Minuscule Schubert Calculus and the Geometric Satake Correspondence

Anderson, Dave; Nigro, Antonio

doi:10.1007/978-981-15-7451-1_6

Cited by 4 publications

(4 citation statements)

References 35 publications

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“…To express the gl( ⋀ k V)-action (1) on B r through a compact formula, a standard philosophy suggests to use generating functions.…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

“…The vertex operators occurring in our description of B r as a representation of gl( ⋀ k V) are defined by means of Schubert derivations, which are distinguished Hasse-Schmidt (HS) derivations on exterior algebras. HS derivations were first introduced in [13] and extensively treated in [18]; see also the survey [1] or [6, p. 116], for more discussions. In a finite dimensional context Schubert derivations are related to Chern and Segre polynomials of the tautological bundle over a Grassmannian.…”

Section: Methods and Their Applicationsmentioning

confidence: 99%

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Polynomial ring representations of endomorphisms of exterior powers

et al. 2021

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An explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

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“…To express the gl( ⋀ k V)-action (1) on B r through a compact formula, a standard philosophy suggests to use generating functions.…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

Section: Methods and Their Applicationsmentioning

confidence: 99%

Polynomial ring representations of endomorphisms of exterior powers

et al. 2021

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“…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].…”

Section: The Universal Decomposition Algebramentioning

confidence: 99%

Universal Decomposition Algebras Represent Endomorphisms

Behzad,

Nejad

2020

Preprint

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The goal of this paper is to supply an explicit description of the universal decomposition algebra of the generic polynomial of degree n into the product of two monic polynomials, one of degree r, as a representation of Lie algebras of n × n matrices with polynomial entries. This is related with the bosonic vertex representation of the Lie algebra gl ∞ due to Date, Jimbo, Kashiwara and Miwa. 0 2020 MSC: 14M15, 14N15, 15A75.

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“…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.…”

mentioning

confidence: 99%

Polynomial ring representations of endomorphisms of exterior powers

Behzad¹,

Contiero²,

Gatto³

et al. 2020

Preprint

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A polynomial ring with rational coefficients is an irreducible representation of Lie algebras of endomorphisms of exterior powers of a infinite countable dimensional Qvector space. We give an explicit description of it, using suitable vertex operators on exterior algebras, which mimick those occurring in the bosonic vertex representation of the Lie algebra gl ∞ , due to Date-Jimbo-Kashiwara and Miwa (DJKM).

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Minuscule Schubert Calculus and the Geometric Satake Correspondence

Cited by 4 publications

References 35 publications

Polynomial ring representations of endomorphisms of exterior powers

Polynomial ring representations of endomorphisms of exterior powers

Universal Decomposition Algebras Represent Endomorphisms

Polynomial ring representations of endomorphisms of exterior powers

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