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Extremal weight projectors II, 𝔤𝔩 N case
Queffelec, Hoel1; Wedrich, Paul2
1 IMAG Univ. Montpellier CNRS Montpellier France
2 Mathematical Sciences Institute The Australian National University Australia
Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 187-223.

We define diagrammatic extremal weight projectors for 𝔤𝔩 N (N≥2), a refinement of Jones–Wenzl projectors and Kuperberg’s clasps. As by-products, we obtain compatible diagrammatic presentations of the representation categories of 𝔤𝔩 N and its Cartan subalgebra, and a categorification of power-sum symmetric polynomials.

Received:
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Accepted:
Published online:
DOI: 10.5802/alco.330
Classification: 16S30, 16T05, 18M30
Keywords: General linear Lie algebras, weight spaces, idempotents

Queffelec, Hoel 1; Wedrich, Paul 2

1 IMAG Univ. Montpellier CNRS Montpellier France
2 Mathematical Sciences Institute The Australian National University Australia
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Queffelec, Hoel; Wedrich, Paul. Extremal weight projectors II, $\mathfrak{gl}_{N}$ case. Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 187-223. doi : 10.5802/alco.330. https://alco.centre-mersenne.org/articles/10.5802/alco.330/

[1] Asaeda, Marta M.; Przytycki, Józef H.; Sikora, Adam S. Categorification of the Kauffman bracket skein module of I-bundles over surfaces, Algebr. Geom. Topol., Volume 4 (2004), pp. 1177-1210 | DOI | MR | Zbl

[2] Bar-Natan, Dror Khovanov’s homology for tangles and cobordisms, Geom. Topol., Volume 9 (2005), pp. 1443-1499 | DOI | MR | Zbl

[3] Beliakova, Anna; Putyra, Krzysztof K.; Wehrli, Stephan M. Quantum link homology via trace functor I, Invent. Math., Volume 215 (2019) no. 2, pp. 383-492 | DOI | MR | Zbl

[4] Blanchet, Christian An oriented model for Khovanov homology, J. Knot Theory Ramifications, Volume 19 (2010) no. 2, pp. 291-312 | DOI | MR | Zbl

[5] Bonahon, Francis; Wong, Helen Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality, Quantum Topol., Volume 10 (2019) no. 2, pp. 325-398 | DOI | MR | Zbl

[6] Bousseau, Pierrick Strong positivity for the skein algebras of the 4-punctured sphere and of the 1-punctured torus, Comm. Math. Phys., Volume 398 (2023) no. 1, pp. 1-58 | DOI | MR | Zbl

[7] Brundan, J. Representations of the oriented skein category, 2017 | arXiv

[8] Cautis, Sabin; Kamnitzer, Joel Quantum K-theoretic geometric Satake: the SL n case, Compos. Math., Volume 154 (2018) no. 2, pp. 275-327 | DOI | MR | Zbl

[9] Cautis, Sabin; Kamnitzer, Joel; Morrison, Scott Webs and quantum skew Howe duality, Math. Ann., Volume 360 (2014) no. 1-2, pp. 351-390 | DOI | MR | Zbl

[10] Ehrig, Michael; Tubbenhauer, Daniel; Wedrich, Paul Functoriality of colored link homologies, Proc. Lond. Math. Soc. (3), Volume 117 (2018) no. 5, pp. 996-1040 | DOI | MR | Zbl

[11] Elias, Ben; Williamson, Geordie Soergel calculus, Represent. Theory, Volume 20 (2016), pp. 295-374 | DOI | MR | Zbl

[12] Frohman, Charles; Gelca, Răzvan Skein modules and the noncommutative torus, Trans. Amer. Math. Soc., Volume 352 (2000) no. 10, pp. 4877-4888 | DOI | MR | Zbl

[13] Grigsby, J. Elisenda; Licata, Anthony M.; Wehrli, Stephan M. Annular Khovanov homology and knotted Schur–Weyl representations, Compos. Math., Volume 154 (2018) no. 3, pp. 459-502 | DOI | MR | Zbl

[14] Jones, V. F. R. Index for subfactors, Invent. Math., Volume 72 (1983) no. 1, pp. 1-25 | DOI | MR | Zbl

