4 Conjectures and results 12 4.1 The symmetry conjecture . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Othogonal idempotents and character formulas . . . . . . . . . . 15 4.3 The double coset conjecture . . . . . . . . . . . . . . . . . . . . . 17 4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.5 The case of a single generator . . . . . . . . . . . . . . ...

In this paper, we discuss properties of medial idempotents on abundant semigroups, study quasi-adequate semigroups with a normal medial idempotent and some of their extreme cases, and give a description of structure of every type of such semigroups, respectively.

We study completely π-regular semigroups admitting a decomposition into a semilattice of σn-simple semigroups, and describe them in terms of properties of their idempotents. In the general case, semigroups admitting a decomposition into a semilattice of σn-simple semigroups were characterized by M. Ćirić and S. Bogdanović in [3] (see Theorem 1 below), in terms of paths of length n in the graph ...

The purpose of this paper is twofold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see [2], [3], [4], [5] and [10]) on the structure of the descent algebras of the Coxeter groups of type An and Bn. But we shall also extend these results to the descent algebra of an arbitrary finite Coxeter group W. The descent algebra, introduced by Solomon in [14]...

We extend to several combinatorial Hopf algebras the endomorphism of symmetric functions sending the first power-sum to zero and leaving the other ones invariant. As a “transformation of alphabets”, this is the (1 − E)-transform, where E is the “exponential alphabet,” whose elementary symmetric functions are en = 1 n! . In the case of noncommutative symmetric functions, we recover Schocker’s id...

The Birman–Wenzl–Murakami algebra was first defined and independently studied by Birman andWenzl [1] and Murakami [4]. The Iwahori–Hecke algebras of Type A and the Birman–Wenzl–Murakami algebras naturally arise as centralizer algebras of tensor product corepresentations of quantum groups of Type A and of Type B, C, and D, respectively [5, 6]. Irreducible characters and primitive idempotents of ...

We interpolate between idempotents in the Hecke algebras associated with the Grassmann representation over different local fields. Consequently, we obtain a bijection between some irreducible representations geometrically defined for all local fields.

Some connections between quantum computing and quantum algebra are described, and the Four Color Theorem (4CT) is interpreted within both contexts. keywords: Temperley-Lieb algebra, Jones-Wenzl idempotents, Bose-Einstein condensate, analog computation, R-matrix

It is then easily verified that RG satisfies the ring axioms; in fact, RG is a linear algebra over R. (We write all groups multiplicatively, and denote group identities by 1; we also use 1 for the unit element of R if there is one.) If R, in addition to being a ring, is a Banach algebra (i.e., an algebra over the complex field K, with a submultiplicative norm which makes R a Banach space), then...