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The Michigan Mathematical Journal

Distance-regular graphs of q-Racah type and the q-tetrahedron algebra

Tatsuro Ito and Paul Terwilliger

Source: Michigan Math. J. Volume 58, Issue 1 (2009), 241-254.

Primary Subjects: 05E30
Secondary Subjects: 05E35, 17B37

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Permanent link to this document: http://projecteuclid.org/euclid.mmj/1242071691
Digital Object Identifier: doi:10.1307/mmj/1242071691
Mathematical Reviews number (MathSciNet): MR2526086
Zentralblatt MATH identifier: 05566081

References

E. Bannai and T. Ito, Algebraic combinatorics I: Association schemes, Benjamin/Cummings, Menlo Park, CA, 1984.
Mathematical Reviews (MathSciNet): MR882540
N. Biggs, Algebraic graph theory, 2nd ed., Cambridge Math. Lib., Cambridge Univ. Press, Cambridge, 1993.
Mathematical Reviews (MathSciNet): MR1271140
A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-regular graphs, Ergeb. Math. Grenzgeb. (3), 18, Springer-Verlag, Berlin, 1989.
Mathematical Reviews (MathSciNet): MR1002568
A. E. Brouwer, C. D. Godsil, J. H. Koolen, and W. J. Martin, Width and dual width of subsets in polynomial association schemes, J. Combin. Theory Ser. A 102 (2003), 255--271.
Mathematical Reviews (MathSciNet): MR1979532
Digital Object Identifier: doi:10.1016/S0097-3165(03)00006-2
Zentralblatt MATH: 1018.05108
J. S. Caughman IV, Spectra of bipartite $P$- and $Q$-polynomial association schemes, Graphs Combin. 14 (1998), 321--343.
Mathematical Reviews (MathSciNet): MR1658861
Digital Object Identifier: doi:10.1007/s003730050034
Zentralblatt MATH: 0917.05088
------, The Terwilliger algebras of bipartite $P$- and $Q$-polynomial association schemes, Discrete Math. 196 (1999), 65--95.
Mathematical Reviews (MathSciNet): MR1664514
Digital Object Identifier: doi:10.1016/S0012-365X(98)00196-4
Zentralblatt MATH: 0924.05067
V. Chari and A. Pressley, Quantum affine algebras, Comm. Math. Phys. 142 (1991), 261--283.
Mathematical Reviews (MathSciNet): MR1137064
Digital Object Identifier: doi:10.1007/BF02102063
Project Euclid: euclid.cmp/1104248585
B. Curtin, 2-homogeneous bipartite distance-regular graphs, Discrete Math. 187 (1998), 39--70.
Mathematical Reviews (MathSciNet): MR1630676
Digital Object Identifier: doi:10.1016/S0012-365X(97)00226-4
Zentralblatt MATH: 0958.05143
------, Bipartite distance-regular graphs I, Graphs Combin. 15 (1999), 143--158.
Mathematical Reviews (MathSciNet): MR1701778
Digital Object Identifier: doi:10.1007/s003730050049
Zentralblatt MATH: 0927.05083
------, Bipartite distance-regular graphs II, Graphs Combin. 15 (1999), 377--391.
Mathematical Reviews (MathSciNet): MR1735086
Digital Object Identifier: doi:10.1007/s003730050072
Zentralblatt MATH: 0939.05088
------, Distance-regular graphs which support a spin model are thin, 16th British combinatorial conference (London, 1997), Discrete Math. 197/198 (1999), 205--216.
Mathematical Reviews (MathSciNet): MR1674863
Zentralblatt MATH: 0929.05095
B. Curtin and K. Nomura, Distance-regular graphs related to the quantum enveloping algebra of $\text\rm sl(2),$ J. Algebraic Combin. 12 (2000), 25--36.
Mathematical Reviews (MathSciNet): MR1791444
Digital Object Identifier: doi:10.1023/A:1008707417118
C. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure Appl. Math., 11, Wiley, New York, 1962.
Mathematical Reviews (MathSciNet): MR144979
G. Dickie, Twice $Q$-polynomial distance-regular graphs are thin, European J. Combin. 16 (1995), 555--560.
Mathematical Reviews (MathSciNet): MR1356846
Digital Object Identifier: doi:10.1016/0195-6698(95)90037-3
Zentralblatt MATH: 0852.05085
G. Dickie and P. Terwilliger, A note on thin $P$-polynomial and dual-thin $Q$-polynomial symmetric association schemes, J. Algebraic Combin. 7 (1998), 5--15.
Mathematical Reviews (MathSciNet): MR1606159
Digital Object Identifier: doi:10.1023/A:1008690026999
E. Egge, A generalization of the Terwilliger algebra, J. Algebra 233 (2000), 213--252.
Mathematical Reviews (MathSciNet): MR1793596
Digital Object Identifier: doi:10.1006/jabr.2000.8420
Zentralblatt MATH: 0960.05108
J. T. Go, The Terwilliger algebra of the hypercube, European J. Combin. 23 (2002), 399--429.
Mathematical Reviews (MathSciNet): MR1914480
Digital Object Identifier: doi:10.1006/eujc.2000.0514
Zentralblatt MATH: 0997.05097
J. T. Go and P. Terwilliger, Tight distance-regular graphs and the subconstituent algebra, European J. Combin. 23 (2002), 793--816.
Mathematical Reviews (MathSciNet): MR1932680
Digital Object Identifier: doi:10.1006/eujc.2002.0597
