Webs of Type P

Davidson, Nicholas; Kujawa, Jonathan R.; Muth, Robert

doi:10.48550/arxiv.2109.03410

Cited by 1 publication

(3 citation statements)

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“…The category Web A,a generalizes a number of web category constructions. In particular when A = we recover the symmetric webs for gl n ( ), which are non-quantized versions of those which appear in the literature in [11,35,37,40], and are isomorphic to categories defined in [8,15]. When A = Cl 1 is a rank one Clifford algebra, our constructions recover those of [6], which are a non-quantum version of those given in [5].…”

mentioning

confidence: 71%

“…The category of gl n -webs. If A = a = then Web , {1} can be seen to be isomorphic to the category of gl n -webs defined in [15], the Schur category defined in [8], and the web category introduced in [11]. For n, d ∈ ≥0 , the -superalgebra W , n,d is isomorphic to the well-studied classical Schur algebra S (n, d).…”

Section: Examplesmentioning

confidence: 99%

“…, and (3.2.2) can be checked as in[15, Theorem 6.1.1] (where full details are included in the arXiv version of that paper). Relations (3.1.5)-(3.1.7) are straightforward, as are (3.1.11) and (3.1.12), noting that f = 0 for f -coupons on thick strands.The left side of (3.1.8) is clear.…”

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confidence: 99%

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Superalgebra deformations of web categories: finite webs

Davidson¹,

Kujawa²,

Muth³

et al. 2023

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Let be a characteristic zero domain. For a locally unital -superalgebra A with distinguished idempotents I and even subalgebra a ⊆ A0, we define and study an associated diagrammatic monoidal -linear supercategory Web A,a I . This supercategory yields a diagrammatic description of the generalized Schur algebras T A a (n, d). We also show there is an asymptotically faithful functor from Web A,a I to the monoidal supercategory of gl n (A)-modules generated by symmetric powers of the natural module. When this functor is full, the single diagrammatic supercategory Web A,a I provides a combinatorial description of this module category for all n ≥ 1. We also use these results to establish Howe dualities between gl m (A) and gl n (A) when A is semisimple. I 6.6. Basis results for Web A,a I 6.7. A remark on arbitrary commutative rings 6.8. Good subpairs 6.9. Truncation 7. Connection to Schurification 7.1. The algebra W A,a n,d 7.2. The Schurification of (A, a) 7.3. Connecting algebras 7.4. Monoidal equivalence 8. The wreath category 8.1. The wreath category, I 8.2. The wreath category, II 9. Fullness of representations 9.1. Defining representations 9.2. Fullness for the web category 9.3. Fullness for the wreath category 10. The category U(gl n (A)) 10.1. The category U(gl n (A)) 10.2. A functor from U(gl n (A)) to Web A,a {1},n 11. Polynomial representations and Schur-Weyl duality 11.1. Polynomial representations 11.2. Schur-Weyl duality 11.3. Representations of finite-dimensional semisimple superalgebras 11.4. Representations of wreath product superalgebras 11.5. Schur-Weyl-Berele-Regev-Sergeev duality 11.6. Polynomial representations and generalized Schur-Weyl duality 12. Howe duality 12.1. The symmetric space S m,n 12.2. Miscellanea 12.3. Categorical representations from S m,n 12.4. Categorical and finitary Howe duality 12.5. Strong multiplicity-free decompositions 13. Examples 13.1. The category of gl n -webs 13.2. Cyclotomic Schur algebras 13.3. The category of q n -webs 13.4. Zigzag superalgebras and RoCK blocks References

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