Together with Marco Timpanella (University of Perugia) and Giovanni Zini (University of Modena and Reggio Emilia), the research group organize monthly seminars.
12 March 2026 [4.00 pm - Italian time] - Rakhi Pratihar (IRMAR - Université de Rennes)
Title: Homological Invariants of q-Matroids
Abstract: The matroid complex, the simplicial complex of independent sets of a finite matroid, carries many significant invariants of the matroid by associating topological (T0-Alexandroff space) and algebraic structure (Stanley–Reisner ring) to the matroid. A central theme is to understand the homological invariants, in particular, the simplicial homology groups, Cohen–Macaulayness, and the graded Betti numbers. Foundational work of Reisner (1978) and Stanley (1976) explains how Cohen–Macaulay properties of Stanley–Reisner rings are obtained by the homology of links, while Hochster gives a direct formula relating graded Betti numbers to simplicial homology. On the topological side, Provan (1977) proved that matroid complexes are shellable, which implies that their reduced homology to be concentrated in top dimension. The q-Matroids, which are a natural q-analogue of matroids, obtained by replacing subsets by vector subspaces, and in the representable case, they correspond to rank metric codes. In this talk, I will present q-analogues of the results above, based on the research works [1, 2, 3, 4] that initiate a homological framework for q-matroids. If time permits, I will discuss an application of these results to determining important discrete invariants of linear errorcorrecting codes, namely the higher weight spectra, in both Hamming and rank metrics.
1] T. Johnsen, R. Pratihar, and H. Verdure, Weight spectra of Gabidulin rank-metric codes and Betti numbers, Sao Paulo Journal of Mathematical Sciences, 17(1), 208-241.
[2] S. R. Ghorpade, R. Pratihar, and T. H. Randrianarisoa, Shellability and homology of q-complexes and q-matroids, Journal of Algebraic Combinatorics, 56(4), 1135-1162.
[3] T. Johnsen, R. Pratihar, and T. H. Randrianarisoa, The Euler Characteristic, q-Matroids, and a Mobius Function Journal of Algebra and its Applications, 2025.
[4] S. R. Ghorpade, R. Pratihar, T. H. Randrianarisoa, H. Verdure, and G. Wilson, Homotopy type of shellable q-complexes and their homology groups, arXiv:2403.07102.