Joint Algebra Seminar

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On the Smooth Locus in Flat Linear Degenerations of the Complete Flag Variety

Linear degenerations of the flag variety arise as very natural generalizations of the complete flag Variety, and their geometrical properties very often appear to be linked with interesting combinatorial patterns. The talk will focus on a special class of linear degenerations, the flat degenerations, which have the remarkable property of being equidimensional algebraic varieties of the same dimension as the complete flag variety. In some very recent works of M. Lanini and A. Pütz it is proved that linear degenerations of the flag variety can be endowed with a structure of GKM variety, under the action of a suitable algebraic torus T. The aim of the talk is to show how GKM Theory can be applied to obtain combinatorial criteria to identify T-fixed points in the smooth locus of flat linear degenerations, generalizing a smoothness criterion proved by G. Cerulli Irelli, E. Feigin and M. Reineke for Feigin Degeneration.

Explicit formulas for 1-point functions in certain classes of vertex operator algebras

The study of vertex operator algebras in Mathematics originates in the famous Monstrous Moonshine Conjecture by McKay-Thompson and Conway-Norton, famously proved by Borcherds, building on important previous work by Frenkel-Lepowsky-Meurman. Since then, a general theory has been developed and one important result due to Huang, Zhu, Dong-Li-Mason and others in this context gives a very close connection between VOAs and modular forms: For a sufficiently nice VOA, its character (i.e. the generating function of the dimensions of its graded components) and more generally the so-called 1-point functions of states in the VOA define modular forms on some congruence subgroup. It is however still rather mysterious which congruence subgroup this might be in general or how to determine the modular form in question. In my talk I will give a short introduction to the theory of VOAs and the problem just described and then present joint work with Geoffrey Mason in which we provide a solution to this problem for certain special VOAs in that we give explicit formulas for these 1-point functions on a basis of the respective VOA. The main ingredient is a new variant of so-called Zhu recursion.

Enumerating triply-periodic tangles

Using periodic surfaces as a scaffold is a convenient route to making periodic entanglements. I will present a systematic way of enumerating new tangled periodic structures, using low-dimensional topology and combinatorics, posing the question of how to best characterise these new patterns. I will also give an insight into applications of these structures.

Eigenvalue problem on the Grassmannian

We determine the number of complex solutions to a nonlinear eigenvalue problem on the Grassmannian in its Plücker embedding. In the case of the Grassmannian of lines, we obtain an explicit formula for the number of complex solutions, which involves Catalan numbers and is the volume of the Cayley sum of the Gelfand-Cetlin polytope with simplex. This rests on the geometry of the graph of a birational parametrization of the Grassmannian. We present a squarefree Gröbner basis for this graph, and we develop connections to toric degenerations from representation theory.

The motivation of this problem comes from quantum chemistry, where it represents the truncation to single electrons in coupled cluster theory. This is a joint work with Bernd Sturmfels and Svala Sverrisdóttir.

Arithmetic counts of tropical plane curves and their properties

We are interested in the enumerative problem of finding the number of plane curves with fixed degree and genus through a suitable number of points. When this problem is phrased over an arbitrary base field, the number of such curves depends on the configuration of the point conditions and hence is not an intrinsic invariant of the problem. Instead, one can study a Levine-Welschinger count to obtain an invariant. In recent work by Jaramillo Puentes and Pauli, they have shown that such an arithmetic count can be computed by counting tropical plane curves. In this talk I will explain various properties of these arithmetic tropical counts arising from the combinatorial structure of the tropical enumerative problem. This is based on joint work with Andrés Jaramillo Puentes, Hannah Markwig, and Sabrina Pauli.

Toric supervarieties with one odd dimension

We describe the notion of a toric supervariety, generalizing that of a toric variety from the classical setting. We give a combinatorial interpretation of the category of quasinormal toric supervarieties with one odd dimension using decorated polyhedral fans. We then use this interpretation to calculate some invariants of these supervarieties and extract geometric information from them.

Representation of quivers for limit linear series

We explore the existence of simple bases for certain special quiver representations arising from degenerations of linear series. The existence of a simple basis implies that the representation decomposes into subrepresentations of dimension one and simplifies the calculus of the multivariate Hilbert polynomial of the quiver Grassmannian of the representation. For these quiver representations, we characterise the existence of a simple basis with a local condition. If the time allows, we apply this characterization to show that the linked projective space is local complete intersection (thus Cohen-Macaulay).

