I just started to learn representation theory and I'm interested in the irreducible representations of $\operatorname{GL}_n(\mathbb{C})$. I looked them up in three different books and apparently, I misunderstood something because for me, they are inconsistent statements.
Stanley (Enumerative Combinatorics II, Theorem A2.4):
The irreducible polynomial representations $\varphi^{\lambda}$ of $\operatorname{GL}_n(\mathbb{C})$ can be indexed by partitions $\lambda$ of length at most $n$.Bump (Lie Groups, Theorem 38.3):
The irreducible algebraic representations $\varphi^{\lambda}$ of $\operatorname{GL}_n(\mathbb{C})$ are indexed by integer sequences $\lambda_1\geq\lambda_2\geq\dots\geq\lambda_n$.Fulton, Harris (Representation Theory, Proposition 15.47):
Every irreducible complex representation of $\operatorname{GL}_n(\mathbb{C})$ is isomorphic to $\varphi^{\lambda}$ for a unique index $\lambda=\lambda_1,\dots,\lambda_n$ with $\lambda_1\geq\lambda_2\geq\dots\geq\lambda_n$.
My questions are:
- Is every algebraic irreducible representation indexed by a partition or by an integer sequence with $\lambda_1\geq\lambda_2\geq\dots\geq\lambda_n$?
- Can every irreducible representation be indexed by an integer partition with $\lambda_1\geq\lambda_2\geq\dots\geq\lambda_n$ or only the irreducible alegbraic representations?
Finite dimensional representations of $GL_n(\mathbb{C})$, both as an algebraic group or as a complex Lie group are indexed by $n$-tuples $\lambda_1\geq \lambda_2\geq \cdots \geq\lambda_n$. Such a representation is polynomial if and only if $\lambda_1\geq \lambda_2\geq \cdots \geq\lambda_n\geq 0$, in which case people often think of this as a partition.
As Tobias Kildetoft points out, you can do other crazy stuff if you don't require your representation to be continuous and holomorphic; for example, $A \mapsto |\det(A)|^2$ is a representation of $GL_n(\mathbb{C})$ as a real Lie group or as a topological group, but it's not holomorphic, so it's not a representation of $GL_n(\mathbb{C})$ as a complex Lie group.