## The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two.(English)Zbl 1136.05076

Let $$\Gamma$$ be a distance-regular graph with diameter $$D$$, distance matrices $$A_i$$ and eigenvalues $$\theta_0>\theta_1>\dots >\theta_D$$. Fix a base vertex $$x$$ of $$\Gamma$$. Let $$T=T(x)$$ denote the subalgebra of $$Mat_X({\mathbb C})$$ generated by $$A,E_0^*,\dots ,E_D^*$$, where $$A=A_1$$ and $$E_i^*$$ denotes the projection onto the $$i$$th subconstituent of $$\Gamma$$ with respect to $$x$$. $$T$$ is called the Terwilliger algebra of $$\Gamma$$ with respect to $$x$$. There exist the unique irreducible $$T$$-module $$V_0$$ with endpoint 0 ($$V_1$$ with endpoint 1). Both $$V_0$$ and $$V_1$$ are thin. Assume $$\Gamma$$ is bipartite and $$W$$ is thin irreducible $$T$$-module with endpoint 2. Then $$E_2^*W$$ is a one-dimensional eigenspace for $$E_2^*A_2E_2^*$$ with eigenvalue $$\eta$$ and $$\tilde \theta_1\leq \eta\leq \tilde \theta_d$$, where $$\tilde \theta_i=-1-b_2b_3(\theta_i^2-b_2)^{-1}$$ and $$d=[D/2]$$. To describe the structure of $$W$$ we distinguish four cases:
(i)
$$\eta=\tilde \theta_1$$,
(ii)
$$D$$ is odd and $$\eta=\tilde \theta_d$$,
(iii)
$$D$$ is even and $$\eta=\tilde \theta_d$$,
(iv)
$$\tilde \theta_1<\eta<\tilde \theta_d$$.
Cases $$(i)$$ and $$(ii)$$ were investigated earlier. It is showed that $${\text{ dim}}W=D-1-e$$ ($$e=1$$ in the case $$(iii)$$, $$e=0$$ in the case $$(iv)$$), $$W$$ has orthogonal basis $$E_iv (i\in S)$$, where $$v$$ – a nonzero vector of $$E_2^*W$$, $$E_i$$ – primitive idempotent $$A$$, associated with $$\theta_i$$, and $$S=\{1,2,\dots D-1\}-\{d\}$$ in the case $$(iii)$$, $$S=\{1,2,\dots D-1\}$$ in the case $$(iv)$$, moreover $$W$$ has orthogonal basis $$E_{i+2}^*A_iv (0\leq i\leq D-2-e)$$. It is founded square-norm of each vector of two bases and transition matrix relating this two bases.

### MSC:

 05E30 Association schemes, strongly regular graphs 20D15 Finite nilpotent groups, $$p$$-groups
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### References:

