Let KG be the group ring of a group G over a commutative ring K with unity. The rings KG are described for which xxσ = xσx for all x = ∑ g∈G αgg ∈ KG, where x → xσ = ∑ g∈G αgf(g)σ(g) is an involution of KG; here f : G → U(K) is a homomorphism and σ is an antiautomorphism of order two of G. Let R be a ring with unity. We denote by U(R) the group of units of R. A (bijective) map : R → R is called...

It has been known for a long time that the structure of a finite simple group is intimately connected with the structure of the centralizers of its involutions. An old result of Brauer and Fowler asserts, in fact, that there are at most a finite number of simple groups in which the centralizer of an involution has a given structure. A more specific, pioneering result of Brauer established that ...

Let (g, k) be a reductive symmetric superpair of even type, i.e. so that there exists an even Cartan subspace a ⊂ p. The restriction map S(p) → S(a) where W = W (g0 : a) is the Weyl group, is injective. We determine its image explicitly. In particular, our theorem applies to the case of a symmetric superpair of group type, i.e. (k ⊕ k, k) with the flip involution where k is a classical Lie supe...

Lactation is under homeostatic control, and sustained milk stasis or weaning leads to mammary gland involution. This transition is under the control of systemic and local factors. Of all the phases of mammary gland development, involution is the least well understood. Recent epidemiological observations suggest that involution events promote breast diseases, including breast cancer. This has re...

We study k-Schur functions characterized by k-tableaux, proving combinatorial properties such as a k-Pieri rule and a k-conjugation. This new approach relies on developing the theory of k-tableaux, and includes the introduction of a weight-permuting involution on these tableaux that generalizes the Bender-Knuth involution. This work lays the groundwork needed to prove that the set of k-Schur Li...

This paper considers Darboux transformations of a bispectral operator which preserve its bispectrality. A sufficient condition for this to occur is given, and applied to the case of generalized Airy operators of arbitrary order r > 1. As a result, the bispectrality of a large family of algebras of rank r is demonstrated. An involution on these algebras is exhibited which exchanges the role of s...

The involutory dimension, if it exists, of an involution poset P := (P,≤,′ ) is the minimum cardinality of a family of linear extensions of ≤, involutory with respect to ′, whose intersection is the ordering ≤. We show that the involutory dimension of an involution poset exists iff any pair of isotropic elements are orthogonal. Some characterizations of the involutory dimension of such posets a...

For a connected complex semi-simple Lie group G and a fixed pair (B, B−) of opposite Borel subgroups of G, we determine when the intersection of a conjugacy class C in G and a double coset BwB− is non-empty, where w is in the Weyl group W of G. The question comes from Poisson geometry, and our answer is in terms of the Bruhat order on W and an involution mC ∈ W associated to C. We study propert...