Related papers: Kronecker multiplicities in the $(k,\ell)$ hook ar…

We apply lattice point counting methods to compute the multiplicities in the plethysm of $GL(n)$. Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old…

The asymptotic performance of a polar code under successive cancellation decoding is determined by the exponent of its polarizing matrix. We first prove that the partial distances of a polarizing matrix constructed from the Kronecker…

Let $K_n^{(k)}$ be the complete $k$-graph on $n$ vertices. A $k$-uniform tight cycle is a $k$-graph with its vertices cyclically ordered so that every $k$ consecutive vertices form an edge and any two consecutive edges share exactly $k-1$…

We write $S_{\leq n}(A)$ and $\Part_{\fin}(A)$ for the set of permutations with at most $n$ non-fixed points, where $n$ is a natural number, and the set of partitions whose members are finite, respectively, of a set $A$. Among our results,…

The problem of analyzing the number of number field extensions $L/K$ with bounded (relative) discriminant has been the subject of renewed interest in recent years, with significant advances made by Schmidt, Ellenberg-Venkatesh, Bhargava,…

We study sampling and interpolation arrays with multiplicities for the spaces P_k of holomorphic polynomials of degree at most k. We find that the geometric conditions satisfied by these arrays are in accordance with the conditions…

The concept of $t$-difference operator for functions of partitions is introduced to prove a generalization of Stanley's theorem on polynomiality of Plancherel averages of symmetric functions related to contents and hook lengths. Our…

For every integer $k$ there exists a bound $B=B(k)$ such that if the characteristic polynomial of $g\in \operatorname{SL}_n(q)$ is the product of $\le k$ pairwise distinct monic irreducible polynomials over $\mathbb{F}_q$, then every…

In this paper, we study the extremal behaviour of deep holes in polyominoes. We determine the maximum number, $h_n$ of deep holes that an $n$-omino can enclose, ensuring that the boundary of each hole is disjoint from the boundaries of any…

The article presents results on the well-known problem concerning the structure of integer polynomials $p_n(z; x, y)$, which define multiplication laws in $n$-valued groups $\mathbb{G}_n$ over the field of complex numbers $\mathbb{C}$. We…

The aim of this paper is twofold. First, we study the number of partitions of a positive integer $m$ into at most $n$ parts in a given set $A$. We prove that such a number is bounded by the $n$-th Fibonacci number $F(n)$ for any $m$ and…

We close three open problems in the separation complexity of valid inequalities for the knapsack polytope. Specifically, we establish that the separation problems for extended cover inequalities, (1,k)-configuration inequalities, and weight…

A complementary group to SU(n) is found that realizes all features of the Littlewood rule for Kronecker products of SU(n) representations. This is accomplished by considering a state of SU(n) to be a special Gel'fand state of the…

For an arbitrary infinite field k of characteristic p > 0, we describe the structure of a block of the algebraic monoid M_n(k) (all n x n matrices over k), or, equivalently, a block of the Schur algebra S(n,p), whose simple modules are…

We present a computational approach to studying the structure of the representation ring of the symmetric group in dimension six. The Kronecker coefficients and all power formulae of irreducible representations of $S_6$ are computed using…

This work is about a partition problem which is an instance of the distance magic graph labeling problem. Given positive integers $n,k$ and $p_1\le p_2\le \cdots\le p_k$ such that $p_1+\cdots+p_k=n$ and $k$ divides $\sum_{i=1}^ni$, we study…

We find conditions such that cup products induce isomorphisms in low degrees for extensions between stable polynomial representations of the general linear group. We apply this result to prove generalizations and variants of the Steinberg…

For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot…

We derive universal lower and upper bounds for max-min and min-max problems (also known as polarization) for the potential of spherical $(k,k)$-designs and provide certain examples, including unit-norm tight frames, that attain these…

We investigate the iterated Kronecker product of a square matrix with itself and prove an invariance property for symmetric subspaces. This motivates the definition of an iterated symmetric Kronecker product and the derivation of an…