 [1] Bannai, E.; Ito, T., Algebraic combinatorics I: association schemes, (1984), Benjamin/Cummings London · Zbl 0555.05019 [2] Biggs, N., Algebraic graph theory, (1994), Cambridge University Press London · Zbl 0797.05032 [3] Brouwer, A.E.; Cohen, A.M.; Neumaier, A., Distance-regular graphs, (1989), Springer Berlin · Zbl 0747.05073 [4] Caughman, J.S., The Terwilliger algebras of bipartite $$P$$- and $$Q$$-polynomial association schemes, Discrete math., 196, 65-95, (1999) · Zbl 0924.05067 [5] Caughman, J.S.; MacLean, M.S.; Terwilliger, P., The Terwilliger algebra of an almost-bipartite $$P$$- and $$Q$$-polynomial association scheme, Discrete math., 292, 17-44, (2005) · Zbl 1063.05135 [6] Caughman, J.S.; Wolff, N., The Terwilliger algebra of a distance-regular graph that supports a spin model, J. alg. combin., 21, 289-310, (2005) · Zbl 1064.05152 [7] Collins, B., The girth of a thin distance-regular graph, Graphs combin., 13, 21-30, (1997) · Zbl 0878.05084 [8] Collins, B., The Terwilliger algebra of an almost-bipartite distance-regular graph and its antipodal 2-cover, Discrete math., 216, 35-69, (2000) · Zbl 0955.05113 [9] Curtin, B., 2-homogeneous bipartite distance-regular graphs, Discrete math., 187, 39-70, (1998) · Zbl 0958.05143 [10] Curtin, B., Bipartite distance-regular graphs I, Graphs combin., 15, 143-158, (1999) · Zbl 0927.05083 [11] Curtin, B., Bipartite distance-regular graphs II, Graphs combin., 15, 377-391, (1999) · Zbl 0939.05088 [12] B. Curtin, Distance-regular graphs which support a spin model are thin, in: 16th British Combinatorial Conference, London, 1997, Discrete Math. 197/198 (1999) 205-216. · Zbl 0929.05095 [13] Curtin, B., Almost 2-homogeneous bipartite distance-regular graphs, European J. combin., 21, 865-876, (2000) · Zbl 1002.05069 [14] Curtin, B., The Terwilliger algebra of a 2-homogeneous bipartite distance-regular graph, J. combin. theory ser. B, 81, 125-141, (2001) · Zbl 1023.05139 [15] Curtin, B., Algebraic characterizations of graph regularity conditions, Des. codes cryptogr., 34, 241-248, (2005) · Zbl 1055.05154 [16] Curtin, B.; Nomura, K., Distance-regular graphs related to the quantum enveloping algebra of $$\mathit{sl}(2)$$, J. alg. combin., 12, 25-36, (2000) · Zbl 0967.05067 [17] Curtin, B.; Nomura, K., 1-homogeneous, pseudo-1-homogeneous, and 1-thin distance-regular graphs, J. combin. theory ser. B, 93, 279-302, (2005) · Zbl 1060.05101 [18] Curtis, C.; Reiner, I., Representation theory of finite groups and associative algebras, (1962), Interscience New York · Zbl 0131.25601 [19] Dickie, G., Twice $$Q$$-polynomial distance-regular graphs are thin, European J. combin., 16, 555-560, (1995) · Zbl 0852.05085 [20] Dickie, G.; Terwilliger, P., A note on thin $$P$$-polynomial and dual-thin $$Q$$-polynomial symmetric association schemes, J. alg. combin., 7, 5-15, (1998) · Zbl 0898.05084 [21] Egge, E., A generalization of the Terwilliger algebra, J. algebra, 233, 213-252, (2000) · Zbl 0960.05108 [22] Egge, E., The generalized Terwilliger algebra and its finite dimensional modules when $$d = 2$$, J. algebra, 250, 178-216, (2002) · Zbl 1003.05107 [23] Go, J.T., The Terwilliger algebra of the hypercube $$Q_D$$, European J. combin., 23, 399-429, (2002) · Zbl 0997.05097 [24] Go, J.T.; Terwilliger, P., Tight distance-regular graphs and the subconstituent algebra, European J. combin., 23, 793-816, (2002) · Zbl 1014.05070 [25] Godsil, C.D., Algebraic combinatorics, (1993), Chapman & Hall Inc., New York · Zbl 0814.05075 [26] Hobart, S.A.; Ito, T., The structure of nonthin irreducible $$T$$-modules: ladder bases and classical parameters, J. alg. combin., 7, 53-75, (1998) · Zbl 0911.05059 [27] Jurišić, A.; Koolen, J.; Terwilliger, P., Tight distance-regular graphs, J. alg. combin., 12, 163-197, (2000) · Zbl 0959.05121 [28] MacLean, M.S.; Terwilliger, P., Taut distance-regular graphs and the subconstituent algebra, Discrete math., 306, 1694-1721, (2006) · Zbl 1100.05104 [29] Pascasio, A.A., On the multiplicities of the primitive idempotents of a $$Q$$-polynomial distance-regular graph, European J. combin., 23, 1073-1078, (2002) · Zbl 1017.05107 [30] Tanabe, K., The irreducible modules of the Terwilliger algebras of Doob schemes, J. alg. combin., 6, 173-195, (1997) · Zbl 0868.05056 [31] Terwilliger, P., The subconstituent algebra of an association scheme I, J. alg. combin., 1, 363-388, (1992) · Zbl 0785.05089 [32] Terwilliger, P., The subconstituent algebra of an association scheme II, J. alg. combin., 2, 73-103, (1993) · Zbl 0785.05090 [33] Terwilliger, P., The subconstituent algebra of an association scheme III, J. alg. combin., 3, 177-210, (1993) · Zbl 0785.05091 [34] Terwilliger, P., The subconstituent algebra of a distance-regular graph; thin modules with endpoint one, Linear algebra appl., 356, 157-187, (2002) · Zbl 1011.05066 [35] Terwilliger, P., An inequality involving the local eigenvalues of a distance-regular graph, J. alg. combin., 19, 143-172, (2004) · Zbl 1047.05045 [36] Tomiyama, M.; Yamazaki, N., The subconstituent algebra of a strongly regular graph, Kyushu J. math., 48, 323-334, (1998) · Zbl 0842.05098
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