Braid group actions via categorified  Heisenberg complexes

Cautis, Sabin; Licata, Anthony; Sussan, Joshua

doi:10.1112/s0010437x13007367

Cited by 10 publications

(10 citation statements)

References 23 publications

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“…However, an analogue of [CLS,Prop. 4.7], can be used to show that there is a unique way (up to homotopy) to choose these multiples so that (18) becomes a complex.…”

Section: Aside: Complexes and Projectorsmentioning

confidence: 99%

On a Categorical Boson–Fermion Correspondence

Cautis

Sussan

2015

Commun. Math. Phys.

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“…However, an analogue of [CLS,Prop. 4.7], can be used to show that there is a unique way (up to homotopy) to choose these multiples so that (18) becomes a complex.…”

Section: Aside: Complexes and Projectorsmentioning

confidence: 99%

On a Categorical Boson–Fermion Correspondence

Cautis

Sussan

2015

Commun. Math. Phys.

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“…More recently, autoequivalences of the (bounded) derived categories D b (X [n] ) of the Hilbert schemes were intensively studied; see [Plo07], [Add11], [PS12], [Mea12], [Kru13], [CLS14], [KS14]. In particular, Addington [Add11] defined the notion of a P n -functor as a Fourier-Mukai transform F : D b (M ) → D b (N ) between derived categories of varieties with rightadjoint F R : D b (N ) → D b (M ) and the main property that In [Kru13] it was shown that for every smooth surface X and n ≥ 2 there is a P n−1 -functor H 0,n := H : D b (X) → D b…”

Section: Introductionmentioning

confidence: 99%

$\mathbb{P}$-functor versions of the Nakajima operators

Krug¹

2019

Alg. Geom.

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For a smooth quasi-projective surface X we construct a series of P n−1 -functors H ℓ,n : D b (X × X [ℓ] ) → D b (X [n+ℓ] ) for n > ℓ and n > 1 using the derived McKay correspondence. They can be considered as analogues of the Nakajima operators. The functors also restrict to P n−1 -functors on the generalised Kummer varieties. We also study the induced autoequivalences and obtain, for example, a universal braid relation in the groups of derived autoequivalences of Hilbert squares of K3 surfaces and Kummer fourfolds.given by the Fourier-Mukai transform along the ideal sheaf of the universal family Ξ ⊂ X ×X [n] for X a K3 surface and n ≥ 2. Similarly, for X = A an abelian surface andΞ ⊂ A×K n A the universal family of the generalised Kummer variety K n A ⊂ A [n+1] the functorF n = FM IΞ : D b (A) → D b (K n A) is a P n−1 -functor for n ≥ 2 and a P 1 -functor for n = 1; see [Mea12] and [KM14]. We will refer to these examples of P-functors as the universal ideal functors.An important tool for the investigation of the derived categories of the Hilbert schemes of points on surfaces is the Bridgeland-King-Reid-Haiman equivalence (also known as derived McKay correspondence; see [BKR01] and [Hai01])Sn (X n ) . This provides an identification of D b (X [n] ) with the derived category of equivariant coherent sheaves on the product X n , i.e. of coherent sheaves equipped with a linearisation by the symmetric group S n .

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“…(b) We expect that the categorical braid group actions of [CLS14] can be generalized to the setting of the current paper.…”

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confidence: 99%

A general approach to Heisenberg categorification via wreath product algebras

Rosso

Savage

2016

Math. Z.

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We associate a monoidal category H B , defined in terms of planar diagrams, to any graded Frobenius superalgebra B. This category acts naturally on modules over the wreath product algebras associated to B. To B we also associate a (quantum) lattice Heisenberg algebra h B . We show that, provided B is not concentrated in degree zero, the Grothendieck group of H B is isomorphic, as an algebra, to h B . For specific choices of Frobenius algebra B, we recover existing results, including those of Khovanov and Cautis-Licata. We also prove that certain morphism spaces in the category H B contain generalizations of the degenerate affine Hecke algebra. Specializing B, this proves an open conjecture of Cautis-Licata.

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Braid group actions via categorified Heisenberg complexes

Cited by 10 publications

References 23 publications

On a Categorical Boson–Fermion Correspondence

On a Categorical Boson–Fermion Correspondence

$\mathbb{P}$-functor versions of the Nakajima operators

A general approach to Heisenberg categorification via wreath product algebras

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