Knowlpedia

Definition

Koszul sign rule

The sign convention in which exchanging homogeneous factors of parities p and q contributes (-1)^(pq).

In the , the Koszul sign rule assigns the sign

(−1)∣v∣∣w∣(-1)^{|v||w|}

whenever homogeneous pieces vv and ww are interchanged. It is the symmetry

v⊗w⟼(−1)∣v∣∣w∣w⊗vv\otimes w\longmapsto(-1)^{|v||w|}w\otimes v

of the , rather than a correction added after a calculation.

For homogeneous maps f:V→V′f:V\to V' and g:W→W′g:W\to W', the is therefore evaluated by

(f⊗g)(v⊗w)=(−1)∣g∣∣v∣f(v)⊗g(w).(f\otimes g)(v\otimes w) =(-1)^{|g||v|}f(v)\otimes g(w).

In particular, two odd pieces acquire a minus sign when exchanged, while an even piece can cross any homogeneous piece without a sign.

References
  1. P. Deligne and J. W. Morgan, “Notes on supersymmetry (following Joseph Bernstein),” in Quantum Fields and Strings: A Course for Mathematicians, Volume 1, American Mathematical Society, 1999, pp. 41–97. Relevant: Sections 1–2.
  2. C. Carmeli, L. Caston, and R. Fioresi, Mathematical Foundations of Supersymmetry, European Mathematical Society, 2011. Publisher record. Relevant: Chapter 1.