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The combinatorial structure of a modular curve modulo a prime number

If we reduce a modular curve modulo a prime p dividing the level, then we often obtain a non-smooth curve with various irreducible components. This special fiber can be turned into a combinatorial object using the notion of a dual intersection graph, whose structure determines many important arithmetic properties of the curve if the used model is suitably nice. If the model is semistable, then we also call this graph a Berkovich skeleton for the modular curve. This skeleton for the modular curve X_{0}(p) was for instance an important ingredient in Ribet's proof on level-lowering for modular forms, as well as Mazur's isogeny theorem for elliptic curves.

In this talk, I will explain how to obtain a deformation retract of this skeleton for modular curves associated to arbitrary congruence subgroups in PSL2(Z). The recipe is in terms of double coset spaces in this group, which often reduce to orbits on flag varieties and symmetric spaces defined over rings such as Z/NZ. From this we deduce the following:

- A simple group-theoretic formula for the first Betti number of the skeleton of a modular curve.
- A determinantal formula for the geometric Tamagawa numbers of the modular curve.
- An explicit fractal representation of the skeleta for X_{0}(N), X_{1}(N), X_{sp}(N) and X^{+}_{sp}(N).

I will shortly explain the proofs behind these methods, as well as go through several examples.

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Computing with the monster and maximal subgroups

The monster group is the largest of the 26 sporadic simple groups with order approximately 8 x 10^53. Apart from its extremely large order, one of the challenges of computing with the monster is that it does not have a small matrix or permutation representation. In this talk, I will discuss the history of computing with the monster, some recent breakthroughs in this area, and some joint work on maximal subgroups with Heiko Dietrich and Tomasz Popiel.

A Pieri formula for affine Demazure modules

In this talk I will report on recent work with Deniz Kus on an analogue of the Pieri formula for affine Demazure modules.
The classical Pieri formula gives a multiplicity free expansion of an irreducible module with a fundamental one for the complex general linear group. We replace the tensor product by the fusion product and prove an analogue Pieri formula for higher level Demazure modules for the affine special linear Lie algebra. To be more precise, we show that the fusion product of an arbitrary stable Demazure module with a fundamental module admits a multiplicity free excellent filtration and the successive quotients are described explicitly. As a consequence, we derive recurrence relations for the generating series encoding the numerical multiplicities in Demazure flags of level one Demazure modules.

Local models and quiver Grassmannians for cyclic quivers.

We will describe certain quiver Grassmannians for the cyclic quivers popping up in the theory of local models of Shimura varieties. The quiver Grassmannians in question have many nice geometric, topological and combinatorial properties. In particular, they admit a realization in terms of affine Schubert varieties, thus providing a finitization of the Gaitsgory central degeneration of affine Grassmannians. Joint work with Martina Lanini and Alexander Puetz.

Buildings, valuated matroids and tropical linear spaces

Affine Bruhat--Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of PGL parametrizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite-dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise-linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space. Inspired by Payne's result that analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear subspaces of rank r (as the embedding and the dimension of the ambient projective space vary), and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid, extending a result of Dress and Terhalle. This is joint work with Luca Battistella, Kevin Kühn, Martin Ulirsch and Alejandro Vargas.

Positive del Pezzo Geometry

Real, complex, and tropical algebraic geometry join forces in a new branch of mathematical physics called positive geometry. We develop the positive geometry of del Pezzo surfaces and their moduli spaces, viewed as very affine varieties. Their connected components are positive geometries derived from highly symmetric polytopes. We study their canonical forms and scattering amplitudes, and we solve the likelihood equations. This is work in progress with Nick Early, Marta Panizzut, Bernd Sturmfels and Claudia Yun.

Amalgamating groups via linear programming

A compact group $A$ is called an amalgamation basis if, for every way of embedding $A$ into compact groups $B$ and $C$, there exist a compact group $D$ and embeddings $B\to D$ and $C\to D$ that agree on the image of $A$. Bergman in a 1987 paper studied the question of which groups can be amalgamation bases. A fundamental question that is still open is whether the circle group $S^1$ is an amalgamation basis in the category of compact Lie groups. Further reduction shows that it suffices to take $B$ and $C$ to be the special unitary groups. In our work, we focus on the case when $B$ and $C$ are the special unitary group in dimension three. We reformulate the amalgamation question into an algebraic question of constructing specific Schur-positive symmetric polynomials and use integer linear programming to compute the amalgamation. We conjecture that $S^1$ is an amalgamation basis based on our data. This is joint work with Michael Joswig, Mario Kummer, and Andreas Thom.

