Anton Khoroshkin, HSE
Title: On generalizations of PBW property
The Poincare-Birkhoff-Witt theorem states that the associated graded to the universal enveloping of a Lie algebra is isomorphic to the symmetric algebra of the underlying Lie algebra.
First, I will explain the categorical definition of the PBW property and why it is not true in positive characteristics.
Second I describe the generalizations of the PBW theorem to different algebraic structures and suggest a useful criterion based on the theory of Grobner bases for coloured operads.
All necessary definitions related to the operad theory will be recalled.
Finally, if time permits, generalizations to the derived setting will be mentioned.
Talk is based on arXiv:1807.05873 and arXiv:2003.06055
Evgeny Smirnov, HSE
Title: Danilov-Koshevoy arrays and dual Grothendieck polynomials
The ring of symmetric polynomials has several nice bases; probably the most remarkable of them is formed by Schur polynomials. They appear everywhere: in combinatorics as generating functions for Young tableaux, in representation theory as the characters of GL(n), in geometry as the cohomology classes of Schubert varieties in Grassmannians. The multiplication of Schur polynomials is given by a quite involved combinatorial rule, called the Littlewood-Richardson rule.
In 2005 Danilov and Koshevoy introduced arrays. An array is a rectangular board divided into squares (like a chessboard). These squares can contain balls that can be moved according to certain rules. Arrays provide uniform and simple proofs of various statements about Schur polynomials: the RSK correspondence, the Bender-Knuth involution, the Littlewood-Richardson rule, the crystal operators on Kashiwara crystals etc.
Schur polynomials have numerous generalizations. One of them, the dual stable Grothendieck polynomials, is obtained by replacing the Young tableaux in the combinatorial definition by the so-called reverse plane partitions: tableaux filled by numbers weakly increasing along both rows and columns. They were introduced by Lam and Pylyavskyy in 2007 as a combinatorial gadget for dealing with the K-theory of Grassmannian. I will speak about a generalization of Danilov-Koshevoy arrays that allows us to work with these polynomials: for this, we will put on the board not just single balls, but also balls connected into “strings of beads”.
The talk is based on our joint work with Anastasia Sukacheva.