Related papers: Matrix algebras with degenerate traces and trace i…
In this thesis we study toric degenerations of projective varieties. We compare different constructions to understand how and why they are related as s first step towards developing a global framework. In focus are toric degenerations…
We propose a graph theoretic approach to determine trace of product of two permutation matrices through a weighted digraph representation of the permutation matrices. Consequently, we derive trace-zero doubly stochastic (DS) matrices of…
Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate…
The cone of lower semicontinuous traces is studied with a view to its use as an invariant. Its properties include compactness, Hausdorffness, and continuity with respect to inductive limits. A suitable notion of dual cone is given. The cone…
An algebra $\mathcal{A}$ of $n\times n$ complex matrices is said to be \textit{idempotent compressible} if $E\mathcal{A}E$ is an algebra for all idempotents $E\in\mathbb{M}_n(\mathbb{C})$. Analogously, $\mathcal{A}$ is said to be…
The trace regression model, a direct extension of the well-studied linear regression model, allows one to map matrices to real-valued outputs. We here introduce an even more general model, namely the partial-trace regression model, a family…
We obtain several structure results for a class of spherical subgroups of connected reductive complex algebraic groups that extends the class of strongly solvable spherical subgroups. Based on these results, we construct certain…
Comtrans algebras, arising in web geometry, have two trilinear operations, commutator and translator. We determine a Gr\"obner basis for the comtrans operad, and state a conjecture on its dimension formula. We study multilinear polynomial…
Given $d,n \in \mathbb{N}$, we write a polynomial $F \in \mathbb{C}[x_1,\dots,x_n]$ to be degenerate if there exist $P\in \mathbb{C}[y_1, \dots, y_{n-1}]$ and $m_j = x_1^{v_{j,1}}\dots x_n^{v_{j,n}}$ with $v_{j,1}, \dots, v_{j,n} \in…
Let H be a finite-dimensional unimodular pivotal quasi-Hopf algebra over a field k, and let H-mod be the pivotal tensor category of finite-dimensional H-modules. We give a bijection between left (resp. right) modified traces on the tensor…
Partially commutative monoids provide a powerful tool to study graphs, viewingwalks as words whose letters, the edges of the graph, obey a specific commutation rule. A particularclass of traces emerges from this framework, the hikes, whose…
A univariate trace polynomial is a polynomial in a variable x and formal trace symbols Tr(x^j). Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses…
In this paper we study stable finiteness of ample groupoid algebras with applications to inverse semigroup algebras and Leavitt path algebras, recovering old results and proving some new ones. In addition, we develop a theory of (faithful)…
Examples of the $\beta$-Jacobi ensemble specify the joint distribution of the transmission eigenvalues in scattering problems. In this context, there has been interest in the distribution of the trace, as the trace corresponds to the…
The main purpose of this paper is to study formal deformations of evolution algebras, determining their existence and classifying them up to equivalence. In addition, we examine degenerations in this setting and provide Hasse diagrams that…
We consider certain functional identities on the matrix algebra $M_n$ that are defined similarly as the trace identities, except that the "coefficients" are arbitrary polynomials, not necessarily those expressible by the traces. The main…
One considers certain degenerations of the generic symmetric matrix over a field $k$ of characteristic zero and the main structures related to the determinant $f$ of the matrix, such as the ideal generated by its partial derivatives, the…
We study the subspace of the exterior algebra of a simple complex Lie algebra linearly spanned by the copies of the little adjoint representation or, in the case of the Lie algebra of traceless matrices, by the copies of the n-th symmetric…
An important building block in all current asymptotically fast algorithms for matrix multiplication are tensors with low border rank, that is, tensors whose border rank is equal or very close to their size. To find new asymptotically fast…
Let $M$ denote the vector space of $2 \times 2$ matrices with coefficients in $\mathbb{F}_3$ and trace zero. Let $G = SL_2(\mathbb{F}_3)$. Then $G$ acts on $M$ via conjugation. Let $R =(S(M^*) \otimes \Lambda(M^*))$ be the algebra of…