[[Representation theory of finite symmetric groups]]
# Associate representation

Multiplying any irrep by the [[alternating character]] $\chi^\mathfrak{a}$ gives another irrep (see [[Tensor product with a 1-dimensional representation]]).
Two irreps $\Gamma^\mu, \Gamma^\tilde{\mu}$ related by
$$
\begin{align*}
\Gamma^\tilde{\mu} \cong \chi^\mathfrak{a} \otimes \Gamma^\mu
\end{align*}
$$
are called **associate representations**. #m/def/rep/sym 
Iff $\Gamma^\nu \cong \chi^\mathfrak{a} \otimes \Gamma^\nu$ then $\Gamma^\nu$ is called **self-associate**.

## Properties

- $\dim \Gamma^\tilde{\mu} = \dim \Gamma^\mu$
- $\Gamma^\mu$ is self-associate iff $\Gamma^\mu(p) = \mathbf{0}$ for odd $p$
- $\chi^\mathfrak{s}$ and $\chi^\mathfrak{a}$ are associate to each other

#
---
#state/tidy | #lang/en | #SemBr