Women in Noncommutative Algebra and Representation Theory workshop 4.
Banff International Research Station (BIRS). March 23 – 28, 2025.
Noncommutative algebra and representation theory are two closely intertwined areas of research that have strong ties to many other fields in mathematics and quantum physics such as conformal field theory, operator algebras, string theory, topological field theory, and the various guises of noncommutative geometry. Noncommutative algebra creates an algebraic framework for generalizing the study of polynomials to other rings that fail to be commutative. Representation theory is a powerful tool for investigating the action of an algebraic object on a space as linear operators and matrices. As in many areas of mathematics, women and non-binary researchers have been underrepresented in noncommutative algebra and representation theory. The goal of this workshop is to bring together some of the best experts and junior participants working on these topics to collaborate on research projects. All research group participants in the workshop will be women and non-binary researchers. This is prompted by the proven success of past BIRS conferences for women in mathematics, including our first event, the WINART1 workshop held at BIRS in 2016, our second event, the WINART2 workshop held at the University of Leeds in 2019, and the third, WINART3 workshop held at BIRS in 2022. For many junior participants, the WINART network has been pivotal to their research programs and careers. This event will also have a great impact on the research careers of the participants in attendance, and as a result, it will have a positive impact on the field as a whole.
Organizers: Darlayne Addabbo, Ellen Kirkman, Maitreyee Kulkarni, Lilit Martirosyan.
Application: Please fill this form to apply to this workshop.
Deadline: August 1, 2024.
See the Research Group Descriptions below and Q & A page for more information. If you have any other questions, feel free to reach out to womenncalgrepthy@gmail.com.
Research Group Descriptions:
Computing weight multiplicities, co-led by Carolina Benedetti and Pamela Harris.
The following problem due to Hermann Weyl, (Mathematische Zeitschrift,1925) arises in representation theory of complex semisimple Lie algebras: What is the multiplicity of the weight $\mu$ in the irreducible representation with dominant highest weight $\lambda$, which we denote by $L(\lambda)$? In 1948, Kostant developed his well-known formula for computing the multiplicity of a weight in an irreducible highest weight representation (Amer. J. Math., 1959). Despite the availability of such a formula, using it for computational purposes can be quite daunting. These complications and the computational complexity involved in such a formula have motivated Pamela Harris’s research in this field.
In her paper On the adjoint representation of $\mathfrak{sl}_n$ and the Fibonacci numbers (C. R. Math. Acad. Sci. Paris 349, 2011) Pamela Harris gives a purely combinatorial proof for Lie type $A$ of the above result of Kostant. More recently, C. Benedetti, C. R. H. Hanusa, P. E. Harris, A. Morales, and A. Simpson established a combinatorial equivalence between Kostant’s partition function and (magic) multiplex juggling sequences, providing a juggling framework to calculate Kostant’s partition functions, and a partition function framework to compute the number of juggling sequences. This equivalence yields applications to polytopes, posets, positroids, and weight multiplicities, thus opening numerous directions for our future research. One initial project is to extend the juggling framework of this paper to the exceptional Lie algebras.
q-rationals in quantized Teichmüller space, co-led by Léa Bittmann and Emine Yıldırım.
Sophie Morier-Genoud and Valentin Ovsienko define the q-deformed rational numbers and q-deformed continued fractions in their celebrated paper titled “q-deformed rationals and q-continued fractions”. These notions are certain quantized versions of ordinary rational numbers and continued fractions. This study led to many other generalizations such as q-deformed Pascal identity for the Gaussian binomial coefficients . After their discovery, q-rationals showed many interesting connections to the other fields. Let’s name a few: Quiver representations, Jones polynomials of rational knots, rank generating functions for posets, finite Schubert varieties etc. The authors also define q-deformed elementary matrices which are quite similar to the matrices used in the context of quantized Teichmüller space and they claim that the notion of q-rational is thus closely related to that of “quantum geodesic length”. As authors mention in their papers, it would be very interesting to investigate this relationship. Therefore, we would like to propose a project to delve into the secrets of this relationship. Léa Bittmann’s expertise on the quantum world will open new ways of seeing this open problem and Emine Yıldırım has been working with a vast variety of combinatorial objects that can give us insight on the intrinsic structure of q-rationals.
Combinatorial tools in Representation Theory, co-led by İlke Çanakçı and Francesca Fedele.
Many representation theoretic problems can be interpreted from a combinatorial point of view. In particular, combinatorial models in terms of surfaces of e.g., module categories, cluster categories, higher cluster categories, derived categories led to progress in their representation theory. Along these lines, as part of WINART3, the leaders have developed representation theoretic interpretation of a super analogue of cluster algebras of type A using the combinatorics of snake graphs. Depending on the background and interests of the participants in this group, we aim to expand the connections between representation theoretic problems and combinatorics.
Vertex operator algebras and Lie group analogs for Borcherds algebras, co-led by Lisa Carbone and Elizabeth Jurisich.
