Appendix: On some Gelfand pairs and commutative association schemes

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• 作者：Eiichi Bannai (1)
  Hajime Tanaka (2)
  
  1. Department of Mathematics ; Shanghai Jiao Tong University ; Shanghai ; 200240 ; China
  2. Research Center for Pure and Applied Mathematics ; Graduate School of Information Sciences ; Tohoku University ; Sendai ; 980-8579 ; Japan
  
• 关键词：Gelfand pair ; commutative association scheme ; 20B05 ; 20G40 ; 05E30
• 刊名：Japanese Journal of Mathematics
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：10
• 期：1
• 页码：97-104
• 全文大小：97 KB
• 参考文献：1. E. Bannai, Character tables of commutative association schemes, In: Finite Geometries, Buildings, and Related Topics, (eds. W.M. Kantor et聽al.), Oxford Sci. Publ., Oxford Univ. Press, New York, 1990, pp. 105鈥?28.
  2. Bannai, E., Hao, S., Song, S.-Y. (1990) Character tables of the association schemes of finite orthogonal groups acting on the nonisotropic points. J Combin Theory Ser A. 54: pp. 164-200 CrossRef
  3. Bannai, E., Ito, T. (1984) Algebraic Combinatorics. I. Association Schemes. Benjamin/Cummings, Menlo Park, CA
  4. Bannai, E., Kawanaka, N., Song, S.-Y. (1990) The character table of the Hecke algebra $${\fancyscript{H}({\rm GL}_{2n}(\mathbf{F}_q),{\rm Sp}_{2n}(\mathbf{F}_q))}$$ H ( GL 2 n ( F q ) , Sp 2 n ( F q ) ) , J. Algebra, 129: pp. 320-366 CrossRef
  5. Bannai, E., Shimabukuro, O., Tanaka, H. (2004) Finite analogues of non-Euclidean spaces and Ramanujan graphs, European J. Combin., 25: pp. 243-259 CrossRef
  6. Bannai, E., Song, S.-Y., Hao, S., Wei, H.Z. (1991) Character tables of certain association schemes coming from finite unitary and symplectic groups. J. Algebra. 144: pp. 189-213 CrossRef
  7. E. Bannai and H. Tanaka, The decomposition of the permutation character \({1_{GL(n,q^2)}^{GL(2n,q)}}\) , J. Algebra, 265 (2003), 496鈥?12.
  8. T. Ceccherini-Silberstein, F. Scarabotti and F. Tolli, Mackey鈥檚 theory of \({\tau}\) -conjugate representations for finite groups, Jpn. J. Math., 10 (2015). doi: 10.1007/s11537-014-1390-8
  9. Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A. (1985) Atlas of Finite Groups, Oxford Univ. Press, Eynsham.
  10. Gow, R. (1984) Two multiplicity-free permutation representations of the general linear group $${GL(n,q^2)}$$ G L ( n , q 2 ). Math. Z., 188: pp. 45-54 CrossRef
  11. Green, J.A. (1955) The characters of the finite general linear groups, Trans. Amer. Math. Soc. 80: pp. 402-447 CrossRef
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  13. N.F.J. Inglis, Multiplicity-free permutation characters, distance transitive graphs and classical groups. Ph. D. thesis, Cambridge 1988
  14. N.F.J. Inglis, M.W. Liebeck and J. Saxl, Multiplicity-free permutation representations of finite linear groups, Math. Z., 192 (1986), 329鈥?37.
  15. Macdonald, I.G. (1995) Symmetric Functions and Hall Polynomials. Oxford Math. Monogr. Oxford Univ. Press, New York
  16. J. Saxl, On multiplicity-free permutation representations, In: Finite Geometries and Designs, (eds. P.J. Cameron et聽al.), London Math. Soc. Lecture Note Ser., 49, Cambridge Univ. Press, Cambridge, 1981, pp. 337鈥?53.
  17. H. Tanaka, Some results on the multiplicity-free permutation group \({{\rm GL}(4, q)}\) acting on \({{\rm GL}(4, q)/{\rm GL}(2, q^2)}\) . (Japanese), In: Codes, Lattices, Vertex Operator Algebras and Finite Groups, RIMS K么ky没roku, 1228, Kyoto Univ., 2001, pp. 127鈥?39.
  18. A. Terras, Fourier Analysis on Finite Groups and Applications, London Math. Soc. Stud. Texts, 43, Cambridge Univ. Press, Cambridge, 1999.
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  History of Mathematics
  
• 出版者：Springer Japan
• ISSN：1861-3624

文摘

This paper is Appendix of the paper of T. Ceccherini-Silberstein, F. Scarabotti and F. Tolli [8]. We pay close attention on a special condition related to Gelfand pairs. Namely, we call a finite group G and its automorphism \({\sigma}\) satisfy Condition ( \({\bigstar}\) ) if the following condition is satisfied: if for \({x,\,y\,\in G}\) , \({x\,\cdot\, x^{-\sigma}}\) and \({y\,\cdot\, y^{-\sigma}}\) are conjugate in G, then they are conjugate in \({K=C_G(\sigma)}\) . The main purpose of the note was to study the meanings of this condition, as well as showing many examples of G and \({\sigma}\) which do (or do not) satisfy Condition ( \({\bigstar}\) ).