Section 2: Determinants

A linear map L from R^m to R^n induce a linear map Star[L] on forms. Specifically, if L:R^m -> R^n is a linear map and w is a
k-form on R^n, then there is a k-form Star[L][w] on R^m (note: not R^n) such that for all vectors v1, ..., vk in R^m, Star[L][w][v1,...vk] = w[L v1, ... Lvk]. The existence and uniqueness of this form can be deduced from the fact that for any linear functional l:R^p->R where an inner product <.,.> has been defined on R^p, there exists a unique u in R^p such that for all v in R^p, l(v) = <v,u>.

To illustrate this, let's take m=3, n=4, and use the 1-forms f, g and h defined in the previous section, with a new 2-form j.

Input := 


x := {x1,x2,x3};
y := {y1,y2,y3};
z := {z1,z2,z3};
f[x_] := 2 x[[1]] + 3 x[[2]] + x[[3]];
g[x_] := x[[1]] + 4 x[[2]] - x[[3]];
h[x_] := x[[1]] - x[[2]] + 5 x[[3]];
j[x_,y_] := x[[1]] y[[2]] - 5 x[[2]] y[[1]] + 2 x[[3]] y[[3]];

Consider a linear transformation L, represnted as a matrix instead of a function of x.

Input := 

L={{2,6,5},{0,12,-3},{6,-5,-1},{-3,4,9}}

Output =

{{2, 6, 5}, {0, 12, -3}, {6, -5, -1}, {-3, 4, 9}}

Input := 

Simplify[Star[L][f][x]]

Output =

10 x1 + 43 x2

Input := 

Simplify[f[L.x]]

Output =

10 x1 + 43 x2

Input := 

Simplify[Star[L][j][x,y]]

Output =

72 x1 y1 - 180 x2 y1 + 18 x3 y1 - 36 x1 y2 - 238 x2 y2 + 
 
  160 x3 y2 - 18 x1 y3 - 308 x2 y3 + 62 x3 y3

Input := 

Simplify[j[L.x,L.y]]

Output =

72 x1 y1 - 180 x2 y1 + 18 x3 y1 - 36 x1 y2 - 238 x2 y2 + 
 
  160 x3 y2 - 18 x1 y3 - 308 x2 y3 + 62 x3 y3

Input := 

Simplify[Star[L][g^f][x,y]]

Output =

6 (127 x2 y1 - 10 x3 y1 - 127 x1 y2 - 43 x3 y2 + 10 x1 y3 + 
 
    43 x2 y3)

Input := 

Simplify[(g^f)[L.x,L.y]]

Output =

6 (127 x2 y1 - 10 x3 y1 - 127 x1 y2 - 43 x3 y2 + 10 x1 y3 + 
 
    43 x2 y3)

If m=n=k, we have some nice results. So let's set m=n=k=3 and define a new linear map L and use the 3-form f^g^h.

Input := 

Clear[L];
L={{2,6,5},{6,-5,-1},{-3,4,9}};
               

Input := 

Simplify[Star[L][f^(g^h)][x,y,z]]

Output =

5145 (x3 y2 z1 - x2 y3 z1 - x3 y1 z2 + x1 y3 z2 + 
 
    x2 y1 z3 - x1 y2 z3)

Input := 

Simplify[(f^(g^h))[L.x,L.y,L.z]]

Output =

5145 (x3 y2 z1 - x2 y3 z1 - x3 y1 z2 + x1 y3 z2 + 
 
    x2 y1 z3 - x1 y2 z3)

Input := 

Simplify[Det[L] (f^(g^h))[x,y,z]]

Output =

5145 (x3 y2 z1 - x2 y3 z1 - x3 y1 z2 + x1 y3 z2 + 
 
    x2 y1 z3 - x1 y2 z3)

We see that Star[L][w][v1,...vk] = w[L v1, ... Lvk] = Det[L] (w[v1,...vk]) if m=n=k. Hence, in this case the star opeartor amounts to a scalar multiplication.

Input := 

Simplify[Star[L][(f^g)][x,y]]

Output =

7 (53 x2 y1 + 88 x3 y1 - 53 x1 y2 - 40 x3 y2 - 88 x1 y3 + 
 
    40 x2 y3)

Input := 

Simplify[(f^g)[L.x,L.y]]

Output =

7 (53 x2 y1 + 88 x3 y1 - 53 x1 y2 - 40 x3 y2 - 88 x1 y3 + 
 
    40 x2 y3)

Input := 

Simplify[Det[L] (f^g)[x,y]]

Output =

343 (5 x2 y1 - 3 x3 y1 - 5 x1 y2 - 7 x3 y2 + 3 x1 y3 + 
 
    7 x2 y3)

Input := 

Simplify[Star[L][(h^g)][x,y]]

Output =

557 x2 y1 + 1327 x3 y1 - 557 x1 y2 - 670 x3 y2 - 
 
  1327 x1 y3 + 670 x2 y3

Input := 

Simplify[(h^g)[L.x,L.y]]

Output =

557 x2 y1 + 1327 x3 y1 - 557 x1 y2 - 670 x3 y2 - 
 
  1327 x1 y3 + 670 x2 y3

Input := 

Simplify[Det[L] (h^g)[x,y]]

Output =

343 (5 x2 y1 - 6 x3 y1 - 5 x1 y2 - 19 x3 y2 + 6 x1 y3 + 
 
    19 x2 y3)

We see that for m=n=!=k, we have
Star[L][w][v1,...,vk] = w[L v1, ... L vk] =!= Det[L] w[v1, ..., vk].

Some other properties of the star operation:
1. The star operation preserves wedging: Star[L][w1^w2] = (Star[L][w1])^(Star[L][w2]) for arbitary forms w1 and w2.
2. Star[L] is a linear map from the space of k-forms on R^n to the space of k-forms on R^m. (Note that Star[f] acts in the direction opposite to that of f.)
3. For arbitrary linear maps L1 and L2, Star[L1 L2]= Star[L2] Star[L1], where the product denotes composition.

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).