We prove the following result concerning the inheritance of hyper-duality by block and quotient Bose-Mesner algebras associated with a hyper-dual pair of imprimitive Bose-Mesner algebras. Let M and M̃ denote Bose-Mesner algebras. Suppose there is a hyper-duality ψ from the subconstituent algebra of M with respect to p to the subconstituent algebra of M̃ with respect to p̃. Also suppose thatM is im...

The Brauer semigroup Bn of partitions of a 2n-element set into two-element subsets, can be generalized in various ways. For example we may consider arbitrary partitions to obtain the semigroup Cn. The visualization of elements of these semigroups as chips leads us to the Temperley-Lieb semigroup TLn and its analogue TLCn. We study combinatorial properties of these four semigroups. For example, ...

In this survey paper we give a detailed account of a construction strictly related to the so-called fusion process for the symmetric group Sn . This procedure gives a way of building up the primitive idempotents in C[Sn]—indeed the Young idempotents are obtained—affording the irreducible representations of Sn as a product of factors 1+ ci, j (i, j) ∈ C[Sn]; these factors should be regarded as v...

However, these powers are often hard to calculate in a nice closed form. For this reason we shall consider their Fourier transforms in some suitable context. The idea being that intricate computation of products in the group algebra are replaced by simple point wise products. More precisely, we shall derive new extended versions of a formula of Garsia (see (1) below) and of similar formulas obt...

Recently, the invertibility of linear combinations of two idempotents has been studied by several authors. Let P and Q be idempotents in a Banach algebra. It was shown that the invertibility of P + Q is equivalent to that of aP + bQ for nonzero a, b with a + b 6= 0. In this note, we obtain a similar result for square zero operators and those operators satisfying x = dx for some scalar d. More g...

Let A be a unital C-algebra. Denote by P the space of selfadjoint projections of A. We study the relationship between P and the spaces of projections P a determined by the diierent involutions # a induced by positive invertible elements a 2 A. The maps ' p : P ! P a sending p to the unique q 2 P a with the same range as p and a : P a ! P sending q to the unitary part of the polar decomposition ...

Let A be a unital C∗-algebra. Denote by P the space of selfadjoint projections ofA. We study the relationship between P and the spaces of projections Pa determined by the different involutions #a induced by positive invertible elements a ∈ A. The maps φp : P → Pa sending p to the unique q ∈ Pa with the same range as p and Ωa : Pa → P sending q to the unitary part of the polar decomposition of t...

The classical characteristic map associates symmetric functions to characters of the symmetric groups. There are two natural analogues of this map involving the Brauer algebra. The first of them relies on the action of the orthogonal or symplectic group on a space of tensors, while the second is provided by the action of this group on the symmetric algebra of the corresponding Lie algebra. We c...

By a Boolean inverse semigroup we mean an inverse semigroup whose semilattice of idempotents is a Boolean algebra. We study representations of a given inverse semigroup S in a Boolean inverse semigroup which are tight in a certain well defined technical sense. These representations are supposed to preserve as much as possible any trace of Booleannes present in the semilattice of idempotents of ...