A bijection between evil-avoiding and rectangular permutations

The asymmetric simple exclusion process (ASEP), a type of interacting particle system, has been a topic of recent interest in mathematics and physics, with H.T. Yau describing it as a "default stochastic model for transport phenomena." A kind of ASEP, the inhomogeneous TASEP, has unexpected connections to Schubert polynomials and evil-avoiding permutations, the latter of which was introduced by Kim and Williams in 2022. A related class of permutations, called rectangular permutations, was introduced by Chirivì, Fang, and Fourier in 2021, and arises in the study of Schubert varieties and Demazure modules. Taking a suggestion of Kim and Williams, we supply an explicit bijection between evil-avoiding and rectangular permutations in Sn that preserves the number of recoils. We encode these classes of permutations as regular languages and construct a length-preserving bijection between words in these regular languages. We extend the bijection to another Wilf-equivalent class of permutations, namely the 1-almost-increasing permutations, and exhibit a bijection between rectangular permutations and walks of length 2n-2 in a path of seven vertices starting and ending at the middle vertex.

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Tensor decomposition for a coideal subalgebra and type B and D Jones-Wenzl projectors

One of the most elementary yet beautiful cases of Schur-Weyl duality is that between U_q(gl2) and the Hecke-algebra of the symmetric group acting on tensor powers of the standard representation. Here the Hecke algebra takes a new form as a quotient known as the Temperley-Lieb algebra. This algebra has been well-studied and understood in terms of special idempotents known as the Jones-Wenzl projectors with a diagrammatic meaning. I will explain how to enhance the story from the symmetric group/Temperley-Lieb side to a type B/D version, by replacing U_q(gl2) with a certain coideal subalgebra. I present the type B/D Jones-Wenzl projectors one discovers on the way and many other generalizations from type A to type B/D including convergent powers of type D full-twists. This is based joint work with Catharina Stroppel.

Juggling patterns, Grassmann necklaces, and affine flag varieties

Postnikov constructed a cellular decomposition of the totally nonnegative Grassmannians. The poset of cells can be described (in particular) via Grassmann necklaces. We study certain quiver Grassmannians for the cyclic quiver admitting a cellular decomposition, whose cells are naturally labeled by Grassmann necklaces. These quiver Grassmannians admit explicit embeddings into affine flag varieties which allow us to realize our quiver Grassmannians as a union of Schubert varieties therein.

From Braids to Transverse Slices in Reductive Groups

Slodowy slices and their quantisations play a central role in the geometric Langlands pogramme. We explain how group analogues of Slodowy slices arise by interpreting certain Weyl group elements as braids. Such slices originate from classical work by Steinberg on regular conjugacy classes, and different generalisations recently appeared in work by Sevostyanov on quantum group analogues of W-algebras and in work by He-Lusztig on Deligne-Lusztig varieties. Also building upon recent work of He-Nie, our perspective furnishes a common generalisation and a simple geometric criterion for Weyl group elements to yield strictly transverse slices. We finish by briefly discussing their Poisson structures and quantisations.

Characters and character sheaves of finite groups of Lie type

An important task in the representation theory of finite groups is the determination of their character tables. As the classification of finite simple groups shows, the main difficulties in this context concern the finite groups of Lie type, which arise as an infinite series of finite groups associated to a certain algebraic group over a field of positive characteristic. In order to generically tackle the problem of determining the character tables of finite groups of Lie type, Lusztig developed the theory of character sheaves in the 1980s. In this framework, due to the work of Lusztig and Shoji, the problem is in principle reduced to determining certain roots of unity. We report on some recent progress in this area.

Multiprojective Seshadri stratifications for flag varieties

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 interesting consequences) gives rise to a standard monomial theory on the homogeneous coordinate ring, at least in certain nice cases. We will generalize Seshadri stratifications to the multiprojective setting and give a geometric construction of the usual standard monomial theory for flag varieties in types A/B/C/F/G.

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