A coproduct structure on the formal affine Demazure algebra

In the present paper we generalize the coproduct structure on nil Hecke rings introduced and studied by Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory and its associated formal group law. We then construct an algebraic model of the T-equivariant oriented cohomology of the variety of complete flags.Comment: 28 pages; minor revision of the previous versio

Coloured peak algebras and Hopf algebras

For $G$ a finite abelian group, we study the properties of general equivalence relations on G_n=G^n\rtimes \SG_n, the wreath product of $G$ with the symmetric group \SG_n, also known as the $G$-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of \k G_n as well as graded connected Hopf subalgebras of \bigoplus_{n\ge o} \k G_n. In particular we construct a $G$-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or $G$-coloured descent algebra). We show that the direct sum of the $G$-coloured peak algebras is a Hopf algebra. We also have similar results for a $G$-colouring of the Loday-Ronco Hopf algebras of planar binary trees. For many of the equivalence relations under study, we obtain a functor from the category of finite abelian groups to the category of graded connected Hopf algebras. We end our investigation by describing a Hopf endomorphism of the $G$-coloured descent Hopf algebra whose image is the $G$-coloured peak Hopf algebra. We outline a theory of combinatorial $G$-coloured Hopf algebra for which the $G$-coloured quasi-symmetric Hopf algebra and the graded dual to the $G$-coloured peak Hopf algebra are central objects.Comment: 26 pages latex2

The nil Hecke ring and singularity of Schubert varieties

We give a criterion for smoothness of a point in any Schubert variety in any G/B in terms of the nil Hecke ring.Comment: AMSTE

GALEX J201337.6+092801: The lowest gravity subdwarf B pulsator

We present the recent discovery of a new subdwarf B variable (sdBV), with an exceptionally low surface gravity. Our spectroscopy of J20136+0928 places it at Teff = 32100 +/- 500, log(g) = 5.15 +/- 0.10, and log(He/H) = -2.8 +/- 0.1. With a magnitude of B = 12.0, it is the second brightest V361 Hya star ever found. Photometry from three different observatories reveals a temporal spectrum with eleven clearly detected periods in the range 376 to 566 s, and at least five more close to our detection limit. These periods are unusually long for the V361 Hya class of short-period sdBV pulsators, but not unreasonable for p- and g-modes close to the radial fundamental, given its low surface gravity. Of the ~50 short period sdB pulsators known to date, only a single one has been found to have comparable spectroscopic parameters to J20136+0928. This is the enigmatic high-amplitude pulsator V338 Ser, and we conclude that J20136+0928 is the second example of this rare subclass of sdB pulsators located well above the canonical extreme horizontal branch in the HR diagram.Comment: 5 pages, accepted for publication in ApJ Letter

The Kazhdan-Lusztig conjecture for finite W-algebras

We study the representation theory of finite W-algebras. After introducing parabolic subalgebras to describe the structure of W-algebras, we define the Verma modules and give a conjecture for the Kac determinant. This allows us to find the completely degenerate representations of the finite W-algebras. To extract the irreducible representations we analyse the structure of singular and subsingular vectors, and find that for W-algebras, in general the maximal submodule of a Verma module is not generated by singular vectors only. Surprisingly, the role of the (sub)singular vectors can be encapsulated in terms of a `dual' analogue of the Kazhdan-Lusztig theorem for simple Lie algebras. These involve dual relative Kazhdan-Lusztig polynomials. We support our conjectures with some examples, and briefly discuss applications and the generalisation to infinite W-algebras.Comment: 11 page

Quantum Pieri rules for isotropic Grassmannians

We study the three point genus zero Gromov-Witten invariants on the Grassmannians which parametrize non-maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or skew-symmetric form. We establish Pieri rules for the classical cohomology and the small quantum cohomology ring of these varieties, which give a combinatorial formula for the product of any Schubert class with certain special Schubert classes. We also give presentations of these rings, with integer coefficients, in terms of special Schubert class generators and relations.Comment: 59 pages, LaTeX, 6 figure

Super duality and irreducible characters of ortho-symplectic Lie superalgebras

We formulate and establish a super duality which connects parabolic categories $O$ between the ortho-symplectic Lie superalgebras and classical Lie algebras of $BCD$ types. This provides a complete and conceptual solution of the irreducible character problem for the ortho-symplectic Lie superalgebras in a parabolic category $O$, which includes all finite-dimensional irreducible modules, in terms of classical Kazhdan-Lusztig polynomials.Comment: 30 pages, Section 5 rewritten and shortene

Induction of sustained clinical remission in early axial spondyloarthritis following certolizumab pegol treatment: 48-week outcomes from C-OPTIMISE

INTRODUCTION: Achievement of remission is a key treatment goal for patients with axial spondyloarthritis (axSpA). C-OPTIMISE assessed achievement of sustained clinical remission in patients with axSpA, including radiographic (r) and non-radiographic (nr) axSpA, during certolizumab pegol (CZP) treatment, and subsequent maintenance of remission following CZP dose continuation, dose reduction or withdrawal. Here, we report outcomes from the first 48 weeks (induction period) of C-OPTIMISE, during which patients received open-label CZP. METHODS: C-OPTIMISE (NCT02505542) was a two-part, multicenter, phase 3b study in adult patients with early axSpA (r-/nr-axSpA), including a 48-week open-label induction period followed by a 48-week maintenance period. Patients with active adult-onset axSpA, < 5 years' symptom duration, and fulfilling Assessment of SpondyloArthritis international Society classification criteria, were included. During the induction period, patients received a loading dose of CZP 400 mg at weeks 0, 2, and 4, followed by CZP 200 mg every 2 weeks (Q2W) up to week 48. The main outcome of the 48-week induction period was the achievement of sustained clinical remission (defined as an Ankylosing Spondylitis Disease Activity Score [ASDAS] < 1.3 at week 32 and < 2.1 at week 36 [or vice versa], and < 1.3 at week 48). RESULTS: In total, 736 patients (407 with r-axSpA, 329 with nr-axSpA) were enrolled into the study. At week 48, 43.9% (323/736) of patients achieved sustained remission, including 42.8% (174/407) of patients with r-axSpA and 45.3% (149/329) with nr-axSpA. Patients also demonstrated substantial improvements in axSpA symptoms, MRI outcomes and quality of life measures. Adverse events occurred in 67.9% (500/736) of patients, of which 6.0% (44/736) were serious. CONCLUSIONS: Over 40% of patients with early axSpA achieved sustained remission during 48 weeks of open-label CZP treatment. Additionally, patients across the axSpA spectrum demonstrated substantial improvements in imaging outcomes and quality of life following treatment. No new safety signals were identified. TRIAL REGISTRATION: NCT02505542

Standard Monomial Theory for desingularized Richardson varieties in the flag variety GL(n)/B

We consider a desingularization Gamma of a Richardson variety X_w^v=X_w \cap X^v in the flag variety Fl(n)=GL(n)/B, obtained as a fibre of a projection from a certain Bott-Samelson variety Z. We then construct a basis of the homogeneous coordinate ring of Gamma inside Z, indexed by combinatorial objects which we call w_0-standard tableaux