Personal webpage of Valentina Kiritchenko

Faculty of Mathematics

November 28, Wednesday, 18:30, room 306

shinymath, · Categories: Без рубрики

Valentina Kiritchenko

Newton-Okounkov polytopes of flag varieties for classical groups

For classical groups SL(n), SO(n) and Sp(2n), we define uniformly geometric valuations on the corresponding complete flag varieties. The valuation in every type comes from a natural coordinate system on the open Schubert cell, and is combinatorially related to the Gelfand-Zetlin pattern in the same type. For SL(n) and Sp(2n), we identify the corresponding Newton-Okounkov polytopes with the Feigin-Fourier-Littelmann-Vinberg polytopes. For SO(n), we compute low-dimensional examples and formulate open problems. All necessary definitions will be given in the talk.

November 21, Wednesday, 18:30, room 306

shinymath, · Categories: Без рубрики

Ilya Dumanskiy

PBW filtration and FFLV basis in type A (continued)

Abstract: Last time we constructed a generating set in an irreducible representation of SL(n). This set is compatible with the PBW filtration and its elements are parameterized by lattice points in a convex polytope (called Feigin-Fourier-Littelmann-Vinberg polytope). By counting the number of lattice points in FFLV polytopes we prove that this generating set is in fact a basis. We remind all definitions and constructions from the previous talk.

November 14, Wednesday, 18:30, room 306

shinymath, · Categories: Без рубрики

Alexandr Popkovich

PBW filtration and FFLV basis in type A

Russian version: ПБВ фильтрация и ФФЛВ базис для типа A (см. ниже анонс на русском)

Abstract: We describe the PBW filtration on the irreducible representations  of SL(n). This filtration is induced by the filtration on the universal enveloping algebra of the Lie algebra sl_n. Using this filtration Feigin, Fourier and Littelmann constructed a basis conjectured by Vinberg in an irreducible representation of SL(n). The basis is parameterized by lattice points in a convex polytope (called Feigin-Fourier-Littelmann-Vinberg polytope). We will formulate the main results of their paper and prove some of them if time permits. All necessary definitions will be given in the talk.

Russian version: Фильтрация на универсальной обертывающей алгебре U(sl_n) индуцирует градуировку на каждом неприводимом представлении алгебры Ли sl_n. В статье Фейгина, Фурье и Литтельмана “PBW filtration and bases for irreducible modules in type A” описано построение некоторого однородного относительно этой градуировки базиса для произвольного неприводимого представления, причём элементы полученного базиса соответствуют целым точкам некоторого многогранника. Я сформулирую результаты этой статьи и, насколько позволит время, приведу доказательства основных из них. Предварительно я напомню все необходимые сведения из теории представлений полупростых алгебр Ли.

November 7, Wednesday, 18:30, room 306

shinymath, · Categories: Без рубрики

Alexei Piskunov

Newton-Okounkov polytopes of symplectic flag varieties

Abstract:
This is continuation of the previous talk, however, we remind all necessary definitions and results. We use chain of subgroups Sp(2n) > Sp(2n-2) > to define a valuation on the variety of isotropic flags. Using this valuation we interpret reduction multiplicities of representations of Sp(2n) in terms of lattice points inside a convex polytope.