The displacement and split decompositions for a Q-polynomial distance-regular graph

Graphs and Combinatorics

Volumn 21, Issue 2, 2005, Pages 263-276

  Terwilliger, Paul ^a  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

Author keywords

Association scheme; Distance regular graph; Subconstituent algebra; Terwilliger algebra

Indexed keywords

EID: 21544470933     PISSN: 09110119     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00373-004-0594-8     Document Type: Article

References (41)

1. Bannai, E., Ito, T.: Algebraic Combinatorics I, Association Schemes. London: Benjamin/Cummings 1984.
2. Biggs, N.: Algebraic Graph Theory. Second edition. Cambridge: Cambridge University Press 1993
3. Brouwer, A.E., Cohen, A.M. Neumaier, A.: Distance-Regular Graphs. Berlin Springer-Verlag, 1989
4. Brouwer, A.E., Godsil, C.D. Koolen, J.H., Martin, W. J.: Width and dual width of subsets in polynomial association schemes. J. Comb. Theory, Ser. A 102, 255-271 (2003)
5. Caughman IV, J.S.: Spectra of bipartite P- and Q-polynomial association schemes. Graphs Comb. 14, 321-343 (1998)
6. Caughman IV, J.S.: The Terwilliger algebras of bipartite P- and Q-polynomia.1 association schemes. Discrete Math. 196, 65-95 (1999)
7. Curtin, B.: Bipartite distance-regular graphs I. Graphs Comb. 15, 143-158 (1999)
8. Curtin, B.: Bipartite distance-regular graphs II. Graphs Comb. 15, 377-391 (1999)
9. Curtin, B.: 2-homogeneous bipartite distance-regular graphs. Discrete Math. 187, 39-70 (1998)
10. Curtin, B.: Distance-regular graphs which support a spin model are thin. 16th British Combinatorial Conference (London, 1997). Discrete Math. 197/198, 205-216 (1999)
11. Curtin, B., Nomura, K.: Distance-regular graphs related to the quantum enveloping algebra of sl(2). J. Algebr. Comb. 12, 25-36 (2000)
12. Curtis, C., Reiner, I., Representation Theory of Finite Groups and Associative Algebras. New York: Interscience 1962
13. Delsarte, P.: Association schemes and t-designs in regular semilattices. J. Comb. Theory Ser. A 20, 230-243 (1976)
14. Dickie, G.: Twice Q-polynomial distance-regular graphs are thin. Eur. J. Comb. 16, 555-560 (1995)
15. Dickie, G., Terwilliger, P.: A note on thin P-polynomial and dual-thin Q-polynomial symmetric association schemes. J. Algebr. Comb. 7, 5-15 (1998)
16. Egge, E.: A generalization of the Terwilliger algebra. J. Algebra 233, 213-252 (2000)
17. Go, J.T.: The Terwilliger algebra of the hypercube. Eur. J. Comb. 23, 399-429 (2002)
18. Go, J.T., Terwilliger, P.: Tight distance-regular graphs and the subconstituent algebra. Eur. J. Comb. 23, 793-816 (2002)
19. Godsil, C.D.: Algebraic Combinatorics. New York: Chapman and Hall, Inc., 1993
20. Hobart, S.A., Ito, T.: The structure of nonthin irreducible T-modules: ladder bases and classical parameters. J. Algebr. Comb. 7, 53-75 (1998)
21. Ito, T., Tanabe, K., Terwilliger, P.: Some algebra related to P- and Q-polynomi&l association schemes. Codes and Association Schemes (Piscataway NJ, 1999), 167-192, DIMACS Ser. Discrete Math. Theor. Comput. Sci. 56, Amer. Math. Soc., Providence RI 2001.
22. Ito, T., Terwilliger, P.: The shape of a tridiagonal pair. J. Pure Appl. Algebra. 188, 145-160 (2004)
23. Pascasio, A.A.: On the multiplicities of the primitive idempotents of a Q-polynomial distance-regular graph. Eur. J. Comb. 23, 1073-1078 (2002)
24. Pascasio, A.A.: Tight distance-regular graphs and the Q-polynomial property. Graphs Comb. 17, 149-169 (2001)
25. Roos, C.: On antidesigns and designs in association schemes. Delft Progress Rpt. 7, 98-109 (1982)
26. Tanabe, K.: The irreducible modules of the Terwilliger algebras of Doob schemes. J. Algebr. Comb. 6, 173-195 (1997)
27. Terwilliger, P.: Quantum matroids. Progress in Algebraic Combinatorics (Fukuoka, 1993), 323-441, Adv. Stud. Pure Math., 24, Tokyo: Math. Soc. Japan, 1996
28. Terwilliger, P.: The incidence algebra of a uniform poset. Coding Theory and Design Theory, Part I, 193-212, IMA Vol. Math. Appl., 20, New York: Springer, 1990
29. Terwilliger, P.: The subconstituent algebra of an association scheme I. J. Algebr. Comb. 1, 363-388 (1992)
30. Terwilliger, P.: The subconstituent algebra of an association scheme II. J. Algebr. Comb. 2, 73-103 (1993)
31. Terwilliger, P.: The subconstituent algebra of an association scheme III. J. Algebr. Comb. 2, 177-210 (1993)
32. Terwilliger, P.: Two linear transformations each tridiagonal with respect to an eigenbasis of the other. Linear Algebra Appl. 330, 149-203 (2001)
33. Terwilliger, P.: Two relations that generalize the q-Serre relations and the Dolan-Grady relations. In Physics and Combinatorics 1999 (Nagoya), 377-398, World Scientific Publishing, River Edge, NJ, 2001
34. Terwilliger, P.: Leonard pairs from 24 points of view. Rocky Mt. J. Math. 32, 827-888 (2002)
35. Terwilliger, P.: Two linear transformations each tridiagonal with respect to an eigenbasis of the other; the TD-D and the LB-UB canonical form. J. Algebra. Submitted
36. Terwilliger, P.: Introduction to Leonard pairs. OPSFA Rome 2001. J. Comput. Appl. Math. 153, 463-475 (2003)
37. Terwilliger, P.: Introduction to Leonard pairs and Leonard systems. Sürikaisekikenkyüsho Kökyüroku, (1109):67-79, 1999. Algebraic combinatorics (Kyoto, 1999)
38. Terwilliger, P.: Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the split decomposition. OPSFA 7 Copenhagen: 2003. J. Comput. Appl. Math, (to appear)
39. Terwilliger, P.: Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array. Geometric and Algebraic Combinatorics 2, Oisterwijk, The Netherlands 2002. (to appear)
40. Terwilliger, P.: Leonard pairs and the cr-Racah polynomials. Linear Algebra Appl. 387, 235-276 (2004)
41. Terwilliger, P., Vidunas, R.: Leonard pairs and the Askey-Wilson relations. J. Algebra Appl. (to appear)