R. Merris, Generalized matrix functions: a research problem, Linear & Multilinear Algebra 8 (1979), 83-86. Research supported by NSF grant MCS 77-28437. Math Reviews 80j: 15005.

Suppose n is a fixed positive integer. Let d be a generalized matrix function (GMF) of n-by-n matrices based on an irreducible (complex) character of a finite permutation group G of degree n. Then d is said to be of

  • class 1 if there is an invertible matrix A such that d(A) = 0 >< d(A^-1).
  • class 2 if it is not of class 1.
  • class MPW if d(A)det(A^-1) = d(A^-1)det(A), for all invertible A.

    • The main point of the article is to introduce the problem of characterizing class MPW in terms of permutation groups and characters. When G is the symmetric group of degree n, the problem was solved in [R. Merris, Representations of GL(n,R) and generalized matrix functions of class MPW, Linear & Multilinear Algebra 11 (1982), 131-141]. The solution involves characters associated with partitions of n whose largest part is at most 2, the so-called "unsaturated" characters.* The rest of the problem was disposed of in a series of papers (below) culminating in [L. B. Beasley, Generalized matrix functions of class MPW, iii, Linear & Multilinear Algebra 15 (1984), 175-186], where it was shown that, in fact, class 2 = class MPW.** Among the publications citing this article are

  • W. Taylor, J. Pure Appl. Math. 21 (1981), 75-98.
  • L. B. Beasley and L. J. Cummings, Linear & Multilinear Algebra 11 (1982), 23-31.
  • L. B. Beasley and L. J. Cummings, Proc. Amer. Math. Soc. 87 (1983), 229-232.
  • L. B. Beasley, Linear & Multilinear Algebra 12 (1983), 273-280.
  • L. B. Beasley, Linear & Multilinear Algebra 13 (1983), 87-95.
  • L. B. Beasley, Linear & Multilinear Algebra 15 (1984), 175-186.
  • R. D. Poshusta, International J. Quantum Chemistry 25 (1991), 225-234.
  • X. J. Wang, J. Huazhong Univ. Sci. Tech. 23 (1995), 124-128.

    • * In considering symmetries of spacial functions for a system of electrons, I. V. Schensted [A Course on the Applications of Group Theory to Quantum Mechanics, NEO Press, Peaks Island, 1976] remarked that "the Pauli Principle permits only [those] symmetries corresponding to [unsaturated characters]."

    ** It had been asserted in the original article that class 2 >< class MPW.