Web bases for the general linear groups
Bruce W. Westbury
DOI: 10.1007/s10801-011-0294-4
Abstract
Let V be the representation of the quantized enveloping algebra of \mathfrak gl( n) \mathfrak{gl}(n) which is the q-analogue of the vector representation and let V \ast be the dual representation. We construct a basis for Ä r( V Å V *) \bigotimes^{r}(V \oplus V^{*}) with favorable properties similar to those of Lusztig's dual canonical basis. In particular our basis is invariant under the bar involution and contains a basis for the subspace of invariant tensors.
Pages: 93–107
Keywords: keywords Schur-Weyl duality; invariant tensors
Full Text: PDF
References
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3. Benkart, G., Doty, S.: Derangements and tensor powers of adjoint modules for sln. J. Algebr. Comb. 16(1), 31-42 (2002)
4. Bigelow, S.: Skein theory for the ADE planar algebras. J. Pure Appl. Algebra 214(5), 658-666 (2010)
5. Cox, A., De Visscher, M., Doty, S., Martin, P.: On the blocks of the walled Brauer algebra. J. Algebra 320(1), 169-212 (2008)
6. Dipper, R., Doty, S., Stoll, F.: Quantized mixed tensor space and Schur-Weyl duality I, 2009
7. Dipper, R., Doty, S., Stoll, F.: Quantized mixed tensor space and Schur-Weyl duality II, 2009
8. Doty, S.: New versions of Schur-Weyl duality. In: Finite Groups 2003, pp. 59-71. Walter de Gruyter, Berlin (2004)
9. Doty, S.: Schur-Weyl duality in positive characteristic. In: Representation Theory. Contemp. Math., vol. 478, pp. 15-28. Am. Math. Soc., Providence (2009)
10. Frenkel, I.B., Khovanov, M.G.: Canonical bases in tensor products and graphical calculus for Uq (sl2). Duke Math. J. 87(3), 409-480 (1997)
11. Jeong, M.-J., Kim, D.: Quantum sl(n, C) link invariants.
12. Jones, V.F.R.: The annular structure of subfactors. In: Essays on geometry and related topics, Vols. 1,
2. Monogr. Enseign. Math., vol. 38, pp. 401-463. Enseignement Math., Geneva (2001)
13. Khovanov, M., Kuperberg, G.: Web bases for sl(3) are not dual canonical. Pac. J. Math. 188(1), 129- 153 (1999)
14. Kim, D.: Graphical Calculus on Representations of Quantum Lie Algebras. Ph.D. thesis, University of California, Davis. ProQuest LLC, Ann Arbor (2003).
15. Kuperberg, G.: Spiders for rank 2 Lie algebras. Commun. Math. Phys. 180(1), 109-151 (1996).
16. Leclerc, B., Thibon, J.-Y.: The Robinson-Schensted correspondence, crystal bases, and the quantum straightening at q =
0. Electron. J. Comb. 3(2), 11 (1996)
17. Lusztig, G.: Introduction to Quantum Groups. Progress in Mathematics, vol.
110. Birkhäuser Boston, Boston (1993)
18. Mihailovs, A.: Tensor invariants of SL(n), wave graphs and L-tris.
19. Morrison, S.: A diagrammatic category for the representation theory of U_q(sl_n). PhD thesis, Uni- versity of California, Berkeley, 2007
20. Morrison, S., Peters, E., Snyder, N.: Skein theory for the D2n planar algebras. J. Pure Appl. Algebra 214(2), 117-139 (2010)
21. Murakami, H., Ohtsuki, T., Yamada, S.: Homfly polynomial via an invariant of colored plane graphs. Enseign. Math. 44(3-4), 325-360 (1998)
22. Ohtsuki, T.: Problems on invariants of knots and 3-manifolds. In: Invariants of Knots and 3-Manifolds, Kyoto,
2001. Geom. Topol. Monogr., vol. 4, pp. 377-572. Geom. Topol. Publ., Coventry (2002). With an introduction by J. Roberts, pp. i-iv
23. Petersen, T.K., Pylyavskyy, P., Rhoades, B.: Promotion and cyclic sieving via webs. J. Algebr. Comb. 30(1), 19-41 (2009)
24. Rhoades, B.: Cyclic sieving, promotion, and representation theory. J. Comb. Theory, Ser. A 117(1), 38-76 (2010)
25. Sagan, B.: The cyclic sieving phenomenon: a survey, 2010
26. Westbury, B.W.: The representation theory of the Temperley-Lieb algebras. Math. Z. 219(4), 539-565 (1995)
27. Westbury, B.W.: Enumeration of non-positive planar trivalent graphs. J. Algebr. Comb. 25(4), 357- 373 (2007)
28. Westbury, B.W.: Invariant tensors for the spin representation of so(7). Math. Proc. Camb. Philos. Soc.
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