Matroids combinatorially abstract independence properties from finite dimensional linear algebra and play a central role in modern combinatorics, yielding connections between graph theory, algebra, polyhedral geometry, optimization, and beyond. The subscheme of the Grassmannian Gr(k,n) consisting of vector spaces that realize a fixed matroid M is the realization space of M, and Gr(k,n) can be decomposed into such realization spaces. This stratification can be given equivalently by identifying each stratum with the multi-intersection of all possible Schubert cells with respect to a fixed basis. Quiver Grassmannians are projective varieties parametrizing subrepresentations of quiver representations, and every projective variety arises as the quiver Grassmannian of a wild quiver. In this talk, we introduce matroid stratifications of quiver Grassmannians. We consider classes of quiver Grassmannians that admit Schubert decompositions and investigate the relationship between such cells and the matroid strata.
An important theme of research on cluster algebras, indeed one of the original motivations for their introduction, is the study of bases for them satisfying
a certain pointedness condition. Indeed, several constructions have been proposed leveraging a diverse set of tools (mirror symmetry, theichmüller
theory, combinatorics, representation theory, etc.).
More recently a unified approach has been proposed by Qin who showed that any two pointed bases of a given cluster algebra are related by a unitriangular
transformation. In his construction he provided, in an abstract way, a description of the moduli space of pointed bases using the so called dominance
order.
In this seminar will explain how, In collaboration with N. Reading and D. Rupel, we recasted this order as a geometric region to describe explicitly the
moduli space of pointed bases in all cluster algebras with at most linear growth.
The study of trace identities, closely related to the invariant theory of $n \times n$ matrices, plays a significant role in contemporary mathematics. Foundational contributions to this area were made by Procesi and Razmyslov.
The aim of this talk is to explain how trace identities for matrix algebras are governed by the representation theory of the symmetric group, emphasizing the constructions arising from the combinatorics of Young tableaux.
In the final part, we present recent results on trace identities for upper triangular matrices equipped with multiple trace functions.
This presentation is partially based on a joint work with Antonio Ioppolo.
Let B_n be the braid group with n-strands and Z(B_n) its center. The (integral) homology of B_n was computed in the seventies by F. Cohen. In this talk we will see how to compute the homology of H_*(B_n/Z(B_n); F_p) for any n natural number and p prime. The approach will be topological, since the classifying space of B_n/Z(B_n) can be realized as the (homotopy) quotient C_n(R^2)//S^1, where C_n(R^2) is the unordered configuration space of point in the plane. Combining the results of F. Cohen with techniques from equivariant cohomology we can do the computation. This talk is based on https://arxiv.org/abs/2404.10639.
Let G<GL(V) be a finite group generated by reflections. The hyperplane arrangement given by the set of fixed spaces of the reflections in G has
been an object of interest from different perspectives. For instance, for the symmetric group G=S_n one gets the Braid arrangement, whose complement space M in V is known to be a classifying space for the Pure Braid group. The cohomology H*(M) is well known (also as a representation), and the cohomology of the Braid group is given by the invariants H*(M)^G. The latter object has been studied beyond real and complex reflection groups: Poincaré series of the invariants were computed with topological methods by Brieskorn for finite Coxeter groups, while for finite, complex reflection groups this is due to Douglass, Pfeiffer and Röhrle. In this talk we briefly present analogous results for finite quaternionic reflection groups, which extend thecomplex case and were classified by Cohen in 1980. This is joint work with Röhrle and Schmitt.
The Bruhat order on permutations is a central object in algebraic combinatorics, but it also raises basic questions from the point of view of poset theory. In this talk, I will focus on poset-theoretic properties of the Bruhat order and related orders, including lattice structure and completions. In particular, the lattice of alternating sign matrices arises as the minimal lattice extension of the Bruhat order on permutations, namely its Dedekind–MacNeille completion. In the second part of the talk, I will discuss extensions of these ideas to higher-dimensional analogues, where permutations are replaced by Latin squares and a Bruhat order is proposed for these objects. I will describe current work on the resulting posets and related lattices; this is ongoing joint work with Cian O'Brien (MIC Limerick).
We study the Chow ring of De Concini–Procesi wonderful models of types A_n and B_n with respect to the minimal building set. In the former case, the model coincides with the Deligne–Mumford compactification of M_0,n . Our focus is on the exponential generating function of the Poincaré polynomials, for which we provide several explicit formulas. These results recover the classical formulas of Keel, Getzler, Manin, and Yuzvinsky, and yield new ones in type B. This is joint work with L. Ferroni and L. Vecchi.