Geometric realizations of the basic representation of $$\widehat{\mathfrak {gl}}_r$$ gl ^ r

Lemay, Joel

doi:10.1007/s00029-015-0200-3

Cited by 1 publication

(1 citation statement)

References 21 publications

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“…Similarly, the geometry of quiver varieties [Nak98] can be used to build categorifications of quantum group representations (see for example [VV11,Zhe14,CKL13,Web]). It is thus natural to expect that the results of the current paper are related to geometric constructions relating these spaces, such as [Sav06,Nag09,LS10,Lem16].…”

Section: Action Of a The Compositionmentioning

confidence: 99%

An equivalence between truncations of categorified quantum groups and Heisenberg categories

Queffelec¹,

Savage²,

Yacobi³

2018

Journal De L’École Polytechnique — Mathématiques

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We introduce a simple diagrammatic 2-category A that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of sl∞. We show that A is equivalent to a truncation of the Khovanov-Lauda categorified quantum group U of type A∞, and also to a truncation of Khovanov's Heisenberg 2-category H . This equivalence is a categorification of the principal realization of the basic representation of sl∞.As a result of the categorical equivalences described above, certain actions of H induce actions of U , and vice versa. In particular, we obtain an explicit action of U on representations of symmetric groups. We also explicitly compute the Grothendieck group of the truncation of H .The 2-category A can be viewed as a graphical calculus describing the functors of i-induction and i-restriction for symmetric groups, together with the natural transformations between their compositions. The resulting computational tool is used to give simple diagrammatic proofs of (apparently new) representation theoretic identities. Contents 1. Introduction 2.1. Bosonic Fock space and the category A 2.2. The basic representation 2.3. A Kac-Moody presentation of A 2.4. Notation and conventions for 2-categories 3. Modules for symmetric groups 3.1. Module categories 3.2. Decategorification 3.3. Biadjunction and the fundamental bimodule decomposition 3.4. The Jucys-Murphy elements and their eigenspaces 3.5. Combinatorial formulas 4. The 2-category A 4.1. Definition 4.2. Truncated categorified quantum groups 4.3. 1-morphism spaces 4.4. 2-morphism spaces 4.5. Decategorification 5. The 2-category H tr 5.1. Definition 5.2. A decomposition H tr = H ǫ ⊕ H δ 5.3. Region shifting

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