Here are two lemmas, one from number theory and one from finite reflection groups.
1) [HW,p.74] Let p be an odd prime. Partition the least nonzero residues (mod p) into positive (P) and negative (N) halves [so N = -P ]. One can compute the Legendre symbol (a/p) by multiplying each element in P by a and then counting up Neg(aP) = number of negative residues appearing in { aP } .
The result is (a/p) = (-1)^Neg(aP) . [Gauss' lemma].
2) [JH,p.14] Let W be a finite reflection group. Partition the simple roots into positive (P) and negative (N) halves [ so N = -P ]. For w in W one can compute sign(w) by applying w to each root in P and then counting up Neg(wP) = number of negative roots in { wP } .
The result is sign(w) = (-1)^Neg(wP) .
Questions:
a) Are there any deeper connections linking these two similar sounding ideas?
b) Do situations similar to the above arise in other areas of math?
References:
[HW] Hardy and Wright , Introduction to the Theory of Numbers .
[JH] Humphreys , Reflection groups and Coxeter groups .