Modified traces on Deligne's category
Jonathan Comes
and Jonathan R. Kujawa
Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA
DOI: 10.1007/s10801-012-0349-1
Abstract
Deligne has defined a category which interpolates among the representations of the various symmetric groups. In this paper we show Deligne's category admits a unique nontrivial family of modified trace functions. Such modified trace functions have already proven to be interesting in both low-dimensional topology and representation theory. We also introduce a graded variant of Deligne's category, lift the modified trace functions to the graded setting, and use them to recover the well-known invariant of framed knots known as the writhe.
Pages: 541–560
Keywords: ribbon category; Deligne's category; symmetric groups; modified traces
Full Text: PDF
References
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21. American Mathematical Society, Providence (2001) Boe, B.D., Kujawa, J.R., Nakano, D.K.: Complexity for modules over the classical Lie superalgebra $\mathfrak{gl}(m|n)$ . Compositio Math. (to appear) Comes, J., Ostrik, V.: On Deligne's category $\underline{\operatorname{Re}}\,\mathrm{p}^{ab}(S_{t})$ , in preparation Comes, J., Ostrik, V.: On blocks of Deligne's category $\underline{\operatorname{Re}}\,\mathrm{p}(S_{t})$ . Adv. Math. $226(2)$, 1331-1377 (2011) CrossRef Comes, J., Wilson, B.: Deligne's category $\underline{\operatorname{Re}}\,\mathrm{p}(GL_{δ})$ and representations of general linear supergroups. arXiv:1108.0652 Deligne, P.: La catégorie des représentations du groupe symétrique St, lorsque t n'est pas un entier naturel. Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai (2007), pp. 209-273 Del Padrone, A.: Deligne's representation theory in complex rank and objects of integral type. Manuscr. Math. 136(3-4), 339-343 (2011) CrossRef Etingof, P.: Representation Theory in Complex Rank. Newton Institute of Mathematical Sciences (2009) Geer, N., Kujawa, J.R., Patureau-Mirand, B.: Generalized trace and modified dimension functions on ribbon categories. Sel. Math. $17(2)$, 453-504 (2011) CrossRef Geer, N., Kashaev, R., Turaev, V.: Tetrahedral forms in monoidal categories and 3-manifold invariants. J. Reine Angew. Math. (to appear) Geer, N., Patureau-Mirand, B.: An invariant supertrace for the category of representations of Lie superalgebras of type I. Pac. J. Math. $238(2)$, 331-348 (2008) CrossRef Geer, N., Patureau-Mirand, B.: Topological invariants from non-restricted quantum groups (2010). arXiv:1009.4120 Geer, N., Patureau-Mirand, B., Turaev, V.: Modified quantum dimensions and re-normalized link invariants. Compos. Math. $145(1)$, 196-212 (2009) CrossRef Kassel, C.: Quantum Groups. Graduate Texts in Mathematics, vol.
155. Springer, New York (1995) CrossRef Kassel, C., Turaev, V.: Double construction for monoidal categories. Acta Math. $175(1)$, 1-48 (1995) CrossRef Knop, F.: A construction of semisimple tensor categories. C. R. Math. Acad. Sci. Paris $343(1)$, 15-18 (2006) CrossRef Knop, F.: Tensor envelopes of regular categories. Adv. Math. $214(2)$, 571-617 (2007) CrossRef Mathew, A.: Categories parametrized by schemes and representation theory in complex rank (2010). arXiv:1006.1381 Serganova, V.: On superdimension of an irreducible representation of a basic classical lie superalgebra. In S. Ferrara, R. Fioresi, and V.S. Varadarajan (eds.) Conference Proceedings “Supersymmetry in Mathematics and Physics”. Springer LNM (to appear) Turaev, V.G.: Quantum Invariants of Knots and 3-Manifolds. de Gruyter Studies in Mathematics, vol. 18. de Gruyter, Berlin (2010). Revised ed. CrossRef