Related papers: Soergel calculus and Schubert calculus
We review some recent advances in modular representation theory of symmetric groups and related Hecke algebras. We discuss connections with Khovanov-Lauda-Rouquier algebras and gradings on the blocks of the group algebras $F\Sigma_n$, which…
We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ${\mathbb Q}$, acting on…
We give several new formulas which are useful for Schubert Calculus associated with the orthogonal groups and related orthogonal degeneracy loci.
We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and…
In this paper we compute Hochschild homology of certain Soergel bimodules. Moreover, we describe explicitly the graded bimodule maps between Soergel bimodules. This computations are motivated by the categorifications of the colored…
We investigate the structure of the Schrodinger algebra and its representations in a Fock space realized in terms of canonical Appell systems. Generalized coherent states are used in the construction of a Hilbert space of functions on which…
In this paper, we develop 2-dimensional algebraic theory which closely follows the classical theory of modules. The main results are giving definitions of 2-module and the representation of 2-ring. Moreover, for a 2-ring $\cR$, we prove…
We exhibit algorithms to compute systems of Hecke eigenvalues for spaces of Hilbert modular forms over a totally real field. We provide many explicit examples as well as applications to modularity and Galois representations.
We utilize effective algorithms for computing in the cohomology of a Shimura curve together with the Jacquet-Langlands correspondence to compute systems of Hecke eigenvalues associated to Hilbert modular forms over a totally real field.
We generalise the construction of Rouquier complexes to the setting of singular Soergel bimodules by taking minimal complexes of the restriction of Rouquier complexes. We show that they retain many of the properties of ordinary Rouquier…
We investigate the representations and the structure of Hecke algebras associated to certain finite complex reflection groups. We first describe computational methods for the construction of irreducible representations of these algebras,…
The main classical result of Schubert calculus is that multiplication rules for the basis of Schubert cycles inside the cohomology ring of the Grassmannian $G(n,m)$ are the same as multiplication rules for the basis of Schur polynomials in…
We classify simple transitive $2$-representations of certain $2$-sub\-ca\-te\-go\-ri\-es of the $2$-category of Soergel bimodules over the coinvariant algebra in Coxeter types $B_2$ and $I_2(5)$. In the $I_2(5)$ case it turns out that…
We announce some new results regarding the classification problem for separable von Neumann algebras. Our results are obtained by applying the notion of Borel reducibility and Hjorth's theory of turbulence to the isomorphism relation for…
We generalize Soergel's tilting algorithm to singular weights and deduce from this the validity of the Lascoux-Leclerc-Thibon conjecture on the connection between the canonical basis of the basic submodule of the Fock module and the…
These notes are based on a course given at the EPFL in May 2005. It is concerned with the representation theory of Hecke algebras in the non-semisimple case. We explain the role that these algebras play in the modular representation theory…
We adapt the diagrammatic presentation of the Hecke category to the dg category formed by the standard representatives for the Rouquier complexes. We use this description to recover basic results about these complexes, namely the…
We study the rings of invariants for the indecomposable modular representations of the Klein four group. For each such representation we compute the Noether number and give minimal generating sets for the Hilbert ideal and the field of…
We outline an abstract approach to the pseudo-differential Weyl calculus for operators in function spaces in infinitely many variables. Our earlier approach to the Weyl calculus for Lie group representations is extended to the case of…
A natural Hasse-Schmidt derivation on the exterior algebra of a free module realizes the (small quantum) cohomology ring of the grassmannian $G_k(\CC^n)$ as a ring of operators on the exterior algebra of a free module of rank $n$. Classical…