Evgeny Feigin

The well‐known Cauchy identity expresses the product of terms for indexing entries of a rectangular ‐matrix as a sum over partitions of products of Schur polynomials: . Algebraically, this identity comes from the decomposition of the symmetric algebra of the space of rectangular matrices, considered as a ‐‐bimodule. We generalize the Cauchy decomposition by replacing rectangular matrices with arbitrary staircase‐shaped matrices equipped with the left and right actions of the Borel upper‐triangular subalgebras. For any given staircase shape , we describe left and right “standard” filtrations on the symmetric algebra of the space of shape matrices. We show that the subquotients of these filtrations are tensor products of Demazure and opposite van der Kallen modules over the Borel subalgebras. On the level of characters, we derive two distinct expansions for the product for written as sums of products of key polynomials and (opposite) Demazure atoms .

In earlier work by three of the authors of the present paper, a supercommutative quadratic algebra was associated to each symmetric quiver, and a new proof of positivity of motivic Donaldson–Thomas invariants of symmetric quivers was given using the so-called numerical Koszul property of these algebras. It was furthermore conjectured that for each symmetric quiver, such an algebra is Koszul. In this work, we lift the linking and unlinking operations on symmetric quivers of Ekholm, Longhi and the third author to the level of quadratic algebras, and use those lifts to prove the Koszulness conjecture.

We study a class of Grassmannians of sub-bimodules over the path algebras of quivers. Our quiver Grassmannians include Escobar's brick manifolds as well as Labelle's generalizations. We give an explicit construction of the varieties in question, provide examples and clarify connection with the quiver representation spaces. We also prove smoothness of our Grassmannians, construct cellular decompositions and derive a realization as framed moduli spaces. The framed moduli realization leads to a recursive formula for the motives.

Positroid subvarieties of complex Grassmannians are the images of the Richardson subvarieties of the full flag varieties under the natural projection map. Positroid varieties admit natural embedding into certain quiver Grassmannians for equioriented cyclic quivers. Varying representations of the quiver, one defines global positroid varieties inside the type A global affine Grassmannian. General fiber of a family is isomorphic to the corresponding classical positroid variety and the special fiber is a subvariety of the juggling variety. We show that the global positroid families are flat and describe their (conjecturally reduced) scheme structures. We also describe the special fibers inside the product of classical positroid varieties and realize the irreducible components of the special fibers in terms of the affine Richardson varieties.

We study the algebra of functions on the Iwahori group via the category of graded bounded representations of the Iwahori Lie algebra. In particular, we identify the standard and costandard objects in this category with certain generalized Weyl modules. Using this identification we express the characters of the standard and costandard objects in terms of specialized nonsymmetric Macdonald polynomials. We also prove that our category of interest admits a generalized highest weight structure (known as stratified structure). We show, more generally, that such a structure on a category of representations of a Lie algebra implies the Peter-Weyl type theorem for the corresponding algebraic group. In the Iwahori case, standard filtrations of indecomposable projective objects correspond to new “reciprocal” Macdonald-type identities.

An arc space of an affine cone over a projective toric variety is known to be nonreduced in general. It was demonstrated recently that the reduced scheme structure of arc spaces is very meaningful from algebro-geometric, representation-theoretic and combinatorial points of view. In this paper we develop a general machinery for the description of the reduced arc spaces of affine cones over toric varieties. We apply our techniques to a number of classical cases and explore some connections with representation theory of current algebras.

The goal of this paper is to extend the quiver Grassmannian description of certain degenerations of Grassmann varieties to the symplectic case. We introduce a symplectic version of quiver Grassmannians studied in our previous papers and prove a number of results on these projective algebraic varieties. First, we construct a cellular decomposition of the symplectic quiver Grassmannians in question and develop combinatorics needed to compute Euler characteristics and Poincaré polynomials. Second, we show that the number of irreducible components of our varieties coincides with the Euler characteristic of the classical symplectic Grassmannians. Third, we describe the automorphism groups of the underlying symplectic quiver representations and show that the cells are the orbits of this group. Lastly, we provide an embedding into the affine flag varieties for the affine symplectic group.

The goal of this paper is to clarify the connection between certain structures from the theory of totally nonnegative Grassmannians, quiver Grassmannians for cyclic quivers and the theory of local models of Shimura varieties. More precisely, we generalize the construction from our previous paper relating the combinatorics and geometry of quiver Grassmannians to that of the totally nonnegative Grassmannians. The varieties we are interested in serve as realizations of local models of Shimura varieties. We exploit quiver representation techniques to study the quiver Grassmannians of interest and, in particular, to describe explicitly embeddings into affine flag varieties which allow us to realize our quiver Grassmannians as a union of Schubert varieties therein.