Theorem 2.5 in the well-known FLOTW paper [1] and Theorem 2.1 in Uglov's paper [2] both refer to the original JMMO paper [3] to define quantum Fock spaces, i.e. Fock spaces for $U_q(\widehat{\mathfrak{sl}}_n)$, the affine quantum group. While Uglov even refers to the FLOTW paper as his source, I do not quite see how their definitions are equivalent: Let $\lambda$ be a multipartition and $v_\lambda$ the corresponding basis vector of a Fock space for a charge $\mathbf{s}=(s_1,\dots, s_r)\in\mathbb{Z}^r$.
In FLOTW, the element $d$ of $U_q(\widehat{\mathfrak{sl}}_n)$ acts as $q^{-N_0(\lambda)}$ on $v_\lambda$, where $N_0(\lambda)$ is the number of nodes of $\lambda$ which have residue $0$.
In Uglov's paper, it seems that this element acts as the scalar $-(\Delta(\mathbf{s})+N_0(\lambda))$.
The summand $\Delta(\mathbf{s})$ is some integer depending on $\mathbf{s}$. This integer is fine, as FLOTW only consider more restrictive $\mathbf{s}$, in which case $\Delta(\mathbf{s})=0$.
What does cause my problem is that in FLOTW the element $d$ acts as a power of $q$ and in Uglov's paper it does not. Why is that?

On a related note: In my understanding, a weight vector of weight $\Lambda$ would in particular be a vector on which $d$ acts as $q^{\Lambda(d)}$. However, this definition is not compatible with the action given in Uglov and the fact that all basis vectors $v_\lambda$ are weight vectors.

[1] FLOTW: https://arxiv.org/pdf/q-alg/9710007.pdf
[2] Uglov: https://arxiv.org/pdf/math/9905196.pdf
[3] JMMO: https://projecteuclid.org/download/pdf_1/euclid.cmp/1104202436


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