Related papers: A full complexity dichotomy for immanant families

Let $(X,T)$ be a Cantor minimal system, and let $\Gamma$ denote either its associated topological full group or the full group of a Bratteli diagram associated with $(X,T)$. In this paper we describe the structure of indecomposable…

Consider the $2n$-by-$2n$ matrix $M=(m_{i,j})_{i,j=1}^{2n}$ with $m_{i,j} = 1$ for $i,j$ satisfying $|2i-2n-1|+|2j-2n-1| \leq 2n$ and $m_{i,j} = 0$ for all other $i,j$, consisting of a central diamond of 1's surrounded by 0's. When $n \geq…

We extend the classical notion of solvability to a lambda-calculus equipped with pattern matching. We prove that solvability can be characterized by means of typability and inhabitation in an intersection type system P based on…

Studying medical imaging, Peter Kuchment and Sergey Lvin encountered an countable family of differential identities for sine, cosine, and the exponential function. Specifically if for a smooth function $u$ and a complex number $\lambda$ the…

The fingerprint invariant of partitions can be used to describe the Kazhdan-Lusztig map for the classical groups. We discuss the basic properties of fingerprint. We construct the fingerprints of rigid partitions in the $B_n$, $C_n$, and…

For an irreducible orthogonal character $\chi $ of even degree there is a unique square class $\det({\chi })$ in the character field such that the invariant quadratic forms in any $L$-representation affording $\chi $ have determinant in…

Permutation Pattern Matching (or PPM) is a decision problem whose input is a pair of permutations $\pi$ and $\tau$, represented as sequences of integers, and the task is to determine whether $\tau$ contains a subsequence order-isomorphic to…

Let $N_\sigma(\pi)$ denote the number of occurrences of a permutation pattern $\sigma\in S_k$ in a permutation $\pi\in S_n$. Gaetz and Ryba (2021) showed using partition algebras that the $d$-th moment $M_{\sigma,d,n}(\pi)$ of $N_\sigma$ on…

We consider the problem of evaluating certain exponential sums. These sums take the form $\sum_{x_1,...,x_n \in Z_N} e^{f(x_1,...,x_n) {2 \pi i / N}} $, where each x_i is summed over a ring Z_N, and f(x_1,...,x_n) is a multivariate…

An ordinary character $\chi $ of a finite group is called orthogonally stable, if all non-degenerate invariant quadratic forms on any module affording the character $\chi $ have the same discriminant. This is the orthogonal discriminant,…

Let n be a positive integer and let Sigma_n be the symmetric group of degree n. Let S^lambda be the Specht module for Sigma_n corresponding to a partition lambda of n, defined over a field F of odd characteristic. We find the indecomposable…

In this paper we study colorings (or tilings) of the two-dimensional grid $\mathbb{Z}^2$. A coloring is said to be valid with respect to a set $P$ of $n\times m$ rectangular patterns if all $n\times m$ sub-patterns of the coloring are in…

This article studies the equation $[A,B]^k = {\rm Id}_n$ for matrices over $\mathbb{C}$, characterizing the pairs $(k,n)$ for which solutions exist via a classical result of Lam and Leung on sums of roots of unity. The problem is next…

We derive several identities that feature irreducible characters of the general linear, the symplectic, the orthogonal, and the special orthogonal groups. All the identities feature characters that are indexed by shapes that are "nearly"…

We consider families of chain-cochain infinite complexes $\mathcal C$ of spaces with elements depending on a number of parameters, and endowed with a converging associative multiple product. The existence of left/right local/non-local…

Let $\Omega=\{1,2,...,n\}$ where $n \ge 2$. The {\em shape} of an ordered set partition $P=(P_1,..., P_k)$ of $\Omega$ is the integer partition $\lambda=(\lambda_1,...,\lambda_k)$ defined by $\lambda_i = |P_i|$. Let G be a group of…

Let $B$ be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition $\lambda$. Then it is known that its nilpotent commutator $N_B$ is an irreducible variety and that there is a unique partition $\mu$ such…

We prove a version of the classical $\lambda$-lemma for holomorphic families of Riemann surfaces. We then use it to show that critical loci for complex H\'{e}non maps that are small perturbations of quadratic polynomials with Cantor Julia…

For any two partitions $\lambda$ and $\mu$ of a positive integer $N$, let $\chi_{\lambda}(\mu)$ be the value of the irreducible character of the symmetric group $S_{N}$ associated with $\lambda$, evaluated at the conjugacy class of elements…

Consider the Wronskians of the classical Hermite polynomials $$H_{\lambda, l}(x):=\mathrm{Wr}(H_l(x),H_{k_1}(x),\ldots, H_{k_n}(x)), \quad l \in \mathbb Z_{\geq 0},$$ where $k_i=\lambda_i+n-i, \,\, i=1,\dots, n$ and $\lambda=(\lambda_1,…