A diagrammatic categorification of the $q$-Schur algebra

Vaz, Pedro

doi:10.4171/qt/34

Cited by 35 publications

(106 citation statements)

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“…Specifically, we quotient the 2-morphism spaces by the identity 2-morphisms of the identity 1-morphisms of such weights. (This type of truncation has appeared before in the categorification literature, for example, in [MSV13,QR16].) The resulting 2-morphism spaces of U tr are nonnegatively graded, and we show that the degree zero part U 0 of U tr is equivalent to the 2-category A (Theorem 4.4).…”

mentioning

confidence: 62%

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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confidence: 62%

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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“…By Lemma 4.2 we know that End S(n,n) (E i ) is the surjective image of the analogous endomorphism ring in U ( sl n ), for any signed sequence i. By Khovanov and Lauda's Theorem 1.1 [KL10] and some general arguments which were explained in detail in [MSV13], and also used in [MT13], this implies that the Grothendieck classes of all direct summands of E i in Kar S(n, n) are contained in the image of γ n . This concludes the proof that γ n is surjective.…”

Section: The Grothendieck Groupmentioning

confidence: 95%

A diagrammatic categorification of the affineq-Schur algebra S(n,n) forn≥ 3

Mackaay

Thiel

2015

Pacific J. Math.

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“…Lusztig's modified quantum group is a key ingredient to the categorification of the quantum group associated to sl(n) (for example, as explained in [14]). Also see [18] and references therein for a discussion of categorifications of the q-Schur algebras. The categorification of quantum supergroups is currently an open problem and a super analogue of Lusztig's modified quantum group may be useful.…”

Section: Future Directionsmentioning

confidence: 99%