Let Γ be a group, and let CΓ be the group ring of Γ over C . We first give a simplified and self-contained proof of Zalesskii’s theorem [23] that the canonical trace on CΓ takes rational values on idempotents. Next, we contribute to the conjecture of idempotents by proving the following result: for a group Γ , denote by PΓ the set of primes p such that Γ embeds in a finite extension of a pro-p-...

Let P and Q be two idempotents on a Hilbert space. In 2005, J. Giol in [Segments of bounded linear idempotents on a Hilbert space, J. Funct. Anal. 229(2005) 405-423] had established that, if P +Q− I is invertible, then P and Q are homotopic with s̃(P,Q) ≤ 2. In this paper, we have given a necessary and sufficient condition that s̃(P,Q) ≤ 2, where s̃(P,Q) denotes the minimal number of segments requ...

Orthogonal sets of idempotents are used to design sets of unitary matrices, known as constellations, such that the modulus of the determinant of the difference of any two distinct elements is greater than 0. It is shown that unitary matrices in general are derived from orthogonal sets of idempotents reducing the design problem to a construction problem of unitary matrices from such sets. The qu...

Let p be a prime. We denote by Sn the symmetric group of degree n, by An the alternating group of degree n and by Fp the field with p elements. An important concept of modular representation theory of a finite group G is the notion of a block. The blocks are in one-to-one correspondence with block idempotents, which are the primitive central idempotents of the group ring FqG, where q is a prime...

A logarithmic residue is a contour integral of the (left or right) logarithmic derivative of an analytic Banach algebra valued function. Logarithmic residues are intimately related to sums of idempotents. The present paper is concerned with logarithmic residues and sums of idempotents in the Banach algebra generated by the compact operators and the identity in the case when the underlying Banac...

Bent functions as optimal combinatorial objects are difficult to characterize and construct. In the literature, bent idempotents are a special class of bent functions and few constructions have been presented, which are restricted by the degree of finite fields and have algebraic degree no more than 4. In this paper, several new infinite families of bent functions are obtained by adding the the...

We consider rings equipped with a closure operation defined in terms of a collection of commuting idempotents, generalising the idea of a topological closure operation defined on a ring of sets. We establish the basic properties of such rings, consider examples and construction methods, and then concentrate on rings which have a closure operation defined in terms of their lattice of central ide...

The representation theory (idempotents, quivers, Cartan invariants, and Loewy series) of the higher-order unital peak algebras is investigated. On the way, we obtain new interpretations and generating functions for the idempotents of descent algebras introduced in Saliola (J. Algebra 320:3866, 2008).

Let [Formula: see text] be a finite group and splitting field of characteristic text]. Denote by the algebra over center text]-subspace trace-zero elements We give some numerical sufficient necessary conditions for text], respectively, to Mathieu subspaces in terms degrees irreducible representations The same also characterize groups that has no nonzero idempotents central idempotents, respecti...