Soundararajan on divisibility properties of the entries of the character tables of symmetric groups, we investigate the question: For a finite group G, when are two columns of the character table of G…

A character table $X$ for a saturated fusion system $\mathcal{F}$ on a finite $p$-group $S$ is the square matrix of values associated to a basis of the lattice of virtual $\mathcal{F}$-stable ordinary characters of $S$. We investigate a…

Let B be a block of a finite group G with defect group D. We prove that the exponent of the center of D is determined by the character table of G. In particular, we show that D is cyclic if and only if B contains a "large" family of…

In this lecture, we survey a number of recent results and developments regarding the representation theory of infinite-dimensional quantum groups (quantum affine algebras and related algebras), as well as their connections with cluster…

Two groups are said to have the same character table if a permutation of the rows and a permutation of the columns of one table produces the other table. The problem of determining when two groups have the same character table is…

By exploiting relationships between the values taken by ordinary characters of symmetric groups we prove two theorems in the modular representation theory of the symmetric group. 1. The decomposition matrices of symmetric groups in odd…

Let $\FF$ be a finite field and $(Q,\bfd)$ an acyclic valued quiver with associated exchange matrix $\tilde{B}$. We follow Hubery's approach \cite{hub1} to prove our main conjecture of \cite{rupel}: the quantum cluster character gives a…

In this investigation of character tables of finite groups we study basic sets and associated representation theoretic data for complementary sets of conjugacy classes. For the symmetric groups we find unexpected properties of characters on…

We define a notion of complexity, which quantifies the nonlinearity of the computation of a neural network, as well as a complementary measure of the effective dimension of feature representations. We investigate these observables both for…

General algebraic properties of the algebras of vector fields over quantum linear groups $GL_q(N)$ and $SL_q(N)$ are studied. These quantum algebras appears to be quite similar to the classical matrix algebra. In particular, quantum…

The category of finite dimensional (type 1) representations of a quantum affine algebra $U_q(\hat{{\mathfrak g}})$ is not semisimple. However, as any abelian category with finite-length objects, it admits a unique decomposition into a…