Hasse-Schmidt derivations tools for algebra and algebraic geometry

• Grant number: 16/03161-3
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Start date: August 01, 2016
• End date: February 28, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Parham Salehyan
• Grantee: Parham Salehyan
• Visiting researcher: Letterio Gatto
• Visiting researcher institution: Politecnico di Torino, Italy
• Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil

Abstract

The purpose of this research project is to develop further the potential for applications to other branches of mathematics of the powerful notion of Hasse-Schmidt derivations (or higher derivations) on a Grassmann Algebras. It has served, from the time being, as a tool to put in a common picture frame many seemingly unrelated subject, such as, e.g., Schubert Calculus and the boson-fermion correspondence arising in the representation theory of the Heisenberg algebra. We want to recover more results regarding the Vertex operators, also in more general cases, and connect our previous results with the Heisenberg algebra arising in studying Hilbert schemes of points exploiting the ADHM description of framed vector bundles.We want to explore also the sheafification of our theory, for instance considering fields of higher derivations on the exterior algebra of the tangent bundle of a smooth algebraic variety. Connections with Riemann Surfaces ofinfinite genus will be explored as well, as the partitions parametrizing generators of the fermionic Fock spaces can be interpreted as gap partitions of Weierstrass points on curves of infinite genus described by means of their Wronskians. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

GATTO, LETTERIO; SALEHYAN, PARHAM. The cohomology of the Grassmannian is a gl(n)-module. COMMUNICATIONS IN ALGEBRA, v. 48, n. 1, p. 17-pg., 2019-08-22. (16/03161-3)

GATTO, LETTERIO; SALEHYAN, PARHAM. The cohomology of the Grassmannian is a gl(n)-module. COMMUNICATIONS IN ALGEBRA, AUG 2019. (16/03161-3)

GATTO, LETTERIO; SALEHYAN, PARHAM. Schubert Derivations on the Infinite Wedge Power. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. 52, n. 1, JAN 2020. (16/03161-3)