Let $(\lambda_1 , \cdots , \lambda_d) \vdash k$ be a partition of $k$ of length $d$. Is there any way to decide if $0 \in \text{Conv}\{(\underbrace{\alpha_1, \cdots, \alpha_1}_{\lambda_1}, \cdots , \underbrace{\alpha_d , \cdots, \alpha_d}_{\lambda_d}) \in \mathbb{Z}^k: \, (\alpha_1, \cdots , \alpha_d) \in \mathbb{Z}^d,\, \sum_{i=1}^d \alpha_i = 0\}$?

Let $(\lambda_1 , \cdots , \lambda_d) \vdash k$ be a partition of $k$ of length $d$. Is there any way to decide if $0 \in \text{Conv}\{(\underbrace{\alpha_1, \cdots, \alpha_1}_{\lambda_1}, \cdots , \underbrace{\alpha_d , \cdots, \alpha_d}_{\lambda_d}) \in \mathbb{Z}^k: \, (\alpha_1, \cdots , \alpha_d) \in \mathbb{Z}^d,\, \sum_{i=1}^d \alpha_i = 0\}$, where not all $\alpha_i$s are zero?

Let $(\lambda_1 , \cdots , \lambda_d) \vdash k$ be a partition of $k$ of length $d$ and suppose $\alpha_1, \cdots , \alpha_d$ are integers such that $\sum_{i=1}^d \alpha_i = 0$. Is there any way to decide if $0 \in \text{Conv}(\lambda_i \alpha_i) \subset \mathbb{Z}^d$?

Let $(\lambda_1 , \cdots , \lambda_d) \vdash k$ be a partition of $k$ of length $d$. Is there any way to decide if $0 \in \text{Conv}\{(\underbrace{\alpha_1, \cdots, \alpha_1}_{\lambda_1}, \cdots , \underbrace{\alpha_d , \cdots, \alpha_d}_{\lambda_d}) \in \mathbb{Z}^k: \, (\alpha_1, \cdots , \alpha_d) \in \mathbb{Z}^d,\, \sum_{i=1}^d \alpha_i = 0\}$?