Trivial representation of $SL_n$ in $V(\alpha_1)\otimes \cdots \otimes V(\alpha_m)$

Question

Suppose $\alpha_1,\cdots, \alpha_m$ are partitions of lenght at most $n$ with $c = \frac{1}{n}\sum_s\sum_i \alpha_s(i)$ an integer. If the representation of $SL_n$ $V(\alpha_1)\otimes \cdots \otimes V(\alpha_m)$ contains the trivial representation, then as a representation of $GL_n$ it contains the representation $(\wedge^n \mathbb C^n)^{\otimes c}$.

I don't understand why the power of the determinant representation is $c$.

Thanks!

Answer 1

$(\wedge^n \mathbb C^n)^{\otimes k}$ is irreducible (of dimension 1) and it's $V((k,k,\cdots, k))$, it must have $nc$ boxes, thus $nc = kn \implies k=c$.