Köln Algebra and Representation Theory Seminar

Abstract: We introduce a generalization of $K$-$k$-Schur functions and $k$-Schur functions via the Pieri rule. Then we obtain the Murnaghan-Nakayama rule for the generalized functions. The rules are described explicitly in the cases of $K$-$k$-Schur functions and $k$-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for $k$-Schur functions, and explains it as a degeneration of the rule for $K$-$k$-Schur functions. In particular, many other special cases promise to be detailed in the future.

Abstract: Let V be an n-dimensional vector space over a non-archimedean local field K and fix a partition λ of an integer d. We study the problem of determining the invariant lattices in the Schur module S_{λ}(V) under the action of the maximal compact subgroup of GL(n,K). 

Abstract: A Hibi variety is a projective toric variety which depends on a poset. We will start with the general definition and provide several examples. We will then explain how to use Hibi varieties to produce toric degenerations of Grassmannians due to Sturmfels/Gonciulea–Lakshmibai/Kogan–Miller. After that we will give the definition of the semi-infinite (a.k.a. quantum) Grassmannians. Finally, we will construct (an infinite) poset responsible for toric degenerations in the semi-infinite settings. Joint work with Igor Makhlin and Alexander Popkovich. 

Abstract: In 2021 Rocco Chirivi, Xin Fang and Peter Littelmann introduced the notion of a Seshadri stratification on an embedded projective variety, which (among many other useful consequences) induce a filtration on the homogeneous coordinate ring with one-dimensional subquotients, hence giving an index set for a basis of this algebra. So called normal and balanced stratifications even give rise to a standard monomial theory. We will generalize Seshadri stratifications to the multiprojective setting and construct a stratification on partial flag varieties to get a geometric interpretation of the usual standard monomial theory of their multihomogeneous coordinate rings. 

Abstract: (This is a joint work with Martina Lanini and Rui Xiong.)

We prove that the structure algebra of a Bruhat moment graph of a finite real root system is a Hopf algebroid (in the language of 2-monoidal categories) with respect to the Hecke and the Weyl actions.  In particular, it implies that the natural Hopf-algebra structure on the algebraic oriented cohomology $A(G)$ of Levine-Morel of a split semi-simple linear algebraic group G can be lifted to a `bi-Hopf' structure on the T-equivariant algebraic oriented cohomology of the complete flag variety. We discuss various applications of this result to generalized Schubert calculus, to Coxeter groups and finite real root systems.

Abstract: We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context (after H. Bass) is an algebra built on the data of two algebras, two bimodules and two morphisms.  For a strong stratifying Morita context - or equivalently for a strong stratifying ideal - we show  that Han's conjecture holds if and only if it holds for the diagonal subalgebra. The main tool is the Jacobi-Zariski long exact sequence. This is a work in collaboration with Claude Cibils, Marcelo Lanzilotta and Eduardo Marcos.

Abstract 2: In this talk I will describe a general philosophy, due to Patimo, for constructing positive combinatorial formulas for Kostka-Foulkes polynomials beyond type A. This amounts to constructing atomic decompositions for crystals as well as swapping functions which allow to define charge statistics. Then I will explain such a construction for crystals of type C_2 and point out future directions to follow. This is joint work with Leonardo Patimo.

Abstract: 

For every finite dimensional algebra, there are correspondences between support $\tau$-tilting modules, functorially finite torsion pairs, and left finite wide subcategories of the module category. The first two classes of objects have "mirror" versions in the category of projective presentations, namely, 2-term silting complexes and cotorsion pairs. In this talk, we propose that the analog of the third class of objects is that of thick subcategories. We will recall the notion of a thick subcategory of the category of projective presentations and show that those with enough injectives are in bijection with left finite wide subcategories. We will explain how thick subcategories arise from an attempt to define semistability for projective presentations.

Abstract: An algebra $A$ is called Koszul if every simple module admits a linear projective resolution. Typical examples of categories of the form $A-Mod$ for a Koszul algebra $A$ are blocks of category $\mathcal{O}$ as shown by Beilison-Ginzburg-Soergel. One main motivation to study Koszul algebras and their duals is that they give rise to derived equivalences under some finiteness assumptions. I will report on joint work with Nehme and Stroppel how to prove Koszulity for "large" algebras which admit special truncations to a finite setting. As examples I will treat a) representations of the Deligne category $Rep(O_t), t \in \mathbb{C}$, b) Khovanov algebras and c) modules over certain infinite quivers. 

Abstract: The semi-infinite flag manifold Q associated with a simple algebraic group G is a "level-zero" variant of the affine flag variety of G. The geometry of Q is quite different from that of the standard affine flag variety; in particular, its Schubert varieties are both infinite-dimensional and infinite-codimensional in Q. Despite this difficulty, a notion of equivariant K-group for Q has been introduced recently by Kato, Naito, and Sagaki and, through further work of Kato, this K-group has played an important role in applications to the quantum K-theory of (partial) flag varieties G/P. In this talk, I will discuss a nil-DAHA action on the equivariant K-group of Q and explain how it can be used to find algebraic and combinatorial "inverse" Chevalley formulas in this setting (for G of type ADE). This is partly based on joint works with Kouno, Lenart, Naito, and Sagaki. 

Abstract:

Abstract: I will report on recent joint work in progress with Takuya Murata. Motivated by work of Harada—Kaveh and the desire to construct moment map type maps to Newton-Okounkov polytopes we study maps to toric varieties. As algebro-geometric methods turn out to not be appropriate we use results of Mather involving Whitney stratifications to obtain a collapsing map. Although the result is more general our main application concerns toric degenerations where we construct a map from the general to the special fibre. Moreover, we generalize a result of Harada-Kaveh constructing an integrable system on a projective variety that admits a toric degeneration induced by the moment map on the toric variety. A first preprint of the work is available on arxiv:2210.13137.