A Pieri-type formula for isotropic flag manifolds

We give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from a Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a type $B$ (respectively, type $C$) Schubert polynomial by the Schur $P$-polynomial $p_m$ (respectively, the Schur $Q$-polynomial $q_m$). Geometric constructions and intermediate results allow us to ultimately deduce this from formulas for the classical flag manifold. These intermediate results are concerned with the Bruhat order of the Coxeter group ${\mathcal B}_\infty$, identities of the structure constants for the Schubert basis of cohomology, and intersections of Schubert varieties. We show these identities follow from the Pieri-type formula, except some `hidden symmetries' of the structure constants. Our analysis leads to a new partial order on the Coxeter group ${\mathcal B}_\infty$ and formulas for many of these structure constants

Non-commutative Pieri operators on posets

We consider graded representations of the algebra NC of noncommutative symmetric functions on the Z-linear span of a graded poset P. The matrix coefficients of such a representation give a Hopf morphism from a Hopf algebra HP generated by the intervals of P to the Hopf algebra of quasi-symmetric functions. This provides a unified construction of quasi-symmetric generating functions from different branches of algebraic combinatorics, and this construction is useful for transferring techniques and ideas between these branches. In particular we show that the (Hopf) algebra of Billera and Liu related to Eulerian posets is dual to the peak (Hopf) algebra of Stembridge related to enriched P-partitions, and connect this to the combinatorics of the Schubert calculus for isotropic flag manifolds.Comment: LaTeX 2e, 22 pages Minor corrections, updated references. Complete and final version, to appear in issue of J. Combin. Th. Ser. A dedicated to G.-C. Rot

Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras

We present a family of rank symmetric diamond-colored distributive lattices that are naturally related to the Fibonacci sequence and certain of its generalizations. These lattices re-interpret and unify descriptions of some un- or differently-colored lattices found variously in the literature. We demonstrate that our symmetric Fibonaccian lattices naturally realize certain (often reducible) representations of the special linear Lie algebras, with weight basis vectors realized as lattice elements and Lie algebra generators acting along the order diagram edges of each lattice. We present evidence that each such weight basis is, in a certain sense, uniquely associated with its lattice. We provide new descriptions of the lattice cardinalities and rank generating functions and offer several conjectures/open problems. Throughout, we make connections with integer sequences from the OEIS.Comment: 18 page

Some Generalizations of Classical Integer Sequences Arising in Combinatorial Representation Theory

There exists a natural correspondence between the bases for a given finite-dimensional representation of a complex semisimple Lie algebra and a certain collection of finite edge-colored ranked posets, laid out by Donnelly, et al. in, for instance, [Don03]. In this correspondence, the Serre relations on the Chevalley generators of the given Lie algebra are realized as conditions on coefficients assigned to poset edges. These conditions are the so-called diamond, crossing, and structure relations (hereinafter DCS relations.) New representation constructions of Lie algebras may thus be obtained by utilizing edge-colored ranked posets. Of particular combinatorial interest are those representations whose corresponding posets are distributive lattices. We study two families of such lattices, which we dub the generalized Fibonaccian lattices LFⁱᵇpn`1, kq and generalized Catalanian lattices LCᵃᵗpn, kq. These respectively generalize known families of lattices which are DCS-correspondent to some special families of representations of the classical Lie algebras An`₁ and Cn. We state and prove explicit formulae for the vertex cardinalities of these lattices; show existence and uniqueness of DCS-satisfactory edge coefficients for certain values of n and k; and report on the efficacy of various computational and algorithmic approaches to this problem. A Python library for computationally modeling and “solving” these lattices appears as an appendix

Growth hormone response to arginine test distinguishes multiple system atrophyfrom Parkinson's disease and idiopathic late-onset cerebellar ataxia

OBJECTIVE: Multiple system atrophy (MSA) is difficult to distinguish from idiopathic Parkinson's disease (PD) and idiopathic late-onset cerebellar ataxia (ILOCA). This study aimed to evaluate GH response to three different GH stimulation tests in order to establish a reliable test to differentiate these degenerative disorders. DESIGN: Twelve patients with MSA, 10 with PD, eight with ILOCA and 30 healthy controls entered the study. They were submitted to clonidine, arginine, and GH-releasing-hormone (GHRH) + arginine tests in a random manner on three different nonconsecutive days. The peak serum GH response was used as a primary variable for analysis of stimulation tests. By ROC analysis, the optimum cut-off level was considered as the cut-off with the maximal sum of sensitivity and specificity. RESULTS: After clonidine administration, GH peak was significantly lower in patients with MSA than in those with ILOCA (P < 0.05) and in the controls (P < 0.001). At the optimum cut-off level of 5 mU/l, the clonidine test distinguished patients with MSA from those with PD with a sensitivity and specificity of 78%. Moreover, this test distinguished patients with MSA from those with ILOCA with a sensitivity of 100% and a specificity of 75% at a cut-off level of 5 mU/l, and with a sensitivity of 75% and a specificity of 100% at the cut-off level of 7.6 mU/l. After arginine administration, the GH peak was significantly lower in patients with MSA than in those with ILOCA (P = 0.001) and in controls (P < 0.001). At the optimum cut-off level of 5 mU/l, the arginine test distinguished patients with MSA from those with PD with a sensitivity and a specificity of 100%. At a GH peak cut-off value of 3.6 mU/l the arginine test distinguished patients with MSA from those with ILOCA with a sensitivity and specificity of 100%. After GHRH + arginine administration, a significant GH increase was found in all groups of patients and controls. CONCLUSIONS: The GH response to arginine administration is impaired in MSA. Therefore, the arginine test showed the highest diagnostic accuracy to distinguish MSA from both PD and ILOCA, and could be used in the clinical practice of these neurodegenerative diseases