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Том 51, № 1 (2017)
- Год: 2017
- Статей: 6
- URL: https://journal-vniispk.ru/0016-2663/issue/view/14568
Article
50th anniversary of our journal
1-1
Infinite-dimensional Lie algebras determined by the space of symmetric squares of hyperelliptic curves
Аннотация
We construct Lie algebras of vector fields on universal bundles of symmetric squares of hyperelliptic curves of genus g = 1, 2,.. For each of these Lie algebras, the Lie subalgebra of vertical fields has commuting generators, while the generators of the Lie subalgebra of projectable fields determines the canonical representation of the Lie subalgebra with generators L2q, q = −1, 0, 1, 2,.., of the Witt algebra. As an application, we obtain integrable polynomial dynamical systems.
2-21
On the relationship between combinatorial functions and representation theory
Аннотация
The paper is devoted to the study of well-known combinatorial functions on the symmetric group Sn—the major index maj, the descent number des, and the inversion number inv—from the representation-theoretic point of view. We show that these functions generate the same ideal in the group algebra C[Sn], and the restriction of the left regular representation of the group Sn to this ideal is isomorphic to its representation in the space of n×n skew-symmetric matrices. This allows us to obtain formulas for the functions maj, des, and inv in terms of matrices of an exceptionally simple form. These formulas are applied to find the spectra of the elements under study in the regular representation, as well as derive a series of identities relating these functions to one another and to the number fix of fixed points.
22-31
A criterion of smoothness at infinity for an arithmetic quotient of the future tube
Аннотация
Let Γ be an arithmetic group of affine automorphisms of the n-dimensional future tube T. It is proved that the quotient space T/Γ is smooth at infinity if and only if the group Γ is generated by reflections and the fundamental polyhedral cone (“Weyl chamber”) of the group dΓ in the future cone is a simplicial cone (which is possible only for n ≤ 10). As a consequence of this result, a smoothness criterion for the Satake–Baily–Borel compactification of an arithmetic quotient of a symmetric domain of type IV is obtained.
32-47
48-65
Spectral properties of the complex airy operator on the half-line
Аннотация
We prove a theorem on the completeness of the system of root functions of the Schrödinger operator L = −d2/dx2 + p(x) on the half-line R+ with a potential p for which L appears to be maximal sectorial. An application of this theorem to the complex Airy operator Lc = −d2/dx2 + cx, c = const, implies the completeness of the system of eigenfunctions of Lc for the case in which |arg c| < 2π/3.We use subtler methods to prove a theorem stating that the system of eigenfunctions of this special operator remains complete under the condition that |arg c| < 5π/6.
66-79