R. Merris, On vanishing decomposable symmetrized tensors, Linear & Multilinear Algebra 5 (1977), 79-86. AMS Notices 24 (1977), A-342. Math. Reviews 58 #28048. Research supported by NSF grant MCS 76-05946.

Let chi be an (ordinary) irreducible character of the symmetric group S_m. Then chi corresponds to a partition lambda = (p1,p2, ... ,pt) of m, where p1 >= p2 >= ... >= pt > 0. In this article, the main result of publication 32 is applied to the immanant corresponding to S_m and chi. This immanant, call it f, is a complex valued function of the m-by-m matrices defined by f(A) = the sum, as s ranges over S_m, of chi(s) times the product of the elements on the diagonal of A corresponding to s. For example, if t = m and p1 = p2 = ... = pm = 1, then chi = epsilon is the alternating (signum) character, and f = det is the determinant. The two applications of most interest are these: (1) If A has more than p1 equal rows, then f(A) = 0, generalizing the well known fact that the det(A) = 0 for any matrix A having two equal rows; and (2) if rank(A) < t, then f(A) = 0, generalizing the equally well known fact that the det(A) = 0 for any singular matrix A. Among the publications citing this article are:

  • G. H. Chan, Linear Algebra Appl. 25 (1979), 139-149.
  • G. H. Chan, Nanta Math. 12 (1979), 7-15.
  • G. H. Chan and M. H. Lim, Canad. J. Math. 32 (1980), 957-968.
  • R. Grone, Linear Algebra Appl. 68 (1985), 252-254.
  • W. Hartmann, Linear & Multilinear Algebra 18 (1985), 127-140.
  • G. H. Chan and M. H. Lim, Linear & Multilinear Algebra 19 (1986), 1-10.
  • _____, Linear & Multilinear Algebra 19 (1986), 149-165.
  • J. A. Dias da Silva and A. Fonseca, Linear & Multilinear Algebra 20 (1987), 203-218.
  • C. Gamas, Linear Algebra Appl. 108, (1988), 83-119.
  • T. H. Pate, Linear & Multilinear Algebra 28 (1990), 175-184.
  • P. Burgisser, SIAM J. Computing 30 (2000), 1010-1022.
  • P. Burgisser, SIAM J. Computing 30 (2000), 1023-1040.