In the proof of Lemma 15.3 in Fulton's Intersection Theory, there appears the formula $$c(\Lambda^\bullet D^\vee \otimes E) = \prod_{p=0}^d \prod_{j=1}^e \prod_{i_1 < \dotsb < i_p} (1 + y_j - x_{i_1} - \dotsc -x_{i_p})^{(-1)^p}.$$ Here $D$ and $E$ are vector bundles of rank $d$ and $e$ on a non-singular variety, with Chern roots $x_1, \dotsc, x_d$ and $y_1, \dotsc, y_e$ respectively. I do not understand where the exponent $(-1)^p$ comes from (even though it is crucial in the proof of the lemma), and I think it should just not be there:
The Chern roots of the dual bundle $D^\vee$ are $\{-x_1, \dotsc, -x_d\}$, and so the Chern roots of $\Lambda^p D^\vee$ are $\{-x_{i_1} - \dotsc - x_{i_p} \mid i_1 < \dotsb < i_p \}$ (I think this is Remark 3.2.3). Hence the Chern roots of $\Lambda^\bullet D^\vee = \bigoplus_{p=0}^d \Lambda^p D^\vee$ should just be $$\bigcup_{p=0}^d \{-x_{i_1} - \dotsc - x_{i_p} \mid i_1 < \dotsb < i_p \},$$ and tensorizing with $E$ just adds the $y_j$'s to those.
Did I do anything wrong here? Any help would be appreciated!
Reading a bit above the lemma, this just seems to be a convention of the definition of $c(\Lambda^\bullet D^\vee \otimes E)$, namely if $F_\bullet$ is a complex (or collection) of vector bundles, then $$c(F_\bullet) := \prod_i c(F_i)^{(-1)^i}.$$ This definition makes the total Chern class into a group homomorphism $$c: K(X) \to A^\times(X) = \left\{1+a_1+a_2+\dotsc \,\mid\, a_i \in A^i(X)\right\}.$$