R. Merris, The dimensions of certain symmetry classes of tensors II, Linear & Multilinear Algebra 4 (1976), 205-207. Math Reviews 55 #2967. Research supported by NSF grant MPS 75-05799.

If lambda is an (ordinary) irreducible character of the full symmetric group Sm of degree m, then lambda corresponds to a partition p = (p1,p2, ... ,pr) of m, where p1 >= p2 >= ... >= pr > 0. Denote by Q(x) the product, as i goes from 1 to r, of the product, as j goes from 1 to pi, of (x - i + j). If, e.g., lambda = (4,3,1), then

Q(x) = (x-2)(x-1)x^2(x+1)^2(x+2)(x+3)

Suppose G is some fixed but arbitrary permutation group of degree m. Let chi be an irreducible character of G and denote by (chi,lambda) the number of occurrences of chi in the restriction of lambda to G. If V is a vector space of dimension dim(V) = n, denote by V[G,chi] the symmetry class of tensors corresponding to G and chi. Then the main result of the article is that dim(V[G,chi]) is the sum, over the irreducible characters lambda of Sm, of

chi(id)lambda(id)(chi,lambda)Q(n)/m!.

If chi = 1, the principal (identically 1) character of G, then dim(V[G,1]) is the number of color patterns enumerated by Polya's theory of counting. Thus, [R. Merris, Pattern enumeration and Young diagrams, AMS Notices 25 (1978), A-571] the results of this article may be viewed as an alternative to Polya's theory.

Among the publications citing this article are:

  • G. H. Chan and M. H. Lim, Canad. J. Math. 32 (1980), 957-968.
  • G. H. Chan and M. H. Lim, Linear & Multilinear Algebra 19 (1986), 1-10.
  • M. Shahryari and M. A. Shahabi Shojaei, Pure Math & Appl. 7 (1996), 323-333.
  • M. R. Darafsheh and M. R. Pournaki, J. Algebra 205 (1998), 317-325.
  • M. R. Darafsheh and M. R. Pournaki, Applicable Algebra in Engr., Commun., and Comput. 19 (2000), 237-250.