An algorithm for the second immanant

TLDR: In this paper, an algorithm for computing d(A) when X corresponds to the partition (2, n-2) was presented. But it is not known whether the algorithm can be used to compute a generalized matrix function.

Abstract: Let X be an irreducible character of the symmetric group S,. For A = (a,J) an n-by-n matrix, define the immanant of A corresponding to X by d(A) = E X(gf)Ha[lt uGSn t=1 The article contains an algorithm for computing d(A) when X corresponds to the partition (2, n-2). Introduction. Denote by Xk the (irreducible, characteristic zero) character of the symmetric group Sn, corresponding to the partition (k, In-k), for k = 1, 2,.. ,n. If A = (ai1) is an n-by-n matrix, define n dk(A) = , Xk(Uf) H at,(t) a(S, t=l Then, for example, dl(A) = det(A) and dn(A) = per(A), the permanent of A. In general, dk is known as an immanant or a generalized matrix function. (An immanant is a generalized matrix function based on Sn.) Suppose G is a (simple) graph on n vertices. Denote by L(G) the Laplacian matrix corresponding to some labeling of the vertices of G, i.e., L(G) is an n-by-n matrix, the (i, j) entry of which is the degree of vertex i when i = j, -1 if i 7 j but vertex i is adjacent to vertex j, and zero otherwise. It is shown in [5] that the number of Hamiltonian circuits in G is given by the formula I n k (1) ~h(G)= (-1) dk(L(G)). 2nk= 2 While there is an immense literature on generalized matrix functions, Eq. (1) is already sufficient motivation to seek "fast" algorithms for their actual computation. The main result of this note is an algorithm for computing d2. (See the next section.) It seems that d2 may be especially appropriate for the study of Laplacian matrices for the following reason: If G is a graph on n vertices, then L(G) is positive semidefinite symmetric and singular. Moreover, G is connected if and only if rank L(G) = n 1. For arbitrary positive semidefinite symmetric matrices without a zero row, it was established in [3, Corollaries 5 and 6] that d2(A) > 0 with equality if and only if rank(A) < n 1. Received August 22, 1983. 1980 Mathematics Subject Classification. Primary 65F30, 15A15, 05C50. *Work of the second author was supported by the National Science Foundation under Grant No. MCS-8300097. ?01984 American Mathematical Society 0025-5718/84 $1.00 + $.25 per page