Imprimitive Q-polynomial Association Schemes
Hiroshi Suzuki
DOI: 10.1023/A:1008660421667
Abstract
It is well known that imprimitive P-polynomial association schemes X = ( X,{ R i } 0 \leqslant i \leqslant d ) \mathcal{X} = (X,\{ R_i \} _{0 \leqslant i \leqslant d} ) with $$ " align="middle" border="0"> are either bipartite or antipodal, i.e., intersection numbers satisfy either 
" align="middle" border="0"> are either bipartite or antipodal, i.e., intersection numbers satisfy either i,\text or b i = c d - i i,{\text{or }}b_i = c_{d - i} for all i \textonesuperior [ d/\text2] i \ne [d/{\text{2}}] . In this paper, we show that imprimitive Q Q -polynomial association schemes X = ( X,{ R i } 0 \leqslant i \leqslant d ) \mathcal{X} = (X,\{ R_i \} _{0 \leqslant i \leqslant d} ) with $$ " align="middle" border="0"> are either dual bipartite or dual antipodal, i.e., dual intersection numbers satisfy either 
" align="middle" border="0"> are either dual bipartite or dual antipodal, i.e., dual intersection numbers satisfy either PDF
" align="middle" border="0"> are either bipartite or antipodal, i.e., intersection numbers satisfy either i,\text or b i = c d - i i,{\text{or }}b_i = c_{d - i} for all i \textonesuperior [ d/\text2] i \ne [d/{\text{2}}] . In this paper, we show that imprimitive Q Q -polynomial association schemes X = ( X,{ R i } 0 \leqslant i \leqslant d ) \mathcal{X} = (X,\{ R_i \} _{0 \leqslant i \leqslant d} ) with $$ " align="middle" border="0"> are either dual bipartite or dual antipodal, i.e., dual intersection numbers satisfy either " align="middle" border="0"> are either dual bipartite or dual antipodal, i.e., dual intersection numbers satisfy either PDF
References
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4. G. A. Dickie, “Q-polynomial structures for association schemes and distance-regular graphs,” Ph.D. Thesis, University of Wisconsin, 1995.
5. G. A. Dickie, “A note on Q-polynomial association schemes,” preprint.
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