Related papers: Totally positive matrices and dilogarithm identiti…
In order to compute the Schmidt decomposition of $A\in M_k\otimes M_m$, we must consider an associated self-adjoint map. Here, we show that if $A$ is positive under partial transposition (PPT) or symmetric with positive coefficients (SPC)…
We classify involutions acting on spherical 3-manifolds up to conjugacy. Our geometric approach provides insights into numerous topological properties of these involutions.
Let $F$ be an algebraically closed field of characteristic zero, and $G$ be a finite abelian group. If $A=\oplus_{g\in G} A_g$ is a $G$-graded algebra, we study degree-inverting involutions on $A$, i.e., involutions $*$ on $A$ satisfying…
Many finite groups, including all finite non-abelian simple groups, can be symmetrically generated by involutions. In this paper we give an algorithm to symmetrically represent elements of finite groups and to transform symmetrically…
The orthogonal group acts on the space of several $n\times n$ matrices by simultaneous conjugation. For an infinite field of characteristic different from two, relations between generators for the algebra of invariants are described. As an…
We prove that the only entrywise transforms of rectangular matrices which preserve total positivity or total non-negativity are either constant or linear. This follows from an extended classification of preservers of these two properties…
The invariants of solvable Lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively. Bases of the invariant sets of all such algebras are…
There are left and right actions of the 0-Hecke monoid of the affine symmetric group $\tilde{S}_n$ on involutions whose cycles are labeled periodically by nonnegative integers. Using these actions we construct two bijections, which are…
K3-surfaces with antisymplectic involution and compatible symplectic actions of finite groups are considered. In this situation actions of large finite groups of symplectic transformations are shown to arise via double covers of Del Pezzo…
We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in $GL_n$ form a real semi-algebraic cell of dimension $n-1$. Furthermore we prove a natural cell decomposition for its closure. The proof uses properties…
We show that the dihedral group $ Z_2\times D_3$ of order twelve acts faithfully on the set LR, either consisting of Littlewood-Richardson tableaux, or their companion tableaux, or Knutson-Tao hives or Knutson-Tao-Woodward puzzles,via…
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…
We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to $S^3.$ Such involutions are called hyperelliptic as the manifolds admitting such an action. We consider finite groups acting on…
We prove two isomorphism-invariance theorems for groupoids associated with ultragraphs. These theorems characterize ultragraphs for which the topological full group of an associated groupoid is an isomorphism invariant. These results extend…
There are three main components to this article: (i) A formula for the eta invariant of the signature complex for any finite subgroup of ${\rm{SO}}(4)$ acting freely on $S^3$ is given. An application of this is a non-existence result for…
We determine precisely when the branching coefficients arising from the restriction of irreducible representations of the symmetric group $S_n$ to the dihedral subgroup $D_n$ are nonzero, and we establish uniform linear lower bounds outside…
We prove that meromorphic differentials $\omega^{(0)}_n(z_1,...,z_n)$ which are recursively generated by an involution identity are symmetric in all their arguments $z_1,...,z_n$. The proof involves an intriguing combinatorial identity…
We consider a three-dimensional (3D) lattice model associated with the intertwiner of the quantized coordinate ring $A_q(sl_3)$, and introduce a family of layer to layer transfer matrices on $m\times n$ square lattice. By using the…
An involution on a semigroup S (or any algebra with an underlying associative binary operation) is a function f:S->S that satisfies f(xy)=f(y)f(x) and f(f(x))=x for all x,y in S. The set I(S) of all such involutions on S generates a…
With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions…