Webs of type Q
Algebraic Combinatorics, Volume 4 (2021) no. 6, pp. 1027-1072.

Howe dualities lead to diagrammatic categories which describe the representations of Lie-type objects as a monoidal category (that is, via generators and relations). Applying this philosophy to the type Q Howe duality of Cheng–Wang and Sergeev, we introduce diagrammatic web supercategories of type Q via generators and relations and show they describe the full subcategory of supermodules for the Lie superalgebra of type Q given by the tensor products of supersymmetric tensor powers of the natural supermodule.

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Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.191
Classification: 17B10,  18D10
Keywords: Monoidal supercategories, diagrammatic categories, web categories, Lie superalgebras.
Brown, Gordon C. 1; Kujawa, Jonathan R. 2

1. Fort Worth, TX, USA
2. Department of Mathematics University of Oklahoma Norman, OK 73019, USA
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Brown, Gordon C.; Kujawa, Jonathan R. Webs of type Q. Algebraic Combinatorics, Volume 4 (2021) no. 6, pp. 1027-1072. doi : 10.5802/alco.191. https://alco.centre-mersenne.org/articles/10.5802/alco.191/

[1] Beliakova, Anna; Guliyev, Zaur; Habiro, Kazuo; Lauda, Aaron D. Trace as an alternative decategorification functor, Acta Math. Vietnam., Volume 39 (2014) no. 4, pp. 425-480 | Article | MR 3319701 | Zbl 1331.81153

[2] Benkart, Georgia; Guay, Nicolas; Jung, Ji Hye; Kang, Seok-Jin; Wilcox, Stewart Quantum walled Brauer-Clifford superalgebras, J. Algebra, Volume 454 (2016), pp. 433-474 | Article | MR 3473434 | Zbl 1342.17002

[3] Brown, Gordon C. Webs for permutation supermodules of type Q, Comm. Algebra, Volume 47 (2019) no. 11, pp. 4763-4790 | Article | MR 3991050 | Zbl 1418.05130

[4] Brown, Gordon C.; Davidson, Nicholas J.; Kujawa, Jonathan R. Quantum Webs of Type Q (2020) (https://arxiv.org/abs/2001.00663)

[5] Brundan, Jonathan; Comes, Jonathan; Kujawa, Jonathan Robert A basis theorem for the degenerate affine oriented Brauer–Clifford supercategory, Canad. J. Math., Volume 71 (2019) no. 5, pp. 1061-1101 | Article | MR 4010422 | Zbl 07106605

[6] Brundan, Jonathan; Comes, Jonathan; Nash, David; Reynolds, Andrew A basis theorem for the affine oriented Brauer category and its cyclotomic quotients, Quantum Topol., Volume 8 (2017) no. 1, pp. 75-112 | Article | MR 3630282 | Zbl 1419.18011

[7] Brundan, Jonathan; Davidson, Nicholas Type A blocks of super category 𝒪, J. Algebra, Volume 473 (2017), pp. 447-480 | Article | MR 3591159 | Zbl 1396.17005

[8] Brundan, Jonathan; Davidson, Nicholas Type C blocks of super category 𝒪, Math. Z., Volume 293 (2019) no. 3-4, pp. 867-901 | Article | MR 4024570 | Zbl 1446.17029

[9] Brundan, Jonathan; Ellis, Alexander P. Monoidal supercategories, Comm. Math. Phys., Volume 351 (2017) no. 3, pp. 1045-1089 | Article | MR 3623246 | Zbl 1396.17012

[10] Brundan, Jonathan; Kleshchev, Alexander Projective representations of symmetric groups via Sergeev duality, Math. Z., Volume 239 (2002) no. 1, pp. 27-68 | Article | MR 1879328 | Zbl 1029.20008

[11] Cautis, Sabin; Kamnitzer, Joel; Morrison, Scott Webs and quantum skew Howe duality, Math. Ann., Volume 360 (2014) no. 1-2, pp. 351-390 | Article | MR 3263166 | Zbl 1387.17027

[12] Chang, Zhihua; Wang, Yongjie Howe duality for quantum queer superalgebras, J. Algebra, Volume 547 (2020), pp. 358-378 | Article | MR 4040727 | Zbl 1461.17015

[13] Chen, Chih-Whi Reduction method for representations of queer Lie superalgebras, J. Math. Phys., Volume 57 (2016) no. 5, Paper no. 051703, 12 pages | Article | MR 3498782 | Zbl 1386.17011

[14] Cheng, Shun-Jen; Kwon, Jae-Hoon; Wang, Weiqiang Character formulae for queer Lie superalgebras and canonical bases of types A/C, Comm. Math. Phys., Volume 352 (2017) no. 3, pp. 1091-1119 | Article | MR 3631400 | Zbl 1406.17014

[15] Cheng, Shun-Jen; Wang, Weiqiang Remarks on the Schur–Howe–Sergeev duality, Lett. Math. Phys., Volume 52 (2000) no. 2, pp. 143-153 | Article | MR 1786858 | Zbl 0974.17031

[16] Cheng, Shun-Jen; Wang, Weiqiang Dualities and representations of Lie superalgebras, Graduate Studies in Mathematics, 144, American Mathematical Society, Providence, RI, 2012, xviii+302 pages | Article | MR 3012224 | Zbl 1271.17001

[17] Dipper, Richard; Doty, Stephen; Stoll, Friederike The quantized walled Brauer algebra and mixed tensor space, Algebr. Represent. Theory, Volume 17 (2014) no. 2, pp. 675-701 | Article | MR 3181742 | Zbl 1368.17017

