Consider a finite group acting linearly on a vector space. Each group element can be represented as a linear map whose graph is a subspace and the collection of these graphs gives a subspace arrangement. From an algebraic point of view, the vanishing ideal of the union of the subspaces can be used to find generators for the ring of polynomial invariants under the action of the group. Geometrically, we can also study the intersection lattice of the subspace arrangement, which can give cohomological information about the same ideal.
When one considers permutation actions, new combinatorial perspectives on this intersection lattice arise. In particular, for the regular action of the group on itself, the intersection lattice of the subspace arrangement is isomorphic to the coset poset, a known object of study in geometric and topological combinatorics. For other permutation actions, we establish that stabiliser subgroups of orbit partitions and their cosets can be used to label the vertices of the lattice. We then study these lattices with combinatorial techniques and computational tools to better understand the various algebraic structures associated with them.
Algebra and Combinatorics Seminar
We meet weekly on Wednesday at 3:00 PM (Central time), GI-126(A)
Organizers: Alessandra Constantini and Kalina Mincheva.
Fall 2026
September 9-
Francesca Gandini, St. Olaf College.
Combinatorial aspects of lattices associated to subspace arrangements arising from group actions. .
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Ngoc Trinh Le, Tulane University.
Relations between higher level Hurwitz class numbers. .We connect generalizations of the classical Hurwitz class numbers coming from two different frameworks: one introduced by Pei and Wang, arising from the generalized Cohen--Eisenstein series, and another by Li, Skoruppa, and Zhou, arising from Eichler orders of quaternion algebras. As applications, we obtain new basis for Eisenstein space $E_{3/2}^{+}(4N,\mathrm{id})$, a generalization of recent results of Beckwith and Mono, and a generalization of Gauss' formula. This is joint work with Olivia Beckwith.
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Victoria Schleis, Durham University, UK.
Representation theory over bands. .This talk presents an overview of ongoing research into representation theory over bands, which are combinatorial generalisations of fields, with applications in matroid theory and tropical geometry. A fundamental departure from the classical theory arises from the observation that many natural general linear objects in this setting are not invertible. To construct a general linear monoid, we study the multiplication of bimatroids and matrices over bands, investigate their interrelations, and construct their moduli spaces as combinatorial analogues of the determinantal variety. These constructions provide the foundation for defining band analogues of classical matrix groups and their representations. As a proof of concept, we conclude with a study of Schur functors in this new setting.
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Aniketh Sivakumar, Tulane University.
Interpolation Problems and the Asymptotic Growth of Minimum Degrees. .Interpolation problems study the hypersurfaces in projective space that pass through a given number of points under certain constraints. Among the main questions are determining the minimum degree of such a hypersurface and understanding the number of independent conditions imposed by the points on hypersurfaces of a fixed degree. In this talk, I will begin with the classical formulation of these interpolation problems and introduce some of the tools used to study them. We will then explore the setting where these points are assigned multiplicities and how this is encoded algebraically. This naturally leads us to the Demailly and Chudnovsky conjectures, which describe how the minimum degree grows asymptotically as the multiplicities grow. Finally, we will discuss the recent proofs of these conjectures and their connection to containment problems in commutative algebra.
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Alexander Heaton, Lawrence University.
TBA. .TBA.
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Alexandra Seceleanu, University Nebraska - Lincoln.
TBA. .TBA.
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Matthew Ortiz, University of North Texas.
TBA. .TBA.
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Lisa Seccia, University of Neuchâtel.
TBA. .TBA.
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William Newman, Ohio State University.
TBA. .TBA.
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Griffin Edwards, Georgia Tech.
TBA. .TBA.
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TBA.
TBA. .TBA.
Past seminars.