On the Vertex Operators Representation of Lie Algebras of Matrices

Abstract: Relying on our common skill to multiply square matrices by vectors, recalled in the first part of the talk, we will describe the exterior algebra of a polynomial ring in one indeterminate, and/or the ring of symmetric polynomials, as a representation of the Lie algebra of matrices of infinite size with all but finitely many zero entries.

The description is achieved by bridging classical Schubert calculus on Grassmannians to the vertex operators occurring in the so-called boson-fermion correspondence (Poincaré duality for Grassmannians of infinite dimensional linear spaces) and highlights a substantial generalization of a classical picture drawn in the Eighties by Date, Jimbo, Kashiwara and Miwa, within the framework of algebraic analysis and infinite dimensional completely integrable systems. The talk will survey joint work with (in alphabetical order) O. Behzad, A. Contiero, D. Martins, P. Salehyan, I. Scherbak and R. Vidal Martins.

algebraic geometry

Audience: researchers in the topic

Brazilian algebraic geometry seminar

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