[18] Du, Jie; Wan, Jinkui Presenting queer Schur superalgebras, Int. Math. Res. Not. IMRN (2015) no. 8, pp. 2210-2272 | Article | MR 3344667 | Zbl 1395.20032

[19] Elias, Ben Light ladders and clasp conjectures (2015) (https://arxiv.org/abs/1510.06840)

[20] Joyal, André; Street, Ross Braided tensor categories, Adv. Math., Volume 102 (1993) no. 1, pp. 20-78 | Article | MR 1250465 | Zbl 0817.18007

[21] Kauffman, Louis H.; Lins, Sóstenes L. Temperley-Lieb recoupling theory and invariants of 3-manifolds, Annals of Mathematics Studies, 134, Princeton University Press, Princeton, NJ, 1994, x+296 pages | Article | MR 1280463 | Zbl 0821.57003

[22] Kelly, Gregory M. Basic concepts of enriched category theory, Repr. Theory Appl. Categ. (2005) no. 10, p. vi+137 (Reprint of the 1982 original [Cambridge Univ. Press, Cambridge; MR0651714]) | MR 2177301 | Zbl 1086.18001

[23] Kim, Dongseok Graphical calculus on representations of quantum Lie algebras, ProQuest LLC, Ann Arbor, MI, 2003, 55 pages Thesis (Ph.D.)–University of California, Davis | MR 2704398

[24] Kuperberg, Greg Spiders for rank 2 Lie algebras, Comm. Math. Phys., Volume 180 (1996) no. 1, pp. 109-151 | Article | MR 1403861 | Zbl 0870.17005

[25] Morrison, Scott E. A diagrammatic category for the representation theory of U q (𝔰𝔩 n ), ProQuest LLC, Ann Arbor, MI, 2007, 89 pages Thesis (Ph.D.)–University of California, Berkeley | MR 2710589

[26] Penkov, Ivan; Serganova, Vera Characters of irreducible G-modules and cohomology of G/P for the Lie supergroup G=Q(N), J. Math. Sci. (New York), Volume 84 (1997) no. 5, pp. 1382-1412 (Algebraic geometry, 7) | Article | MR 1465520 | Zbl 0920.17003

[27] Queffelec, Hoel; Rose, David E. V. The 𝔰𝔩 n foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality, Adv. Math., Volume 302 (2016), pp. 1251-1339 | Article | MR 3545951 | Zbl 1360.57025

[28] 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 | Article | MR 3912946 | Zbl 07032784

[29] Rose, David E. V.; Tubbenhauer, Daniel Symmetric webs, Jones-Wenzl recursions, and q-Howe duality, Int. Math. Res. Not. IMRN (2016) no. 17, pp. 5249-5290 | Article | MR 3556438 | Zbl 1404.17025

[30] Rose, David E. V.; Wedrich, Paul Deformations of colored 𝔰𝔩 N link homologies via foams, Geom. Topol., Volume 20 (2016) no. 6, pp. 3431-3517 | Article | MR 3590355 | Zbl 1420.57044

[31] Rumer, Georg; Teller, Edward; Weyl, Hermann Eine für die Valenztheorie geeignete Basis der binären Vektorinvarianten, Nachrichten von der Ges. der Wiss. Zu Göttingen, Math.-Phys. Klasse (1932), pp. 498-504 | Zbl 0006.14901

[32] Sartori, Antonio; Tubbenhauer, Daniel Webs and q-Howe dualities in types BCD, Trans. Amer. Math. Soc., Volume 371 (2019) no. 10, pp. 7387-7431 | Article | MR 3939581 | Zbl 07051088

[33] Sergeev, Alexander N. Tensor algebra of the identity representation as a module over the Lie superalgebras Gl(n,m) and Q(n), Mat. Sb. (N.S.), Volume 123(165) (1984) no. 3, pp. 422-430 | MR 735715

[34] Sergeev, Alexander N. The Howe duality and the projective representations of symmetric groups, Represent. Theory, Volume 3 (1999), pp. 416-434 | Article | MR 1722115 | Zbl 0999.17014

[35] Sergeev, Alexander N. An analog of the classical invariant theory for Lie superalgebras. I, II, Michigan Math. J., Volume 49 (2001) no. 1, p. 113-146, 147–168 | Article | MR 1827078 | Zbl 1002.17002

[36] Stembridge, John R. Shifted tableaux and the projective representations of symmetric groups, Adv. Math., Volume 74 (1989) no. 1, pp. 87-134 | Article | MR 991411 | Zbl 0677.20012

[37] Tubbenhauer, Daniel; Vaz, Pedro; Wedrich, Paul Super q-Howe duality and web categories, Algebr. Geom. Topol., Volume 17 (2017) no. 6, pp. 3703-3749 | Article | MR 3709658 | Zbl 1422.57030

[38] Turaev, Vladimir G. Operator invariants of tangles, and R-matrices, Izv. Akad. Nauk SSSR Ser. Mat., Volume 53 (1989) no. 5, p. 1073-1107, 1135 | Article | MR 1024455 | Zbl 0707.57003

[39] Turaev, Vladimir G. Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics, 18, De Gruyter, Berlin, 2016, xii+596 pages | Article | MR 3617439 | Zbl 1346.57002

[40] Wenzl, Hans On sequences of projections, C. R. Math. Rep. Acad. Sci. Canada, Volume 9 (1987) no. 1, pp. 5-9 | MR 873400 | Zbl 0622.47019

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