Zentralblatt MATH: 1014.05070
C. D. Godsil, Algebraic combinatorics, Chapman & Hall, New York, 1993.
Mathematical Reviews (MathSciNet): MR1220704
B. Hartwig and P. Terwilliger, The Tetrahedron algebra, the Onsager algebra, and the $\frak s\frak l_2$ loop algebra, J. Algebra 308 (2007), 840--863.
Mathematical Reviews (MathSciNet): MR2295093
Digital Object Identifier: doi:10.1016/j.jalgebra.2006.09.011
Zentralblatt MATH: 1163.17026
S. A. Hobart and T. Ito, The structure of nonthin irreducible $T$-modules of endpoint 1: Ladder bases and classical parameters, J. Algebraic Combin. 7 (1998), 53--75.
Mathematical Reviews (MathSciNet): MR1606168
Digital Object Identifier: doi:10.1023/A:1008619211978
T. Ito, K. Tanabe, and P. Terwilliger, Some algebra related to $P$- and $Q$-polynomial association schemes, Codes and association schemes (Piscataway, 1999), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 56, pp. 167--192, Amer. Math. Soc., Providence, RI, 2001.
Mathematical Reviews (MathSciNet): MR1816397
Zentralblatt MATH: 0995.05148
T. Ito and P. Terwilliger, Tridiagonal pairs and the quantum affine algebra $U_q(\widehatsl_2),$ Ramanujan J. 13 (2007), 39--62.
Mathematical Reviews (MathSciNet): MR2281156
Digital Object Identifier: doi:10.1007/s11139-006-0242-4
------, The $q$-tetrahedron algebra and its finite-dimensional irreducible modules, Comm. Algebra 35 (2007), 3415--3439.
Mathematical Reviews (MathSciNet): MR2362663
Digital Object Identifier: doi:10.1080/00927870701509180
Zentralblatt MATH: 1133.17011
------, $q$-inverting pairs of linear transformations and the $q$-tetrahedron algebra, Linear Algebra Appl. 426 (2007), 516--532.
Mathematical Reviews (MathSciNet): MR2350675
Digital Object Identifier: doi:10.1016/j.laa.2007.05.031
Zentralblatt MATH: 1146.17009
------, Distance-regular graphs and the $q$-tetrahedron algebra, European J. Combin. (to appear), $\langle$arXiv:math.CO/0608694$\rangle.$
Mathematical Reviews (MathSciNet): MR2494442
Zentralblatt MATH: 05541742
Digital Object Identifier: doi:10.1016/j.ejc.2008.07.011
------, Evaluation modules for the $q$-tetrahedron algebra (in preparation).
J. Kim, A duality between pairs of split decompositions for a $Q$-polynomial distance-regular graph, Ars Math. Contemp. (submitted).
G. Lusztig, Introduction to quantum groups, Birkhäuser, Boston, 1990.
Mathematical Reviews (MathSciNet): MR1227098
Zentralblatt MATH: 0788.17010
A. A. Pascasio, On the multiplicities of the primitive idempotents of a $Q$-polynomial distance-regular graph, European J. Combin. 23 (2002), 1073--1078.
Mathematical Reviews (MathSciNet): MR1938361
Digital Object Identifier: doi:10.1006/eujc.2002.0607
Zentralblatt MATH: 1017.05107
K. Tanabe, The irreducible modules of the Terwilliger algebras of Doob schemes, J. Algebraic Combin. 6 (1997), 173--195.
Mathematical Reviews (MathSciNet): MR1436534
Digital Object Identifier: doi:10.1023/A:1008647205853
P. Terwilliger, The subconstituent algebra of an association scheme I, J. Algebraic Combin. 1 (1992), 363--388.
Mathematical Reviews (MathSciNet): MR1203683
Digital Object Identifier: doi:10.1023/A:1022494701663
------, The subconstituent algebra of an association scheme II, J. Algebraic Combin. 2 (1993), 73--103.
Mathematical Reviews (MathSciNet): MR1210403
Digital Object Identifier: doi:10.1023/A:1022480715311
------, The subconstituent algebra of an association scheme III, J. Algebraic Combin. 2 (1993), 177--210.
Mathematical Reviews (MathSciNet): MR1229433
Digital Object Identifier: doi:10.1023/A:1022415825656
------, Leonard pairs and the $q$-Racah polynomials, Linear Algebra Appl. 387 (2004), 235--276.
Mathematical Reviews (MathSciNet): MR2069278
Digital Object Identifier: doi:10.1016/j.laa.2004.02.014
Zentralblatt MATH: 1075.05090
------, Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array, Des. Codes Cryptogr. 34 (2005) 307--332.
Mathematical Reviews (MathSciNet): MR2128338
Digital Object Identifier: doi:10.1007/s10623-004-4862-7
------, The displacement and split decompositions for a $Q$-polynomial distance-regular graph, Graphs Combin. 21 (2005), 263--276.
Mathematical Reviews (MathSciNet): MR2191022
Digital Object Identifier: doi:10.1007/s00373-004-0594-8
Zentralblatt MATH: 1065.05097
------, The equitable presentation for the quantum group $U_q(\frak g)$ associated with a symmetrizable Kac--Moody algebra $\frak g,$ J. Algebra 298 (2006), 302--319.
Mathematical Reviews (MathSciNet): MR2215130
Digital Object Identifier: doi:10.1016/j.jalgebra.2005.11.013
Zentralblatt MATH: 1106.17021

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