Tango bundles on Grassmannians

Abstract

The goal of this paper is to prove the existence of indecomposable rank ((k+1)(n−k)−(k+1)) vector bundles on the Grassmannian variety Gr(k,n). We will call them Tango bundles since in the particular case of double-struckPn≅Gr(0,n) they correspond to the celebrated vector bundle discovered by H. Tango in 1974. We will give a geometrical description of Tango bundles and we will prove that they are μ‐stable.

Polynomial ring representations of endomorphisms of exterior powers

“…Hasse-Schmidt derivations on exterior algebras have shown their versatility in applications to improve effectiveness in Schubert Calculus computations (see [4,5]), to equivariant cohomology of Grassmannians (Cf. [22], but also [26]), to generalise the Cayley-Hamilton theorem [16,23], with perspective applications to globalise the local Wronskian as in [15,Section 4.2], or, inspired by [14,17], like in [2,20,21] and in the present paper, to revisit the bosonic vertex representation of Lie algebras of endomorphisms as in [8] (see also [24] and [25,), providing new methods and new insight.…”

Universal Decomposition Algebras Represent Endomorphisms

“…This section is quite necessary because in spite there are very precise reference to Hasse-Schmidt derivations of exterior algebras (see also [13] in addition to the previously cited ones), most of the terminology is not standard yet, although has already been mentioned by other authors like e.g. in [1,2] or, more recently, in [3, p. 116]. In this section we also recall the theory of Laksov and Thorup together with their definition of residue of an r-tuple of Laurent polynomials, as in [19].…”

Polynomial ring representations of endomorphisms of exterior powers

“…The latter are distinguished Hasse-Schmidt derivations on exterior algebras, introduced in [12] and extensively treated in [14]; see also the survey [1] or [5, p. 116], for more related discussions. They have shown their versatility in applications to improve effectiveness in Schubert Calculus computations (see [3,4]), to equivariant cohomology of Grassmannians (Cf. [18], but also [22,23]), to generalise the Cayley-Hamilton theorem [13,19] or, like in [16,17] and in the present paper, to revisit the bosonic vertex representation of Lie algebras of endomorphisms as in [7] (see also [20] and [21, Propositions 5.2-5.3]), providing new methods and new insight.…”