On the Betti numbers of oriented Grassmannians and independent semi-invariants of binary forms

Year, pages: 2017, 1263 - 1268

We present a complete functional formula expressing the $i$th $\mathbb{Z}₂$-Betti number of the oriented Grassmann manifold of oriented $3$-dimensional vector subspaces in Euclidean $n$-space for $i$ from the range determined by the characteristic rank of the canonical oriented $3$-dimensional vector bundle over this manifold. The same formula explicitly exhibits the number of linearly independent semi-invariants of degree $3$ of a binary form of degree $n-3$. Using the approach and data presented in this note, analogous results can be obtained for the oriented Grassmann manifold of oriented $4$-dimensional vector subspaces in Euclidean $n$-space and semi-invariants of degree $4$ of a binary form of degree $n-4$.

Korbaš, J. 2017. On the Betti numbers of oriented Grassmannians and independent semi-invariants of binary forms. In Mathematica Slovaca, vol. 67, no.5, pp. 1263-1268. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0047

Korbaš, J. (2017). On the Betti numbers of oriented Grassmannians and independent semi-invariants of binary forms. Mathematica Slovaca, 67(5), 1263-1268. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0047