Friday, 14 November
01:15 pm
Bielefeld University U2-232
Torsion and complete dualizable objects in tt-categories with a Noetherian ring action
Speaker: Jan Stovicek
Seminar Representation Theory of Algebras
The main object of interest in this talk is a rigidly compactly generated tensor triangulated category with an action of a graded commutative Noetherian ring R making (graded) homomorphism groups between compacts finitely generated R-modules. There are various examples of this setup coming from commutative algebra, modular representation theory of groups or homotopy theory.
In joint work with Jun Maillard, we studied the categories of dualizable objects inside the mutually equivalent categories of a-torsion and a-complete objects with respect to a homogeneous ideal a of R. This is a surprisingly well behaved setup. For example, the categories of a-torsion/complete dualizable objects are always Hom-finite over the a-adic completion of R and we can recover them using a form of Cauchy completions from the categories of a-torsion/comlete compact objects. The best results are achieved when we in addition assume that the category of compact objects is strongly generated (or equivalently, has finite Rouquier dimension). Then so is the category of a-torsion/complete dualizable objects and the so-called local regularity condition introduced by Benson, Iyengar, Krause and Pevtsova is automatically satisfied. Moreover, there are Brown representability results providing a "perfect pairing" between the categories of a-torsion/complete compact objects and a-torsion/complete dualizable objects which are formally very similar to Neeman's arXiv:1804.02240.
Wednesday, 19 November
02:15 pm
Bielefeld University M4-122/126
Rascal Posets
Speaker: Drew Amstrong
Oberseminar Gruppen und Geometrie
We introduce a family of "Rascal posets" R(k,n-k) containing (n choose k) elements. The undirected Hasse diagram of R(k,n-k) is isomorphic to Young's lattice on integer partitions in a rectangle, but the orientation is different. We show that R(k,n-k) is a graded poset of height n-k containing k^{n-k} maximal chains and we derive an explicit formula for rank-selected multichains. The poset R(k,n-k) has two interesting geometric interpretations. First, its order complex is the type C alcove triangulation of a dilated simplex. Second, it is the poset of bounded regions in the cyclic hyperplane arrangement. This is related to work of Karp and Williams on the m=1 amplituhedron.
03:45 pm
Bielefeld University M4-122/126
Linear operators preserving volume polynomials
Speaker: Lukas Grund
Oberseminar Gruppen und Geometrie
Volume polynomials are a class of Lorentzian polynomials coming from algebraic geometry. They measure the self intersection number of linear combinations of positive divisors in projective varieties. Their coefficients satisfy additional inequalities, and the support of a (realizable) volume polynomial is the set of bases of an algebraic polymatroid. We show that Aluffi's covolume polynomials are precisely the polynomial differential operators preserving volume polynomials. This provides us with a sufficiency criterion for linear operators preserving volume polynomials. We furthermore explore applications to algebraic matroids. This is joint work with Huh, Michałek, Süß and Wang.
Thursday, 20 November
02:00 pm
Bielefeld University M4-122/126
Centralizers in Hecke Algebras of Any Coxeter Group
Speaker: Haiyu Chen
Oberseminar Gruppen und Geometrie
We study the centralizer of a parabolic subalgebra in the Hecke algebra associated with an arbitrary (possibly infinite) Coxeter group. While the center and cocenter have been extensively studied in the finite and affine cases, much less is known in the indefinite setting. We describe a basis for the centralizer, generalizing known results about the center. Our approach combines algebraic techniques with geometric tools from the Davis complex, a CAT(0)-space associated to the Coxeter group. As part of the construction, we classify finite partial conjugacy classes in infinite Coxeter groups and define a variant of the class polynomial adapted to the centralizer.
Thursday, 27 November
04:15 pm
Bielefeld University V2-210/216
Zeros of S-characters - Showcasing the Computer Algebra System OSCAR
Speaker: Michael Joswig
Mathematisches Kolloquium (TRR 358)
The concept of $S$-characters of finite groups was introduced by Zhmud' (1995) as a generalization of transitive permutation characters. Any non-trivial $S$-character takes a zero value on some group element. By a deep result depending on the classification of finite simple groups a non-trivial transitive permutation character even vanishes on some element of prime power order. J-P. Serre asked whether this generalizes to $S$-characters. We provide a translation of this question into the language of polyhedral geometry and thereby construct many counterexamples, using the capabilities of the computer algebra system OSCAR.
The work on S-characters is joint with Thomas Breuer und Gunter Malle. OSCAR is joint work with The OSCAR Development Team, currently lead jointly with Simon Brandhorst, Claus Fieker, Tommy Hofmann and Max Horn