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

Thinking of this as a $\mathrm{GL}_n$ representation, the product $V(\alpha_1) \otimes \cdots \otimes V(\alpha_m)$ will decompose into irreducibles $V(\lambda_1) \oplus \cdots \oplus V(\lambda_M)$ where each partition $\lambda_i$ has exactly $cn$ boxes. Since any power of the determinant representation is a partition with $n$ rows and some number of columns, it follows that it must have $c$ columns. Upon restriction to $\mathrm{SL}_n$, any power of the determinant representation becomes trivial.