The irreducible representations of $\mathrm{GL}_n(\mathbb C)$ are indexed by partitions $\lambda$. These representations are denoted by $\mathbb S_{\lambda}(V)$, where $V$ is the standard $n$-dimensional representation. Apparently, the only such representations which are $1$-dimensional are when $\lambda=(k^n)$ for some $k$, and I'm trying to find a reference or a good proof for this fact. (Notation: $k^n$ is the partition of $kn$ into $n$ equal pieces, i.e. $(k,k,k,\ldots,k)$ with $n$ $k$'s.) I can see how to use the Schur polynomial method for computing characters to show that when $\lambda =(k^n)$, we get the $k$th power of the determinant representation. So I understand why these representations are $1$-dimensional, and I'd like to know how to show all other representations are higher dimensional. Using the Schur polynomial technique, it should suffice to show that for any strictly increasing $0\leq k_1<k_2<\ldots<k_n$ where $k_i-k_{i-1}\geq 2$ for at least one $i$, the quotient of two determinants $$\frac{\left|\begin{array}{cccc}x_1^{k_1}&x_1^{k_2}&\cdots&x_n^{k_n}\\ x_2^{k_1}&x_2^{k_2}&\cdots&x_2^{k_n}\\ \vdots&&&\vdots\\ x_n^{k_1}&x_n^{k_2}&\cdots&x_n^{k_n} \end{array}\right|}{\left|\begin{array}{cccc}x_1^{n-1}&x_1^{n-2}&\cdots&1\\ x_2^{n-1}&x_2^{n-2}&\cdots&1\\ \vdots&&&\vdots\\ x_n^{n-1}&x_n^{n-2}&\cdots&1 \end{array}\right|}$$ evaluates to a number greater than $1$ when you plug in $x_1=x_2=\cdots=x_n=1$. For example, it is not that hard to show that when $n=2$, we have $$\frac{\left|\begin{array}{cc}x_1^{k_1}&x_2^{k_2}\\ x_2^{k_1}&x_2^{k_2} \end{array}\right|}{\left|\begin{array}{cc}x_1&1\\ x_2&1 \end{array}\right|}\underset{(x_1,x_2)\mapsto(1,1)}{\longrightarrow} k_2-k_1$$ which indeed is greater than $1$ if the gap between the powers of $x_i$ must be at least $2$.
So I wonder if there is an argument to show that quotient of these determinants is indeed greater than one when you plug in $x_1=x_2=\cdots=x_n$. I'd also be interested in any other method for showing that the dimension $\mathbb S_{\lambda}(V)$ exceeds $1$ if $\lambda\neq (k^n)$ for some $k$.
I do not know much about representation theory, but $SL(n, \mathbb C)$ is the commutator subgroup of $GL(n, \mathbb C)$ and $GL(n, \mathbb C)/SL(n, \mathbb C)\cong \mathbb C^\times$. Thus, all 1-dimensional representations of $GL(n, \mathbb C)$ are of the form $\chi_k: A\mapsto (det(A))^k$ for integers $k$.