Related papers: The wreath matrix

We examine the following version of a classic combinatorial search problem introduced by R\'enyi: Given a finite set $X$ of $n$ elements we want to identify an unknown subset $Y \subset X$ of exactly $d$ elements by testing, by as few as…

Using a construction closely related to Waldhausen's $S_\bullet$-construction, we produce a spectrum $K(\mathbf{Var}_{/k})$ whose components model the Grothendieck ring of varieties (over a field $k$) $K_0 (\mathbf{Var}_{/k})$. We then…

A notion of quantum matrix (QM-) algebra generalizes and unifies two famous families of algebras from the theory of quantum groups: the RTT-algebras and the reflection equation (RE-) algebras. These algebras being generated by the…

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is a vector…

We introduce a new construction of matrix wreath products of algebras that is similar to wreath products of groups. We then use it to prove embedding theorems for Jacobson radical, nil, and primitive algebras. In \S\ref{Section6}, we…

The classical matrix-tree theorem discovered by G.Kirchhoff in 1847 relates the principal minor of the nxn Laplace matrix to a particular sum of monomials of matrix elements indexed by directed trees with n vertices and a single sink. In…

Since the time of Minkowski a basic problem in number theory has been to find lower bounds for the absolute value $\Delta(K)$ of the discriminant of a number field $K$ in terms of the degree $n(K)$ of $K$. In this paper we study another…

In 1989, Rota made the following conjecture. Given $n$ bases $B_{1},\dots,B_{n}$ in an $n$-dimensional vector space $V$, one can always find $n$ disjoint bases of $V$, each containing exactly one element from each $B_{i}$ (we call such…

The MULTICUT IN TREES problem consists in deciding, given a tree, a set of requests (i.e. paths in the tree) and an integer k, whether there exists a set of k edges cutting all the requests. This problem was shown to be FPT by Guo and…

We study the problem of determining a matrix whose $k$th multiplicative compound is a prescribed matrix~$M$. The cardinality of the set of matrices whose $k$th multiplicative compound equals~$M$ is characterized in terms of $\rank(M)$. On…

Let $A$ be a gentle algebra. For every collection of string and band diagrammes, we consider the constructible subset of the variety of representations containing all modules with this underlying diagramme. We study degenerations of such…

It is a well known that, for odd $n$, the number of subsets of $\{1,2,\dots,n\}$ the sum of whose elements is divisible by $n$ equals the number of binary necklaces of length $n$. In this paper generalize this result in two directions. On…

In this paper we give an alternative proof that the family of matrices studied by Daeshik Choi in A proof of Crouzeix's conjecture for a class of matrices, Linear Algebra and its Applications, 438, no. 8 (2013), pp. 3247-3257, satisfy…

We investigate spectral conditions on Hermitian matrices of roots of unity. Our main results are conjecturally sharp upper bounds on the number of residue classes of the characteristic polynomial of such matrices modulo ideals generated by…

We study lowest-weight irreducible representations of rational Cherednik algebras attached to the complex reflection groups G(m,r,n) in characteristic p. Our approach is mostly from the perspective of commutative algebra. By studying the…

We introduce a family of polynomials, which arise in three distinct ways: in the large $N$ expansion of a matrix integral, as a weighted enumeration of factorisations of permutations, and via the topological recursion. More explicitly, we…

We consider the problem of reconstructing compositions of an integer from their subcompositions, which was raised by Raykova (albeit disguised as a question about layered permutations). We show that every composition w of n\ge 3k+1 can be…

Using $\lambda$ operations, we give some results on the kernel of the natural map from the monoid algebra $\mathbb{Z} R$ of a commutative ring $R$ to the ring of $S$-Witt vectors of $R$. As a byproduct we obtain a very natural…

In this paper the authors introduce a new notion called the quantum wreath product, which is the algebra $B \wr_Q \mathcal{H}(d)$ produced from a given algebra $B$, a positive integer $d$, and a choice $Q=(R,S,\rho,\sigma)$ of parameters.…

A Mathieu-Zhao subspace is a generalization of an ideal of an associative algebra $\mathcal A$ over a unital ring $R$ first formalized in 2010. A vertex algebra is an algebraic structure first developed in conjunction with string theory in…