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 <Pine.GSO.3.95.990615202951.7822D-100000@esther.rad.tju.edu>
"Dr. Michael Albert" <albert@esther.rad.tju.edu> 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.