The cohomology of the Grassmannian is a <i>gl<sub>n</sub></i>-module

Gatto, Letterio; Salehyan, Parham

doi:10.1080/00927872.2019.1640240

Cited by 11 publications

(12 citation statements)

References 17 publications

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“…ily imply that This enables to state a second version of 8.5, which works well for practical purposes and generalises[20, Main Theorem 4.3].…”

mentioning

confidence: 56%

“…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.…”

Section: Methods and Their Applicationsmentioning

confidence: 96%

“…If k = 1 , and z = z 1 , one obtains which is precisely [19,Proposition 5.4] or [20,Proposition 3.2]. Their shape there looks more involved because of a different notation.…”

Section: Theoremmentioning

confidence: 98%

“…We have so gotten a description of the gl( ⋀ k V)-module structure of B r , which generalises the case r < ∞ and k = 1 studied in [20]. The output is that the direct sum ⨁ k≥0 gl( ⋀ k V) is a Lie subalgebra of gl( ⋀ V) , represented by B r for all r ≥ 0 .…”

Section: The Boson-fermion Correspondence and The Djkm Representationmentioning

confidence: 99%

“…Certainly this idea is already present in the infinite wedge power context (e.g. [25,Chapter 5]), but the present article, together with [19][20][21], is the first instance of applications of the techniques and ideas in finite dimensional landscapes.…”

Section: Organization Of the Papermentioning

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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“…ily imply that This enables to state a second version of 8.5, which works well for practical purposes and generalises[20, Main Theorem 4.3].…”

mentioning

confidence: 56%

Section: Methods and Their Applicationsmentioning

confidence: 96%

“…If k = 1 , and z = z 1 , one obtains which is precisely [19,Proposition 5.4] or [20,Proposition 3.2]. Their shape there looks more involved because of a different notation.…”

Section: Theoremmentioning

confidence: 98%

Section: The Boson-fermion Correspondence and The Djkm Representationmentioning

confidence: 99%

Section: Organization Of the Papermentioning

confidence: 99%

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Polynomial ring representations of endomorphisms of exterior powers

et al. 2021

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Automorphism Groups of ind-Varieties of Generalized Flags

Ignatyev

Penkov

2022

Transformation Groups

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We compute the group of automorphisms of an arbitrary ind-variety of (possibly isotropic) generalized flags. Such an ind-variety is a homogeneous ind-space for one of the ind-groups $SL(\infty )$ S L ( ∞ ) , $O(\infty )$ O ( ∞ ) or $Sp(\infty )$ S p ( ∞ ) . We show that the respective automorphism groups are much larger than $SL(\infty )$ S L ( ∞ ) , $O(\infty )$ O ( ∞ ) or $Sp(\infty )$ S p ( ∞ ) , and present the answer in terms of Mackey groups. The latter are groups of automorphisms of non-degenerate pairings of (in general infinite-dimensional) vector spaces. An explicit matrix form of the automorphism group of an arbitrary ind-variety of generalized flags is also given. The case of the Sato grassmannian is considered in detail, and its automorphism group is the projectivization of the connected component of unity in the group known as Japanese $GL(\infty )$ G L ( ∞ ) .

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Schubert Derivations on the Infinite Wedge Power

Gatto

Salehyan

2020

Bull Braz Math Soc, New Series

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The Schubert derivation is a distinguished Hasse-Schmidt derivation on the exterior algebra of a free abelian group, encoding the formalism of Schubert calculus for all Grassmannians at once. The purpose of this paper is to extend the Schubert derivation to the infinite exterior power of a free Z-module of infinite rank (fermionic Fock space). Classical vertex operators naturally arise from the integration by parts formula, that also recovers the generating function occurring in the bosonic vertex representation of the Lie algebra gl ∞ (Z), due to Date, Jimbo, Kashiwara and Miwa (DJKM). In the present framework, the DJKM result will be interpreted as a limit case of the following general observation: the singular cohomology of the complex Grassmannian G(r, n) is an irreducible representation of the Lie algebra of n × n square matrices.

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The cohomology of the Grassmannian is a gl_n-module

Cited by 11 publications

References 17 publications

Polynomial ring representations of endomorphisms of exterior powers

Polynomial ring representations of endomorphisms of exterior powers

Automorphism Groups of ind-Varieties of Generalized Flags

Schubert Derivations on the Infinite Wedge Power

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The cohomology of the Grassmannian is a gln-module

Cited by 11 publications

References 17 publications

Polynomial ring representations of endomorphisms of exterior powers

Polynomial ring representations of endomorphisms of exterior powers

Automorphism Groups of ind-Varieties of Generalized Flags

Schubert Derivations on the Infinite Wedge Power

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The cohomology of the Grassmannian is a gl_n-module