Попов Владимир Леонидович

The cone of Hilbert nullforms

2010 · PREPRINT · en

We describe a geometric-combinatorial algorithm that allows one, using solely the system of weights and roots, to determine the Hesselink strata of the null-cone of a linear representation of a reductive algebraic group and calculate their dimensions. In particular, it provides a constructive approach to calculating the dimension of the null-cone and determining all its irreducible components of maximal dimension. In the case of the adjoint representation (and, more generally, a $\theta$-representation), the algorithm turns into the classification algorithm for the conjugacy classes of nilpotent elements in a semisimple Lie algebra (respectively, homogeneous nilpotent elements in a cyclically graded semisimple Lie algebra).

Две орбиты: когда одна лежит в замыкании другой?

2009 · ARTICLE · ru

Пусть G — связная линейная алгебраическая группа, V — конечномерный алгебраический G-модуль и O_1, O_2 — две G-орбиты в V . Мы указываем конструктивный способ выяснить, лежит O_1 в замыкании O_2 или нет.

Алгебраические конусы

2009 · ARTICLE · ru

Дана характеризация алгебраических конусов в терминах действий одномерного мультипликативного моноида M_m и алгебраической группы G_m.

Two orbits: When is one in the closure of the other?

2009 · ARTICLE · en

Let G be a connected linear algebraic group, let V be a finite dimensional algebraic G-module, and let O1 and O2 be two G-orbits in V. We describe a constructive way to find out whether or not O1 lies in the closure of O2 .

Algebraic cones

2009 · ARTICLE · en

Irregular and singular loci of commuting varieties

2008 · ARTICLE · en

Tensor product decompositions and open orbits in multiple flag varieties

2007 · ARTICLE · en

Quasihomogeneous affine threefolds

2007 · ARTICLE · en

Birationally nonequivalent linear actions. Cayley degrees of simple algebraic groups. Singularities of two-dimensional quotients

2007 · ARTICLE · en

Generically multiple transitive algebraic group actions

2007 · CHAPTER · en

With every nontrivial connected algebraic group G we associate a positive integer gtd(G) called the generic transitivity degree of G and equal to the maximal n such that there is a nontrivial action of G on an irreducible algebraic variety X for which the diagonal action of G on Xn admits an open orbit. We show that gtd(G) ≤2 (respectively, gtd(G) = 1) for all solvable (respectively, nilpotent) G, and we calculate gtd(G) for all reductive G. We prove that if G is nonabelian reductive, then the above maximal n is attained for X = G/P where P is a proper maximal parabolic subgroup of G (but not only for such homogeneous spaces of G). For every reductive G and its proper maximal parabolic subgroup P, we find the maximal r such that the diagonal action of G (respectively, a Levi subgroup L of P) on (G/P)r admits an open G-orbit (respectively, L-orbit). As an application, we obtain upper bounds for the multiplicities of trivial components in some tensor product decompositions. As another application, we classify all the pairs (G; P) such that the action of G on (G/P)3 admits an open orbit, answering a question of M. Burger.