Graphs and Combinatorics

, Volume 9, Issue 2–4, pp 117–134 | Cite as

A problem related to Foulkes’s conjecture

  • C. Coker
Graphs and Combinatoric


In this paper we study a class of symmetric matricesT indexed by positive integers m≥ n≥2 and defined as follows: for any positive integersp andq let ℬp,q be the set of partitions ofU = {1,2,3, ...,pq} into p blocks each of sizeq. Letmn ≥ 2 be positive integers. By atransversal of α = A1/A2/.../An ∈ ℬn,m we mean a partitionß = B1/B2/.../Bm m,n such that ‖A i B j = 1 for every i= 1,2, ...,n and everyj = 1,2, ...,m. LetM be the zero-one matrix with rows indexed by the elements of ℬn,m and columns indexed by the elements of ℬm,n such that Mαß = 1 iffß is a transversal of α. We are interested in finding the eigenvalues and eigenspaces of the symmetric matrixT = MMt. The nonsingularity ofT implies Foulkes’s Conjecture (for these values of m andn). In the casen = 2 we completely determine the eigenvalues and eigenspaces of T and in so doing demonstrate the non-singularity ofT. Forn = 3 we develop a fast algorithm for computing the eigenvalues ofT, and give numerical results in the cases m = 3,4, 5, 6.


Symmetric Group Irreducible Character Wreath Product Association Scheme Real Vector Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Black, S., List, R.: A note on plethysm. Eur. J. Comb.10 (1), 111–112 (1989)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Delsarte, P.: Hahn polynomials, discrete harmonics and t-designs. SIAM J. Appl. Math.34, 157–166 (1978)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Foulkes, H. O.: Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form. J. London Math. Soc.25, 205–209 (1950)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gould, H. W.: Combinatorial identities, Morgantown, West Virginia: Morgantown Printing and Binding 1972zbMATHGoogle Scholar
  5. 5.
    Hanlon, P., Wales, D.: Eigenvalues connected with Brauer’s centralizer algebras. J. Algebra121 (2), 446–476 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Horn, R., Johnson, C.: Matrix analysis Cambridge: Cambridge University Press 1985zbMATHGoogle Scholar
  7. 7.
    James, G., Kerber, A.: The representation theory of the symmetric group, Reading, MA: Addison-Wesley 1981zbMATHGoogle Scholar
  8. 8.
    Littlewood, D.E.: Invariant theory, tensors and group characters. Philos Trans. R. Soc. Lond.293, 305–365 (1944)MathSciNetGoogle Scholar
  9. 9.
    Roy, R.: Binomial identities and hypergeometric series. Am. Math. Mon.94, 36–45 (1987)zbMATHCrossRefGoogle Scholar
  10. 10.
    Sloane, N.J.A.: An introduction to association schemes and special functions. In: Askey, R. (ed.): Theory and applications of special functions, pp. 225–260 New York: Academic Press 1975Google Scholar
  11. 11.
    Thrall, R.M.: On symmetrized Kronecker powers and the structure of the free Lie ring. Am. J. Math.64, 371–388 (1942)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • C. Coker
    • 1
  1. 1.SUNY College at OneontaOneontaUSA

Personalised recommendations