[15] Khovanov, Mikhail A categorification of the Jones polynomial, Duke Math. J., Volume 101 (2000) no. 3, pp. 359-426 | DOI | MR | Zbl

[16] Khovanov, Mikhail Categorifications of the colored Jones polynomial, J. Knot Theory Ramifications, Volume 14 (2005) no. 1, pp. 111-130 | DOI | MR | Zbl

[17] Khovanov, Mikhail; Rozansky, Lev Matrix factorizations and link homology, Fund. Math., Volume 199 (2008) no. 1, pp. 1-91 | DOI | MR | Zbl

[18] Khovanov, Mikhail; Sazdanovic, Radmila A categorification of one-variable polynomials, Proceedings of FPSAC 2015 (Discrete Math. Theor. Comput. Sci. Proc.), Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2015), pp. 937-948 | MR | Zbl

[19] Kuperberg, Greg Spiders for rank 2 Lie algebras, Comm. Math. Phys., Volume 180 (1996) no. 1, pp. 109-151 | DOI | MR | Zbl

[20] Lacabanne, Abel; Tubbenhauer, Daniel; Vaz, Pedro Annular webs and Levi subalgebras, J. Comb. Algebra, Volume 7 (2023) no. 3-4, pp. 283-326 | DOI | MR | Zbl

[21] Lê, Thang T. Q. On positivity of Kauffman bracket skein algebras of surfaces, Int. Math. Res. Not. IMRN (2018) no. 5, pp. 1314-1328 | DOI | MR | Zbl

[22] Morton, Hugh; Samuelson, Peter The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra, Duke Math. J., Volume 166 (2017) no. 5, pp. 801-854 | DOI | MR | Zbl

[23] Murakami, Hitoshi; Ohtsuki, Tomotada; Yamada, Shuji Homfly polynomial via an invariant of colored plane graphs, Enseign. Math. (2), Volume 44 (1998) no. 3-4, pp. 325-360 | MR | Zbl

[24] Przytycki, Józef H. Skein modules of 3-manifolds, Bull. Polish Acad. Sci. Math., Volume 39 (1991) no. 1-2, pp. 91-100 | MR | Zbl

[25] Queffelec, Hoel Skein modules from skew Howe duality and affine extensions, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 11 (2015), Paper no. 030, 36 pages | DOI | MR | Zbl

[26] Queffelec, Hoel; Rose, David E. V. Sutured annular Khovanov-Rozansky homology, Trans. Amer. Math. Soc., Volume 370 (2018) no. 2, pp. 1285-1319 | DOI | MR | Zbl

[27] Queffelec, Hoel; Sartori, Antonio Mixed quantum skew Howe duality and link invariants of type A, J. Pure Appl. Algebra, Volume 223 (2019) no. 7, pp. 2733-2779 | DOI | MR | Zbl

[28] Queffelec, Hoel; Wedrich, Paul Extremal weight projectors, Math. Res. Lett., Volume 25 (2018) no. 6, pp. 1911-1936 | DOI | MR | Zbl

[29] Queffelec, Hoel; Wedrich, Paul Khovanov homology and categorification of skein modules, Quantum Topol., Volume 12 (2021) no. 1, pp. 129-209 | DOI | MR | Zbl

[30] Thurston, Dylan Paul Positive basis for surface skein algebras, Proc. Natl. Acad. Sci. USA, Volume 111 (2014) no. 27, pp. 9725-9732 | DOI | MR | Zbl

[31] Tubbenhauer, Daniel; Vaz, Pedro; Wedrich, Paul Super q-Howe duality and web categories, Algebr. Geom. Topol., Volume 17 (2017) no. 6, pp. 3703-3749 | DOI | MR | Zbl

[32] Turaev, Vladimir G. Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. École Norm. Sup. (4), Volume 24 (1991) no. 6, pp. 635-704 | DOI | Numdam | MR | Zbl

[33] Wenzl, Hans On sequences of projections, C. R. Math. Rep. Acad. Sci. Canada, Volume 9 (1987) no. 1, pp. 5-9 | MR | Zbl

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