Joint Algebra Seminar

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Classifications at the intersection of group and graph theory

Group theory and graph theory are deeply interconnected. While groups are used to abstractly analyze various mathematical objects, graphs serve as an indispensable tool for representing them. The primary focus of my PhD research has been to explore this relationship in two complementary directions. In this talk, I will present an overview of this research and highlight the main classification results.

First, we classify certain finite groups for which the so-called power graph, a graph capturing the power structure of a group, has a particular property, namely it is a cograph or it is chordal. Altogether, we provide classifications of the finite non-nilpotent power-cograph groups and finite simple power-chordal groups, relative to typical number theoretical problems. Second, we classify certain finite ultrahomogeneous colored binary relational structures, represented as vertex- and edge-colored graphs. In particular, we classify finite ultrahomogeneous vertex-colored oriented graphs and develop an efficient algorithm to enumerate all finite edge-colored graphs up to a specified order.

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The free tridendriform structure

The notions of dendriform algebras, respectively tridendriform, describe the action of some elements of the symmetric groups called shuffle, respectively quasi-shuffle over the set of words whose letters are elements of an alphabet, respectively of a monoid. A link between dendriform and tridendriform algebras will be made.
Those words algebras are quite nice but they are not free. This means that they satisfy extra properties like commutativity. In this talk, we will describe the free tridendriform algebra. It will be described with planar trees (not necessarily binary) called Schroeder trees. We will describe the tridendriform structure over those trees in a non-recursive way. Then, if time permits it we will give its bialgebra structure and talk about some classical questions in this context.

Recent insights surrounding combinatorial approaches to isomorphism and symmetry problems

Modern practical software libraries that are designed for isomorphism tests and symmetry computation rely on combinatorial techniques combined with techniques from algorithmic group theory. The Weisfeiler-Leman algorithm is such a combinatorial technique. When taking a certain view from descriptive complexity theory, the algorithm is universal. After an introduction to problems arising in symmetry computation and this particular combinatorial technique, I will give an overview of results from recent years. The results give insights into worst-case behavior and computational complexity. In the course of this, I will discuss combinatorial “cops and robbers” games and some lower bound constructions.

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On Frobenius graphs of diameter 3 for finite groups

For a subgroup H of a finite group G, the Frobenius graph Γ(G, H) records the constituents of the restrictions to H of the irreducible characters of G. We investigate when this graph has diameter 3.

(This is joint work with L. Héthelyi, E. Horváth, and B. Külshammer.)

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Disconnected Reductive Groups

A disconnected reductive group is a linear algebraic group whose connected component of the identity is a reductive group. If one is only interested in connected reductive groups, disconnected ones enter the picture as subgroups.

In this talk I will explain how to classify disconnected reductive groups up to an isomorphism. Time permitting, I will also briefly discuss the representation ring of such group. The talk is based on joint work with Dylan Johnston and Diego Martin Duro.

Crystal bases for the Kronecker problem

Kronecker coefficients arise as multiplicities in decompositions of GL(nm) irreducibles after restricting the GL(nm) action to the subgroup GL(n) x GL(m) with tensor product embedding. A long-standing open problem in algebraic combinatorics asks for a combinatorial rule for these coefficients akin to Littlewood-Richardson rule. One approach suggests passing to quantum groups Uq(gl(nm)) and Uq(gl(n)+gl(m)) and using combinatorics of crystal bases to describe Kronecker multiplicities. However the main issue is that the standard quantum version Uq(gl(n)+gl(m)) of the subgroup is no longer Hopf subalgebra of the standard Uq(gl(nm)) and there is no interplay between their crystal bases. I will give an overview of attempts to overcome this and still define and use crystal bases, in particular, via considering nonstandard universal enveloping and Hecke algebras.

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Symmetry-Preserving Operations on Embedded graphs

Symmetry-preserving operations on polyhedra, such as truncation and dual, have been studied for a very long time. However, it was only recently that a general description of all 'local symmetry-preserving operations' (lsp-operations) was presented. With this description it becomes possible to prove general results about all symmetry-preserving operations, instead of studying every operation separately. We use this approach to determine the effect of lsp-operations on the connectivity of embedded graphs, and investigate when these operations not only preserve, but also increase symmetry.

Difference Galois theory in characteristic p

The goal of difference Galois theory is to study the algebraic properties of the solutions to a certain type of functional equations, which are the equivalent of differential equations where the derivative is replaced by a ring endomorphism. Like in the usual Galois theory, we can define a Galois group for such an equation, which is an algebraic subgroup of GL_n, and the equations that define the Galois group tell us about the algebraic relations between the solutions. In the positive characteristic case, an additional problem appears due to inseparability, which can be solved by considering the Galois group as a group scheme. I will explain how to construct this Galois group, the Galois correspondance that we obtain in this general case, and how it can be used to find algebraic properties of the solutions.

Motzkin combinatorics and linear degenerations of flag varieties

Linear degenerations of flag varieties constitute a family of projective varieties defined in linear-algebraic terms whose generic element is a flag variety. Work of X. Fang and M. Reineke has revealed the unexpected appearance of Motzkin numbers in the study of the topology of linear degenerations. In joint work with G. Cerulli Irelli and M. Marietti, we give a combinatorial interpretation of the Fang-Reineke result. If time permits, I will illustrate open problems and the state of ongoing research on these questions.