The Monster Lie algebra is an example of a Borcherds algebra. It was discovered by Borcherdsas part of his work on the Conway-Norton Monstrous Moonshine conjecture. This was a far reaching conjecture connecting a certain elliptic modular invariant, the Moonshine module vertex operator algebra of Frenkel, Lepowsky and Meurman and the irreducible representations of the Monster finite simple group. In joint work Lisa Carbone, Elizabeth Jurisich and Scott Murray, proposed a Lie group analog for the Monster Lie algebra (arxiv/2002.06658). In order to explore connections with the Monster finite simple group and its action on the Moonshine module, in recent joint work, Darlayne Addabbo, Lisa Carbone, Elizabeth Jurisich, Maryam Khaqan and Scott Murray constructed vertex operators corresponding to imaginary simple roots of the Monster Lie algebra. This work has opened up many new avenues for research, including the possibility of the construction of a new Lie group analog for the Monster Lie algebra using primary vectors in the Moonshine module.
The WINART 2025 workshop will be an excellent avenue to follow-up on this exciting direction of research on this topic and on other aspects of infinite-dimensional Lie theory in which Carbone and Jurisich are leading experts.
Springer fibers, Hessenberg varieties, and webs co-led by Laura Colmenarejo and Julianna Tymoczko
Springer fibers and their generalizations, Hessenberg varieties, are at the nexus of combinatorics, geometry, linear algebra, and representation theory. They sit inside the flag variety, which can be viewed either as the collection of nested linear subspaces in a fixed n-dimensional complex vector space, or as the quotient of invertible matrices by upper-triangular matrices. It’s long been known that the geometry of flag varieties is deeply connected with the combinatorics of the symmetric group. This field, which is called Schubert calculus, has led to enormously powerful interactions between representation theory, commutative algebra, algebraic geometry, and combinatorics.
The Springer fiber of a square matrix X consists of the “eigenflags” of that matrix, namely the flags for which X restricts to an endomorphism on each of the nested subspaces. Its geometry is subtle and complicated, interlaying the combinatorics of partitions (inherited from the Jordan decomposition of X) with the combinatorics of permutations (inherited from the flag variety). Adding even more complexity, Hessenberg varieties loosen the condition under which X acts on each subspace using another combinatorial constraint, which is essentially a Dyck path. Our project will use combinatorial tools — especially labeled graphs called webs — to analyze representation theoretic and geometric questions about Springer fibers and Hessenberg varieties.
Algebraic structures on moduli of curves from vertex operator algebras, co-led by Angela Gibney and Svetlana Makarova.
The moduli space of n-pointed (Deligne-Mumford) stable curves of genus g provides a natural environment in which to study smooth curves and their degenerations. These spaces, for different values of g and n, are related to each other through systems of tautological maps. The most useful algebraic structures on the moduli space of curves are reflective of this, and are often governed by recursions, and amenable to inductive arguments.
Sheaves on the moduli space of curves given by representations of vertex operator algebras (VOAs for short) exemplify such algebraic structures. VOAs generalize associative algebras as well as Lie algebras, and have played important roles in both mathematics and physics, in understanding conformal field theories, finite group theory, and in the construction of knot invariants and 3-manifold invariants. Given nice enough VOAs and V-modules, sheaves of coinvariants have good properties. For instance, as is shown in work with Damiolini and Krashen, where we define a series of associative algebras $\mathfrak{A}_d(V)$ for $d\in \mathbb{Z}_{\ge 0}$, if $\mathfrak{A}_d(V)$ have unities that act as identity elements on modules (ie. strong unities), then coherent sheaves are vector bundles.
Grassmannians and Cohen-Macaulay modules, co-led by Sira Gratz and Špela Špenko
Grassmannian cluster categories were introduced by Jensen, King, and Su in the finite case and an analogue was given by August, Cheung, Faber, Gratz, and Scroll in the infinite case. They are given by a category of Cohen-Macaulay modules over some hypersurface singularity. The project would build on these results exploring further directions, using Grassmannian cluster combinatorics, novel techniques on completions of triangulated categories by Neeman and representation theory of reductive groups.
Caldero-Chapoton Map for Gentle Algebras, co-led by Khrystyna Serhiyenko and Yadira Valdivieso-Díaz.
Cluster algebras are a class of commutative rings defined recursively by a set of generators called cluster variables. They have an intricate combinatorial structure, and have been related to numerous areas of mathematics and physics. In terms of representation theory of algebras via categorification, cluster variables correspond to certain $\tau$-rigid indecomposable modules of the corresponding Jacobian algebra by applying the so-called Caldero-Chapoton(CC) map. The connections between the two areas has lead to some important results, in particular the development of $\tau$-tilting theory by Adachi-Iyama-Reiten, which uncovers cluster algebra like structure in the module category of every finite dimensional algebra.
In this more general setting, given a finite dimensional algebra we can apply the CC map to $\tau$-rigid modules and consider the algebra generated by all these images. In the case of Jacobian algebras, this yields a cluster algebra, so it is natural to ask what properties of the cluster algebra pass to this more general setting. For example, we want to explore an analog of the exchange relations or a mutation formula, and find a basis for these algebras.