Discrete Mathematics Seminar
Date and time: Monday, 23rd Jan 2023 at 2.30 pm
Venue: Ramanujan Hall
Host:Sudhir R Ghorpade
Speaker: Rakhi Pratihar
Affiliation: IIIT Delhi
Title: Matroids, Euler characteristics, Möbius functions, and q-analogs
Abstract: For a co-loopless matroid M of rank r, the reduced Euler characteristic of the corresponding matroid complex S_M is determined by a Mobius function via the relation χ(S_M) = (−1)^{r−1} |μ_{L_M} (\hat{0}, \hat{1})|, where L_M is the lattice of cycles of M. The relation can be seen as a link between the poset of independent sets of M, and the geometric lattice of flats of the dual matroid M^*, which has a very interesting application to coding theory. It has been shown that the generalized Hamming weights of a linear code can be determined by the Betti numbers of the Stanley-Reisner ring of an associated matroid. In this talk, I will present a q-analogue of this relation where one consider the Euler characteristic of the order complex associated to a q-matroid. I will also briefly discuss its potential application to the theory of rank metric codes.