Related papers: On symmetric matrices associated with oriented lin…

This is the first paper in a series of three devoted to studying twisted linking forms of knots and three-manifolds. Its function is to provide the algebraic foundations for the next two papers by describing how to define and calculate…

A matrix is totally positive if all of its minors are positive. This notion of positivity coincides with the type A version of Lusztig's more general total positivity in reductive real-split algebraic groups. Since skew-symmetric matrices…

In 2018, Kashaev introduced a square matrix indexed by the regions of a link diagram, and conjectured that it provides a novel way of computing the Levine-Tristram signature and Alexander polynomial of the corresponding oriented link. In…

The Levine-Tristram signature associates to each oriented link $L$ in $S^3$ a function $\sigma_L \colon S^1 \to \mathbb{Z}.$ This invariant can be defined in a variety of ways, and its numerous applications include the study of unlinking…

An ordered and oriented 2-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring the orientation and ordering of the components. Kirk-Livingston showed that if L is achiral then the…

Let $n \ge 3$ be an integer. Let $P_n = \{1, 2, 3, ..., n-1, n \}$ and let $S_n$ be the symmetric group of permutations on $P_n$. Motivated by the theory of discrete dynamical systems on the interval, we associate each permutation $\si_n$…

Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of $(SU(2) \times SU(2), \text{diag})$ are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised…

We study the exponential map of connected symmetric spaces and characterize, in terms of midpoints and of infinitesimal conditions, when it is a diffeomorphism, generalizing the Dixmier-Saito theorem for solvable Lie groups. We then give a…

Based on some analogies with the Hodge theory of isolated hypersurface singularities, we define Hodge-type numerical invariants (called H-numbers) of any, not necessarily algebraic, link in $S^3$. They contain the same information as the…

A symmetric doubly stochastic matrix A is said to be determined by its spectra if the only symmetric doubly stochastic matrices that are similar to A are of the form $P^TAP$ for some permutation matrix P. The problem of characterizing such…

Building upon Dyson's fundamental 1962 article known in random-matrix theory as 'the threefold way', we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important examples are…

A new class of simple symmetric digraphs called $\mathcal{D}$ is defined and studied here. Any digraph in $\mathcal{D}$ has the property that each non-pendant vertex is adjacent to at least one pendant vertex. A graph theoretical…

Using the vertex operator representations for symplectic and orthogonal Schur functions, we define two families of symmetric functions and show thatthey are the skew symplectic and skew orthogonal Schur polynomials defined implicitly by…

We introduce a new duality for $\mathcal{N}=1$ supersymmetric gauged matrix models. This $0d$ duality is an order 4 symmetry, namely an equivalence between four different theories, hence we call it Quadrality. Our proposal is motivated by…

Let $\mathbb F$ be a field of characteristic not $2$, and let $(A,B)$ be a pair of $n\times n$ matrices over $\mathbb F$, in which $A$ is symmetric and $B$ is skew-symmetric. A canonical form of $(A,B)$ with respect to congruence…

Two matrix vector spaces $V,W\subset \mathbb C^{n\times n}$ are said to be equivalent if $SVR=W$ for some nonsingular $S$ and $R$. These spaces are congruent if $R=S^T$. We prove that if all matrices in $V$ and $W$ are symmetric, or all…

A 1989 result of Duarte asserts that for a given tree T on n vertices, a fixed vertex i, and two sets of distinct real numbers L, M of sizes n and n-1, respectively, such that M strictly interlaces L, there is a real symmetric matrix A such…

For a nonsingular real algebraic curve in the 3-dimensional projective space and sphere a new numeric characteristic is introduced. It takes integer values, is invariant under rigid isotopy, multiplied by -1 under mirror reflection. In a…

The characteristic map for the symmetric group is an isomorphism relating the representation theory of the symmetric group to symmetric functions. An analogous isomorphism is constructed for the symmetric space of symplectic forms over a…

We define and study odd analogues of classical geometric and combinatorial objects associated to permutations, namely odd Schubert varieties, odd diagrams, and odd inversion sets. We show that there is a bijection between odd inversion sets…