The Schubert calculus for G/B can be completely determined by a certain matrix related to the Kostant polynomials introduced in section 5 of Bernstein, Gelfand, and Gelfand [Bernstein, I., Gelfand, I. & Gelfand, S. (1973) Russ. Math. Surv. 28, 1-26]. The polynomials are defined by vanishing properties on the orbit of a regular point under the action of the Weyl group. For each element w in the ...

We consider the problem of determining the asymptotic order of the Gelfand numbers of mixed(quasi-)norm embeddings lp(l d q) →֒ l b r(l d u) given that p ≤ r and q ≤ u, with emphasis on cases with p ≤ 1 and/or q ≤ 1. These cases turn out to be related to structured sparsity. We obtain sharp bounds in a number of interesting parameter constellations. Our new matching bounds for the Gelfand number...

The spectrum of a Gelfand pair (K nN, K), where N is a nilpotent group, can be embedded in a Euclidean space. We prove that in general, the Schwartz functions on the spectrum are the Gelfand transforms of Schwartz K-invariant functions on N . We also show the converse in the case of the Gelfand pair (SO(3) n N3,2, SO(3)), where N3,2 is the free two-step nilpotent Lie group with three generators...

The chasing method proposed by Gelfand and Lokutzievskii and implemented inMathematica package ”Numerical Solution of Linear Boundary Value Problems” is described. This package served as a base for a linear boundary value problem solver built in Mathematica 5.0 function NDSolve. Examples of the package usage are presented.

We address two problems regarding the structure and representation theory of finite W -algebras associated with the general linear Lie algebras. Finite W -algebras can be defined either via the Whittaker modules of Kostant or, equivalently, by the quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand-Kirillov conjecture for the skew fields of fractions of the finite W a...

In the topos approach to quantum theory of Doering and Isham the Kochen–Specker Theorem, which asserts the contextual nature of quantum theory, can be reformulated in terms of the global sections of a presheaf characterised by the Gelfand spectrum of a commutativeC∗–algebra. In previous work we showed how this topos perspective can be generalised to a class of categories typically studied withi...

For an admissible affine vertex algebra Vk(g) of type A, we describe a new family of relaxed highest weight representations of Vk(g). They are simple quotients of representations of the affine Kac-Moody algebra ĝ induced from the following g-modules: 1) generic Gelfand-Tsetlin modules in the principal nilpotent orbit, in particular all such modules induced from sl2; 2) all Gelfand-Tsetlin modul...

Let K ⊂ G be a compact subgroup of a real Lie group G. Denote by D(X) thealgebra of G-invariant differential operators on the homogeneous space X = G/K. ThenX is called commutative or the pair (G,K) is called a Gelfand pair if the algebra D(X)is commutative. Symmetric Riemannian homogeneous spaces introduced by Élie Cartanand weakly symmetric homogeneous spaces introduced by Sel...

Gelfand with grandaughter. Israel Moiseevich Gelfand, a mathematician compared by Henri Cartan to Poincaré and Hilbert, was born on September 2, 1913, in the small town of Okny (later Red Okny) near Odessa in the Ukraine and died in New Brunswick, New Jersey, USA, on October 5, 2009. Nobody guided Gelfand in his studies. He attended the only school in town, and his mathematics teacher could off...