Tag - Geometric representation theory

Giovanni Cerulli Irelli: Motzkin combinatorics in linear degenerations of the flag variety

14th March, 2023

In 2021, Fang and Reineke described the support of linear degenerations of flag varieties in terms of Motzkin paths, by using Knight-Zelevinsky multi-segment duality. In a joint project with Esposito and Marietti we give a new characterization of supports in representation-theoretic terms by what we call excessive multi-segments. To do so we consider an algebraic structure on the set of Motzkin paths that we call Motzkin monoid. By using a universal property of the Motzkin monoid, we show that excessive multi segments are parametrized in a natural way  by Motzkin paths. Moreover, we show that this parametrization coincides exactly with the Fang-Reineke parametrization. As a byproduct we have an elementary combinatorial criterion to decide if a multisegment is a support. We have an inductive procedure to describe the inverse of the Fang-Reineke map. In this term, there is a very beautiful (as yet conjectural) formula for the coefficients.

Leonardo Maltoni: Towards a Bernstein presentation of the affine Hecke category

28th November, 2022

The affine Hecke algebra has a remarkable commutative subalgebra corresponding to the coroot lattice in the affine Weyl group. Its nature is encoded in the Bernstein presentation and reveals important representation-theoretic properties of the algebra. If one considers categorifications of the Hecke algebra, for instance the diagrammatic category, the above subalgebra corresponds to a class of complexes in the homotopy category called Wakimoto sheaves, which can be seen as Rouquier complexes. In this talk I will introduce the affine Hecke algebra, the diagrammatic category and the objects mentioned above. Then I will describe some reduced representarives for Rouquier complexes and present some results about the extension groups between Wakimoto sheaves in affine type A1.

Eduardo Esteves: Quiver representations arising from degenerations of linear series

20th October, 2022

We describe all the schematic limits of divisors associated to any family of linear series on any 1-dimensional family of projective varieties degenerating to any connected reduced projective scheme X defined over any field, under the assumption that the total space of the family is regular along X. More precisely, the degenerating family gives rise to a special quiver Q, called a Zn-quiver, a special representation L of Q in the category of line bundles over X, called a maximal exact linked net, and a special subrepresentation V of the representation induced from L by taking global sections, called a pure exact finitely generated linked net. Given g=(Q, L, V) satisfying these properties, we prove that the quiver Grassmanian G of subrepresentations of V of pure dimension 1, called a linked projective space, is Cohen-Macaulay, reduced and of pure dimension. Furthermore, we prove that there is a morphism from G to the Hilbert scheme of X whose image parameterizes all the schematic limits of divisors along the degenerating family of linear series if g arises from one.

Scott Larson: A Categorification of the Lusztig-Vogan Module

4th April, 2022

Continuous actions of real reductive groups are often studied by first linearizing the action to spaces related to functions, then using algebra via Lie algebras and compact groups (cf. Gelfand, Harish-Chandra, Vogan). This paradigm essentially simplifies to the easier problem of studying a complex algebraic group K acting on flag varieties. K-orbit closures are important for representation theory, are generalizations of Schubert varieties, and certain properties are explicitly determined via equivariant resolutions of singularities. In joint work with Anna Romanov, we provide a geometric and algebraic categorification of the Lusztig-Vogan module using the equivariant derived category. Our methods allow us to compute cohomology of all fibres of resolutions constructed quite generally and generalize Soergel bimodule techniques from complex to real reductive algebraic groups.

Ed Richmond: The isomorphism problem for Schubert varieties

28th March, 2022

Schubert varieties in the full flag variety of Kac-Moody type are indexed by elements of the corresponding Weyl group. In this talk, I will discuss recent work with William Slofstra where we give a practical criterion for when two such Schubert varieties (from potentially different flag varieties) are isomorphic, in terms of the Cartan matrix and reduced words for the indexing Weyl group elements. As a corollary, we show that two such Schubert varieties are isomorphic if and only if there is an isomorphism between their integral cohomology rings that preserves the Schubert basis. As an application, we show that the isomorphism classes of Schubert varieties in a given flag variety are controlled by graph automorphisms of the Dynkin diagram.

David Galban: Cohomology and Representation Theory for Lie Superalgebras

14th March, 2022

This talk will consist of two parts. In the first, I will describe the cohomology groups for the subalgebra 𝔫+ relative to the BBW parabolic subalgebras constructed by D. Grantcharov, N. Grantcharov, Nakano and Wu, essentially with these calculations essentially providing the first steps towards an analogue of Kostant’s theorem for Lie superalgebras. In the second part, based on joint work with Nakano, I will analyze the sheaf cohomology groups RI indBG L𝔣(λ), where L𝔣(λ) is an irreducible representation for the detecting subalgebra 𝔣, providing analogues for the BBW theorem and Kempf’s vanishing theorem for sufficiently large λ.

Robert Spencer: (Some) Gram Determinants for An nets

1st March, 2022

The nets giving a diagrammatic description of the category of (tensor products of) fundamental representations of 𝔰𝔩n form a cellular category. We can then ask about the natural inner form on certain cell modules. In this talk, we will calculate the determinant of some of these forms in terms of certain traces of clasps or magic weave elements (for which there is a conjectured formula due to Elias). The method appears moderately general and gives a result which is hopefully illuminating and applicable to other monoidal, cellular categories.

William Graham: Geometry of generalized Springer fibres II

7th February, 2022

Previous work constructed an analogue of the Springer resolution for the universal cover of the principal nilpotent orbit. In joint work with Precup and Russell, we showed that in type A this generalized Springer resolution is closely connected with Lusztig's generalized Springer correspondence. In this talk we discuss the geometry of the fibres of the generalized Springer resolution, and in particular, show that the fibres have an analogue of an affine paving.

William Graham: Geometry of generalized Springer fibres I

31st January, 2022

Previous work constructed an analogue of the Springer resolution for the universal cover of the principal nilpotent orbit. In joint work with Precup and Russell, we showed that in type A this generalized Springer resolution is closely connected with Lusztig's generalized Springer correspondence. In this talk we discuss the geometry of the fibres of the generalized Springer resolution, and in particular, show that the fibres have an analogue of an affine paving.

Marc Besson: Twisted affine Schubert varieties and twisted affine Demazure modules

24th January, 2022

In this talk I will discuss work (joint with J. Hong) on the T-fixed subscheme of twisted affine Schubert varieties. This may be viewed as an extension of results of Zhu in the untwisted case. I will discuss the Frenkel-Kac-Segal isomorphism, as well as Zhu's geometrization of this result. An application of this work is the determination of the smooth locus of twisted affine Schubert varieties.