Hasse-Schmidt Derivations on Grassmann Algebras

Gatto, Letterio; Salehyan, Parham

doi:10.1007/978-3-319-31842-4

Cited by 21 publications

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“…Let V ∶= ⨁ j∈ℤ ℚ ⋅ b j be a vector space with basis ∶= (b j ) j∈ℤ , parameterized by the integers (one may think of V as being the vector space ℚ[X −1 , X] of the Laurent polynomials) and V * its restricted dual with basis ( j ) j∈ℤ . It is well known that V ⊕ V * supports a canonical structure of Clifford algebra C ∶= C(V ⊕ V * ) ([9, p. 85] or [18]) and that the Fermionic Fock space F (also called the semi-infinite wedge power and denoted by ⋀ ∞∕2 V ) is an irreducible representation of C .…”

Section: The Boson-fermion Correspondence and The Djkm Representationmentioning

confidence: 99%

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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Section: The Boson-fermion Correspondence and The Djkm Representationmentioning

confidence: 99%

Polynomial ring representations of endomorphisms of exterior powers

et al. 2021

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“…from which the desired expression of Γ * r (w)∆ λ (H r ). 4 The gl ∞ (Z) structure of B r . First description…”

Section: Schubert Derivations On Mmentioning

confidence: 99%

The cohomology of the Grassmannian is a gl_n-module

Gatto

Salehyan

2019

Communications in Algebra

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The integral singular cohomology ring of the Grassmann variety parametrizing r-dimensional subspaces in the n-dimensional complex vector space is naturally an irreducible representation of the Lie algebra of all the n × n matrices with integral entries. Using the notion of Schubert derivation, a distinguished Hasse-Schmidt derivation on an exterior algebra, we describe explicitly such a representation, indicating its relationship with the celebrated bosonic vertex representation of the Lie algebra of infinite matrices due to Date, Jimbo, Kashiwara and Miwa.

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“…Details are in Section 5, see also [5]. Take the polynomial ring of countably many indeterminates, B = Q[x 1 , x 2 , .…”

Section: Finite-dimensional Approximations Of Bosonic Vertex Operatorsmentioning

confidence: 99%

“…In a number of papers motivated by Schubert Calculus [3,6] (see also the book [5]), one of us proposed to study HS-derivations for exterior algebras.…”

Section: Introductionmentioning

confidence: 99%

Hasse–Schmidt derivations and Cayley–Hamilton theorem for exterior algebras

Gatto¹,

Scherbak²

2019

Functional Analysis and Geometry

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In memory of those participants of the Voronezh Winter Mathematical School who have already passed into another world where all problems are solved Abstract. Using the natural notion of Hasse-Schmidt derivations on an exterior algebra, we relate two classical and seemingly unrelated subjects. The first is the famous Cayley-Hamilton theorem of linear algebra, "each endomorphism of a finite-dimensional vector space is a root of its own characteristic polynomial", and the second concerns the expression of the bosonic vertex operators occurring in the representation theory of the (infinite-dimensional) Heinsenberg algebra.2010 Mathematics Subject Classification. Primary 15A75, 17B69. 050E5.

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Hasse-Schmidt Derivations on Grassmann Algebras

Cited by 21 publications

References 0 publications

Polynomial ring representations of endomorphisms of exterior powers

Polynomial ring representations of endomorphisms of exterior powers

The cohomology of the Grassmannian is a gl_n-module

Hasse–Schmidt derivations and Cayley–Hamilton theorem for exterior algebras

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Hasse-Schmidt Derivations on Grassmann Algebras

Cited by 21 publications

References 0 publications

Polynomial ring representations of endomorphisms of exterior powers

Polynomial ring representations of endomorphisms of exterior powers

The cohomology of the Grassmannian is a gln-module

Hasse–Schmidt derivations and Cayley–Hamilton theorem for exterior algebras

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The cohomology of the Grassmannian is a gl_n-module