m m
. Sp where P = I aP, Is the
1 1=1
representation (2.1) of P In terms of elemental paths. Alter P by
Proof. By (2.2), Sp = £ c^S p wher e P = £ a ^
replacing a consecutive pair of jumps, the first of which is vertical and
the second horizontal, by a horizontal jump followed by a vertical jump.
By the fundamental relation (2.4), Sp is unchanged. Repetition yields
a path all of whose horizontal jumps precede its vertical jumps. Since
S, pv + Sp = 0, we may discard a consecutive pair of horizontal (or
vertical) jumps with opposite orientations without altering Sp. Note
that s(P) and e(P) are also unchanged. Repetition yields a path with
horizontal jumps (all with the same direction) followed by vertical jumps
(all with the same direction). Since s(P) and e(P) determine this
final path, they in fact determine Sp.
A closed path P is one with s(P) = e(P). If the fundamental
relation (2.4) holds, then we may alternatively conclude that Sp = 0
whenever P is closed.
Let n be a given infinite product. The q-difference equations
satisfied by I I are of the form I I = ri(H). The operators n are linear
and they form a group G(n). Our operators have actions which may be
interpreted geometrically: they act on Sp by moving the underlying
path P isometrically. This feature, derived constructively in Sections
3 and 4, is essential to the development of path functions In Section 6.
By moving the entire square (2.5), the linear operator r\ associated to a
q-difference equation moves the fundamental relation (2.4).
pn m
be the path whose itinerary is obtained by adding (n,m)
r 0 1
to the points visited by P. In particular, let P = P ' and
P = P ' be obtained by pushing P one unit to the right or downward,
respectively. An example follows.
t +
p0,1 . .
••; ••}
d _
i,o -t U.
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