Related papers: The Change of Basis Groupoid
A set $B$ is a basis for a vector space $V$ if every element of $V$ can be uniquely written as a linear combination of the elements of $B$. There is a similar definition of a basis for a finite group. We show that certain semidirect…
In this paper we study a symmetry group of vector space. Basis manifold is a homogeneous space of a symmetry group. This concept leads us to the definition of active and passive transformations on basis manifold. Active transformation can…
The quantum completion of the space of connections in a manifold can be seen as the set of all morphisms from the groupoid of the edges of the manifold to the (compact) gauge group. This algebraic construction generalizes an analogous…
By the Fourier transformations, any group-invariant functions over finite Abelian groups are transformed into group-invariant functions over the character groups. In this paper, we calculate matrix elements of this transformations under…
When G is a finite abelian group, we define G-spans of groupoids and their associated matrices with entries in the group ring QG and show that composition of spans corresponds to multiplication of matrices.
The effect of replacing a basis element on the way the basis spans other elements is studied. This leads to a new characterization of binary matroids.
We describe a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. Roughly speaking, we require the…
We study symmetries of bases and spanning sets in finite element exterior calculus, using representation theory. We want to know which vector-valued finite element spaces have bases invariant under permutation of vertex indices. The…
For a matroid with an ordered (or "labelled") basis, a basis exchange step removes one element with label $l$ and replaces it by a new element that results in a new basis, and with the new element assigned label $l$. We prove that one…
The main purpose of this paper is to study the vector groupoids. This is an algebraic structure which combines the concepts of Brandt groupoid and vector space such that these are compatible.
A general proposition is proved relating multiplicities (of restriction of a representation of a group to a subgroup) under basechange, and used to calculate some multiplicities for cuspidal representations which become principal series…
We present abstraction techniques that transform a given non-linear dynamical system into a linear system or an algebraic system described by polynomials of bounded degree, such that, invariant properties of the resulting abstraction can be…
In this paper we study the possibility to define irreducible representations of the symmetric groups with the help of finitely many relations. The existence of finite bases is established for the classes of representations corresponding to…
A simple observation, showing that every groupoid becomes an inverse semigroup after adding one element. In such inverse semigroups all idempotents are mutually orthogonal. This fact implies that every C*-algebra of a discrete groupoid is a…
In this paper we introduce the concept of generalized vector groupoid. Several properties of them are established.
Recently, a geometrical characterization of vector spaces served to generalize them into a new class of algebras. Instead of the algebraic properties of the underlying fields, we generalized the recently discovered property of such spaces…
Let $V$ be a vector space of dimension $d$ over $F_q$, a finite field of $q$ elements, and let $G \le GL(V) \cong GL_d(q)$ be a linear group. A base of $G$ is a set of vectors whose pointwise stabiliser in $G$ is trivial. We prove that if…
A matroid is a combinatorial structure that captures and generalizes the algebraic concept of linear independence under a broader and more abstract framework. Matroids are closely related with many other topics in discrete mathematics, such…
We increase the scope of previous work on change of basis between finite bases of polynomials by defining ascending and descending bases and introducing three techniques for defining them from known ones. The minimum degrees of polynomials…
The group algebra of the permutation group is spanned by a set of elements called projectors. The coordinates of permutations expanded in projectors are matrix elements of irreducible representations. The projectors of the permutation group…