From: mckay@cs.concordia.ca (MCKAY john) Subject: Immanents: (was) Generalization of even/odd permutations? Date: 16 Jun 1999 11:34:42 GMT Newsgroups: sci.math Keywords: determinants, permanents, immanents of a matrix In article "Dr. Michael Albert" writes: >One way to view this is that the even/odd assignment >corresponds to a group homomorphism from the >group of permutations onto Z_2 (the group of two >elements). One might be tempted to look for >other homomorphisms into other groups. However, >for n>=5 the alternating group (the group of even >permutations) is known to be "simple" (allows >no non-trivial homomorphisms) so the group of >all permutatoins permits no no-trivial homomorphisms >besides this the one which assigns even or odd parity >(ie, Z_2={even, odd}). > >Thus generalization in this direction is seen to be >blocked by the classical result. Right - BUT you can have fun defining generalizations of the determinant: The determinant of an n x n matrix is the SIGNED sum of all products of n (distinct) terms. The sign is the sign (+1 if even, -1 if odd) of the permutation of the second subcripts of the entries (when the first subscripts are in order). If this SIGN is replaced by +1 we obtain the permanent. We have, more generally, the immanents of a matrix. They are products as above with the SIGN replaced by a character of the symmetric group. Example (3 x 3): a11 a12 a13 a21 a22 a23 a31 a32 a33 Determinant: a11.a22.a33-a11.a23.a32-a12.a21.a33+a12.a23.a31+a13.a21.a32-a13.a22.a31 Permanent: replace all SIGNs by +1 in the determinant. and [if you are really into it!] here is the other immanent for 3x3: (2,1)-Immanent: 2a11.a22.a33-a12.a23.a31-a13.a21.a32 where the coefficients are [2,0,-1] for permutations of shape (lengths of disjoint cycles) [111,21,3] respectively. John McKay -- But leave the wise to wrangle, and with me the quarrel of the universe let be; and, in some corner of the hubbub couched, make game of that which makes as much of thee.