A simple bijection between standard 3\times n tableaux and irreducible webs for \mathfrak sl 3 \mathfrak{sl}_{3}
Julianna Tymoczko
DOI: 10.1007/s10801-011-0317-1
Abstract
Combinatorial spiders are a model for the invariant space of the tensor product of representations. The basic objects, webs, are certain directed planar graphs with boundary; algebraic operations on representations correspond to graph-theoretic operations on webs. Kuperberg developed spiders for rank 2 Lie algebras and \mathfrak sl 2 \mathfrak {sl}_{2}. Building on a result of Kuperberg, Khovanov-Kuperberg found a recursive algorithm giving a bijection between standard Young tableaux of shape 3\times n and irreducible webs for \mathfrak sl 3 \mathfrak{sl}_{3} whose boundary vertices are all sources.
Pages: 611–632
Keywords: keywords spider; representations of Lie algebras; Young tableau; jeu de taquin; promotion
Full Text: PDF
References
1. Fulton, W.: Young Tableaux. Cambridge UP, New York (1997)
2. Fung, F.: On the topology of components of some Springer fibers and their relation to Kazhdan- Lusztig theory. Adv. Math. 178(2), 244-276 (2003)
3. Jeong, M.-J., Kim, D.: Quantum sl(n, C) link invariants.
4. Khovanov, M., Kuperberg, G.: Web bases for sl(3) are not dual canonical. Pac. J. Math. 188, 129-153 (1999)
5. Kim, D.: Graphical calculus on representations of quantum Lie algebras. PhD thesis, UC Davis (2003). Available at
6. Kuperberg, G.: Spiders for rank 2 Lie algebras. Commun. Math. Phys. 180(1), 109-151 (1996)
7. Morrison, S.: A diagrammatic category for the representation theory of Uq (sln). PhD thesis, UC Berkeley (2007). Available at
8. Petersen, K., Pylyavskyy, P., Rhoades, B.: Promotion and cyclic sieving via webs. J. Algebr. Comb. 30, 19-41 (2009)
9. Petersen, K., Pylyavskyy, P., Speyer, D.: A non-crossing standard monomial theory. J. Algebra 324, 951-969 (2010)
10. Russell, H., Tymoczko, J.: Springer representations on the Khovanov Springer varieties. Math. Proc.
2. Fung, F.: On the topology of components of some Springer fibers and their relation to Kazhdan- Lusztig theory. Adv. Math. 178(2), 244-276 (2003)
3. Jeong, M.-J., Kim, D.: Quantum sl(n, C) link invariants.
4. Khovanov, M., Kuperberg, G.: Web bases for sl(3) are not dual canonical. Pac. J. Math. 188, 129-153 (1999)
5. Kim, D.: Graphical calculus on representations of quantum Lie algebras. PhD thesis, UC Davis (2003). Available at
6. Kuperberg, G.: Spiders for rank 2 Lie algebras. Commun. Math. Phys. 180(1), 109-151 (1996)
7. Morrison, S.: A diagrammatic category for the representation theory of Uq (sln). PhD thesis, UC Berkeley (2007). Available at
8. Petersen, K., Pylyavskyy, P., Rhoades, B.: Promotion and cyclic sieving via webs. J. Algebr. Comb. 30, 19-41 (2009)
9. Petersen, K., Pylyavskyy, P., Speyer, D.: A non-crossing standard monomial theory. J. Algebra 324, 951-969 (2010)
10. Russell, H., Tymoczko, J.: Springer representations on the Khovanov Springer varieties. Math. Proc.
© 1992–2009 Journal of Algebraic Combinatorics
©
2012 FIZ Karlsruhe /
Zentralblatt MATH for the EMIS Electronic Edition