Permutations, bases and low rank groups

Let G = GL(V), where V is a finite-dimensional vector space, and recall that any element in G is uniquely determined by its action on a basis for V. In addition, any two pairs of linearly independent vectors can be mapped to each other by an element of G. These two basic linear algebra properties can be interpreted in the language of permutation groups, which leads us naturally to the definitions of base and rank of a permutation group. In this talk, I will present some of my recent results on bases for primitive permutation groups, and I will report on recent progress with C.H. Li and Y.Z. Zhu towards a classification of the rank three groups.

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h^*-polynomials of Cosmological Polytopes

Cosmological polytopes appear in the study of the wavefunction of the universe and therefore are of high interest in physics. In several recent articles cosmological polytopes have been studied. In one of them, Juhnke, Solus and Venturello showed that cosmological polytopes admit a regular unimodular triangulation by finding a Gröbner basis for its toric ideal. We extend these results and investigate Ehrhart theoretic aspects of cosmological polytopes. For example, we can compute the h^*-polyomials of cosmological polytopes of trees and cycles, that can also have multiple edges. Therefore, geometric as well as algebraic methods are used, which will all be explained in the talk.

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Computing in Wreath Products

Nowadays, computer algebra systems are an integral tool to assist scientists across various disciplines in their research and to solve real-world problems. One of the fundamental algebraic structures we come across are groups, which often arise from studying the symmetries of objects. There are various strategies to deal with groups of large encoding size on a computer but there are many open problems left in this area. One class of groups that is currently difficult to deal with is the class of large base permutation groups, of which a great number can be described as certain wreath products with socle type An.

In my phd thesis, I am working on computing wreath product decompositions of such groups, and coming up with ideas on how this data structure can be used to design efficient algorithms for further investigations about these groups. In this talk, I will introduce the wreath cycle decomposition in (abstract) wreath products analogously to the disjoint cycle decomposition in permutation groups. Building on top of that, I will present concepts for wreath products that can be exploited on the computer to design fast algorithms for solving various problems, one of them the computation of conjugacy classes. Based on joint work with Dominik Bernhardt, Alice C. Niemeyer, and Lucas Wollenhaupt.

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Weyl group twists and representations of Borel subalgebras of quantum affine algebras

Finite-dimensional representations of Lie algebras (and quantum groups) have characters invariant under the Weyl group action. However, this invariance does not hold for infinite-dimensional representations in the category $\mathcal{O}$. Recent studies by E. Frenkel and D. Hernandez have revealed that the Weyl group's action on the q-characters of representations of quantum affine algebras in the category $\mathcal{O}$ exhibits intricate structures and has significant application for previously unsolved questions.
In this talk, I will discuss my research on this subject. We explore various categories $\mathcal{O}^w$ of representations of Borel subalgebras of quantum affine algebras, labeled by elements of the Weyl group. We propose methods for calculating q-characters of representations in these categories. Furthermore, we suggest a conjectural relationship between two of these categories, which may provide a deeper understanding of earlier questions.

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Polyhedral compactifications of Bruhat-Tits buildings

Bruhat-Tits theory associates to any reductive group G over a valued field k a polyhedral complex, called a building, endowed with an action of G(k). In this talk, we will discuss various results surrounding a particular compactification of Bruhat-Tits buildings, sometimes called polyhedral, or maximal Satake. First introduced by Landvogt, it was studied under various guises by several authors.

Signed tropical discriminants and steady state varieties

One of the most fundamental questions in tropical geometry is: How much topological information does the tropicalization of a variety retain.
Although Viro patchworking was one of the earliest achievements in tropical geometry, the area of real tropical geometry remains comparatively unexplored. In this talk we study real tropicalizations of discriminants, complete intersections, and steady state varieties from reaction networks.

Wachspress Objects and the Reconstruction of Polytopes from Partial Metric Data

In how far can a convex polytope be reconstructed from partial combinatorial and geometric data, such as its edge-graph, edge lengths and dihedral angles, up to combinatorial type, affine equivalence or isometry? Questions of this nature have a long history and are intimately linked to both classical rigidity theory and real algebraic geometry.
After a short survey of the state of the art I will focus on one particular reconstruction conjecture: is a polytope uniquely determined by its edge-graph, edge lengths and the distance of each vertex from some interior point? If true, this would generalize and unify a number of known results, such as the Kirszbraun theorem and the reconstruction of matroids from their base exchange graph. It turns out that progress on this question can be made by employing unexpected tools from the intersection of algebraic/convex geometry and spectral graph theory - Wachspress coordinates and the Izmestiev matrix. I will introduce these objects and explain how they allow us to resolve the conjecture in several relevant special cases. If there is time, I will explore the surprising emergence of these "Wachspress objects" across mathematics and their potential to bridge between algebra, geometry and combinatorics.

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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 $B o D$ and $C o D$ 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 $S_n$ 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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