A diagram algebra for Soergel modules corresponding to smooth Schubert varieties

Sartori, Antonio

doi:10.48550/arxiv.1311.6968

Cited by 2 publications

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“…Sartori gave an alternative categorification of V ⊗n using q-presentable quotients of parabolic category O [Sar16b,Sar16a]; this categorification can be described using categories of modules over certain properly stratified algebras. In [LM21, Theorem 7.6] it was shown that the algebras B l (n, k) are graded flat deformations of Sartori's algebras A n,k over the ring of symmetric functions in k variables.…”

Section: Finite Global Dimensionmentioning

confidence: 99%

“…In [LM21, Theorem 7.6] it was shown that the algebras B l (n, k) are graded flat deformations of Sartori's algebras A n,k over the ring of symmetric functions in k variables. Inflations of Sartori's proper standard modules from [Sar16b] along the quotient map were identified with the dual of the standard basis of V ⊗n with respect to the bilinear form on K C(q) 0 (B l (n, k)); inflations of Sartori's standard and indecomposable projective modules gave slightly less natural elements of V ⊗n . In particular, while we would like to categorify at least the canonical and standard bases of V ⊗n and their duals using B l (n, k), only three of these bases were categorified in [LM21]; a categorification of the (most natural) standard basis was missing, unlike in Sartori's picture where his algebras admit a different bilinear form, see Remark 4.12 .…”

Section: Finite Global Dimensionmentioning

confidence: 99%

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Strands algebras and the affine highest weight property for equivariant hypertoric categories

Lauda¹,

Licata²,

Manion³

2021

Preprint

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We show that the equivariant hypertoric convolution algebras introduced by Braden-Licata-Proudfoot-Webster are affine quasi hereditary in the sense of Kleshchev and compute the Ext groups between standard modules. Together with the main result of [LLM20], this implies a number of new homological results about the bordered Floer algebras of Ozsváth-Szabó, including the existence of standard modules over these algebras. We prove that the Ext groups between standard modules are isomorphic to the homology of a variant of the Lipshitz-Ozsváth-Thurston bordered strands dg algebras.

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Section: Finite Global Dimensionmentioning

confidence: 99%

Section: Finite Global Dimensionmentioning

confidence: 99%

Strands algebras and the affine highest weight property for equivariant hypertoric categories

Lauda¹,

Licata²,

Manion³

2021

Preprint

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“…Proposition 3.3. We collect the relevant results from [Sar16b] on Soergel modules and the hom spaces between them.…”

Section: It Is Possible To Enumerate the Monomial Basis By Permutatio...mentioning

confidence: 99%

“…. , 1) respectively [Sar16b]. The structure of these subquotients is accessed through a combinatorial relationship with Soergel modules corresponding to smooth Schubert varieties.…”

Section: Introductionmentioning

confidence: 99%

“…Modules over Ozsváth-Szabó's algebras. One important feature of Sartori's algebra A n,k is that it is graded cellular and properly stratified (for more details, see [Sar16b]). The cellular structure gives us a family of modules over A n,k (cell modules or standard modules) whose classes in an appropriate Grothendieck group correspond to standard tensor product basis elements of V ⊗n .…”

Section: Introductionmentioning

confidence: 99%

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Ozsvath-Szabo bordered algebras and subquotients of category O

Lauda,

Manion

2019

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We show that Ozsváth-Szabó's bordered algebra used to efficiently compute knot Floer homology is a graded flat deformation of the regular block of a q-presentable quotient of parabolic category O. We identify the endomorphism algebra of a minimal projective generator for this block with an explicit quotient of the Ozsváth-Szabó algebra using Sartori's diagrammatic formulation of the endomorphism algebra. Both of these algebras give rise to categorifications of tensor products of the vector representation V ⊗n for Uq(gl(1|1)). Our isomorphism allows us to transport a number of constructions between these two algebras, leading to a new (fully) diagrammatic reinterpretation of Sartori's algebra, new modules over Ozsváth-Szabó's algebra lifting various bases of V ⊗n , and bimodules over Ozsváth-Szabó's algebra categorifying the action of the quantum group element F and its dual on V ⊗n . Contents1. Introduction 1 2. Ozsváth-Szabó's algebras 7 3. Sartori's algebras 10 4. A surjective homomorphism from Ozsváth-Szabó to Sartori 13 5. Characterizing illicit morphisms via generating intervals 14 6. A vanishing ideal in the Sartori algebra 20 7. Fork elements and injectivity 21 8. Categorification of bases and bilinear forms 23 9. Bimodules for quantum group generators 31 References 34

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A diagram algebra for Soergel modules corresponding to smooth Schubert varieties

Cited by 2 publications

References 0 publications

Strands algebras and the affine highest weight property for equivariant hypertoric categories

Strands algebras and the affine highest weight property for equivariant hypertoric categories

Ozsvath-Szabo bordered algebras and subquotients of category O

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