Let G be a simply connected semisimple group over C. We show that a certain involution of an open subset of the affine Grassmannian of G, defined previously by Achar and the author, corresponds to the action of the nontrivial Weyl group element of SL(2) on the framed moduli space of Gmequivariant principal G-bundles on P. As a result, the fixed-point set of the involution can be partitioned int...

If D is a division algebra with center a number field K and with an involution of the second kind, it is unknown if the group SU(1, D)/[U(1, D), U(1, D)] is trivial. We show that, by contrast, if K is a function field in one variable over a number field, and if D is an algebra with center K and with an involution of the second kind, the group SU(1, D)/[U(1, D), U(1, D)] can be infinite in gener...

Abstract Involution Schubert polynomials represent cohomology classes of K -orbit closures in the complete flag variety, where is orthogonal or symplectic group. We show they also $\mathsf {T}$ -equivariant subvarieties defined by upper-left rank conditions spaces symmetric skew-symmetric matrices. This geometry implies that these are positive combinations monomials variables $x_i + x_j$ , and ...

This article presents a unified approach to the abstract notions of partial convolution and involution in $L^p$-function spaces over semi-direct product of locally compact groups. Let $H$ and $K$ be locally compact groups and $tau:Hto Aut(K)$ be a continuous homomorphism.  Let $G_tau=Hltimes_tau K$ be the semi-direct product of $H$ and $K$ with respect to $tau$. We define left and right $tau$-c...

Lactation mastitis is a common, but poorly understood, inflammatory breast disease that is a significant health burden. A better understanding of the aetiology of mastitis is urgently required, and will assist in the development of improved prevention and treatment strategies in both human and animal species. Studies in mice have the potential to greatly assist in identifying new drug candidate...

1 The Setup {s:setup} The starting point is: a group G, a Cartan involution θ, and another involution σ of finite order, commuting with θ. It is natural to consider the coset σK = {σ ◦ int(k) | k ∈ K} ⊂ Aut(G). Every element of this coset commutes with θ. We’re mainly interested when σ is an involution, especially the case σ = θ. Now fix a pinning P = (H,B, {Xα}), and write δ, ǫ ∈ Aut(G) for th...

OBJECTIVE Thymus plays a crucial role in immune system homeostasis, and thymic abnormalities have been previously reported in many autoimmune diseases, including SSc. The aim of this study is to evaluate the frequency of radiological thymus abnormalities in SSc patients and its relationship with various clinical and laboratory features. METHODS Sixty-three female SSc patients (diffuse/limited...

Abstract For a Weyl group W of rank r , the -Catalan number is antichains poset positive roots, and -Narayana numbers refine by keeping track cardinalities these antichains. The are symmetric – that is, cardinality k same as $r-k$ . However, this symmetry far from obvious. Panyushev posed problem defining an involution on root exhibits numbers. Rowmotion rowvacuation two related operators, defi...

Hyperoctahedral homology is the theory associated to hyperoctahedral crossed simplicial group. It defined for involutive algebras over a commutative ring using functor and bar construction of Fiedorowicz. The main result paper proves that related equivariant stable homotopy theory: discrete group odd order, algebra isomorphic fixed points under involution an infinite loop space built from class...