Tiling bijections between paths and Brauer diagrams
Robert J. Marsh
and Paul Martin
DOI: 10.1007/s10801-010-0252-6
Abstract
There is a natural bijection between Dyck paths and basis diagrams of the Temperley-Lieb algebra defined via tiling. Overhang paths are certain generalisations of Dyck paths allowing more general steps but restricted to a rectangle in the two-dimensional integer lattice. We show that there is a natural bijection, extending the above tiling construction, between overhang paths and basis diagrams of the Brauer algebra.
Pages: 427–453
Keywords: keywords Brauer algebra; temperley-Lieb diagram; pipe dream; Dyck path; overhang path; double-factorial combinatorics
Full Text: PDF
References
1. Andrews, G.E., Baxter, R.J., Forrester, P.J.: Eight-vertex SOS model and generalized Rogers- Ramanujan-type identities. J. Stat. Phys. 35(3-4), 193-266 (1984)
2. Baker, T.H., Forrester, P.J.: Random walks and random fixed-point free involutions. J. Phys. A, Math. Gen. 34, L381-L390 (2001)
3. Bergeron, N., Billey, S.: RC-graphs and Schubert polynomials. Exp. Math. 2(4), 257-269 (1993)
4. Boerner, H.: Representations of Groups. With Special Consideration for the Needs of Modern Physics. North-Holland/American Elsevier, Amsterdam/New York (1970). Translated from the German by P.G. Murphy in cooperation with J. Mayer-Kalkschmidt and P. Carr. 2nd. English ed.
5. Brauer, R.: On algebras which are connected with the semisimple continuous groups. Ann. Math. (2) 38(4), 857-872 (1937)
6. Chen, W.Y.C., Deng, E.Y.P., Du, R.R.X., Stanley, R.P., Yan, C.H.: Crossings and nestings of matchings and partitions. Trans. Am. Math. Soc. 359(4), 1555-1575 (2007)
7. Cox, A., De Visscher, M., Martin, P.: A geometric characterisation of the blocks of the Brauer algebra. J. Lond. Math. Soc. (2) 80(2), 471-494 (2009)
8. Cox, A., De Visscher, M., Martin, P.P.: Alcove geometry and a translation principle for the Brauer algebra. Journal of Pure and Applied Algebra, (2010)
9. Cox, A., De Visscher, M., Martin, P.P.: Private communication (2010)
10. Dale, M.R.T., Moon, J.W.: The permuted analogues of three Catalan sets. J. Stat. Plan. Inference 34(1), 75-87 (1993)
11. Fomin, S., Kirillov, A.N.: The Yang-Baxter equation, symmetric functions, and Schubert polynomials. Discrete Math. 153, 123-143 (1996)
12. Halverson, T., Lewandowski, T.: RSK insertion for set partitions and diagram algebras. Electron. J. Comb. 11(2), 24 (2004)
13. Lickorish, W.B.R.: An Introduction to Knot Theory. Graduate Texts in Mathematics, vol.
175. Springer, New York (1997)
14. Marsh, R.J., Martin, P.: Pascal arrays: counting Catalan sets. Preprint [math.CO] (2006)
15. Martin, P.: Potts Models and Related Problems in Statistical Mechanics. Series on Advances in Statistical Mechanics, vol.
5. World Scientific, Singapore (1991)
16. Martin, P.P.: The decomposition matrices of the Brauer algebra over the complex field. Preprint [math.RT] (2009)
17. Martin, P.P., Rollet, G.: The Potts model representation and a Robinson-Schensted correspondence for the partition algebra. Compos. Math. 112(2), 237-254 (1998)
18. Martin, P., Woodcock, D., Levy, D.: A diagrammatic approach to Hecke algebras of the reflection equation. J. Phys. A 33(6), 1265-1296 (2000)
19. Martin, P., Saleur, H.: On algebraic diagonalization of the XXZ chain. Perspectives on solvable models. Int. J. Mod. Phys. B 8(25-26), 3637-3644 (1994)
20. Rubey, M.: Nestings of matchings and permutations and north steps in PDSAWs. In: Discrete Mathematics & Theoretical Computer Science Proceedings, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), pp. 691-704. North America, December
2008. Available at: Preprint [math.CO]
21. Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences (2009).
2. Baker, T.H., Forrester, P.J.: Random walks and random fixed-point free involutions. J. Phys. A, Math. Gen. 34, L381-L390 (2001)
3. Bergeron, N., Billey, S.: RC-graphs and Schubert polynomials. Exp. Math. 2(4), 257-269 (1993)
4. Boerner, H.: Representations of Groups. With Special Consideration for the Needs of Modern Physics. North-Holland/American Elsevier, Amsterdam/New York (1970). Translated from the German by P.G. Murphy in cooperation with J. Mayer-Kalkschmidt and P. Carr. 2nd. English ed.
5. Brauer, R.: On algebras which are connected with the semisimple continuous groups. Ann. Math. (2) 38(4), 857-872 (1937)
6. Chen, W.Y.C., Deng, E.Y.P., Du, R.R.X., Stanley, R.P., Yan, C.H.: Crossings and nestings of matchings and partitions. Trans. Am. Math. Soc. 359(4), 1555-1575 (2007)
7. Cox, A., De Visscher, M., Martin, P.: A geometric characterisation of the blocks of the Brauer algebra. J. Lond. Math. Soc. (2) 80(2), 471-494 (2009)
8. Cox, A., De Visscher, M., Martin, P.P.: Alcove geometry and a translation principle for the Brauer algebra. Journal of Pure and Applied Algebra, (2010)
9. Cox, A., De Visscher, M., Martin, P.P.: Private communication (2010)
10. Dale, M.R.T., Moon, J.W.: The permuted analogues of three Catalan sets. J. Stat. Plan. Inference 34(1), 75-87 (1993)
11. Fomin, S., Kirillov, A.N.: The Yang-Baxter equation, symmetric functions, and Schubert polynomials. Discrete Math. 153, 123-143 (1996)
12. Halverson, T., Lewandowski, T.: RSK insertion for set partitions and diagram algebras. Electron. J. Comb. 11(2), 24 (2004)
13. Lickorish, W.B.R.: An Introduction to Knot Theory. Graduate Texts in Mathematics, vol.
175. Springer, New York (1997)
14. Marsh, R.J., Martin, P.: Pascal arrays: counting Catalan sets. Preprint [math.CO] (2006)
15. Martin, P.: Potts Models and Related Problems in Statistical Mechanics. Series on Advances in Statistical Mechanics, vol.
5. World Scientific, Singapore (1991)
16. Martin, P.P.: The decomposition matrices of the Brauer algebra over the complex field. Preprint [math.RT] (2009)
17. Martin, P.P., Rollet, G.: The Potts model representation and a Robinson-Schensted correspondence for the partition algebra. Compos. Math. 112(2), 237-254 (1998)
18. Martin, P., Woodcock, D., Levy, D.: A diagrammatic approach to Hecke algebras of the reflection equation. J. Phys. A 33(6), 1265-1296 (2000)
19. Martin, P., Saleur, H.: On algebraic diagonalization of the XXZ chain. Perspectives on solvable models. Int. J. Mod. Phys. B 8(25-26), 3637-3644 (1994)
20. Rubey, M.: Nestings of matchings and permutations and north steps in PDSAWs. In: Discrete Mathematics & Theoretical Computer Science Proceedings, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), pp. 691-704. North America, December
2008. Available at: Preprint [math.CO]
21. Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences (2009).
© 1992–2009 Journal of Algebraic Combinatorics
©
2012 FIZ Karlsruhe /
Zentralblatt MATH for the EMIS Electronic Edition