Representation theory of sl(n; C)

representation_theory toy_model

The goal of this note is to give a concise account of representation theory of \(\fsl_n = \fsl(n; \cplx)\), and to show that the dimension of a irreducible finite-dimensional representation of \(\fsl_n\) is the number of semistandard Young of certain shapes with entries in \([n]\), and that the characters of these representations are the Schur functions.

We mostly follow the amazing textbook [{hall03}].

We first give an overview of the representation theory of complex semisimple (matrix) Lie algebras containing the classical Lie algebras of \(\fsl_n\), \(\fso_n\) and \(\fsp_n\) by listing all its elements that are necessary in deriving the connection with Young tableaux and Schur functions, and then we match all these elements in the toy model of \(\fg = \fsl_n\).

Representations of complex semisimple Lie algebras

Let \(\Ad_H\) be an operator defined by \(\Ad_H X := [H, X]\).

Here are the elements of representation theory of complex semisimple Lie algebra \(\fg\):

The Big Theorem. Irreducible finite-dimensional representation (IFDR) of \(\fg\) are characterised by highest weights:

Theorem (The Weyl character and dimensional formulas). Let \((\pi, V_\mu)\) be the IFDR of \(\fg\) with highest weight \(\mu\), and let \(\chi_\mu\) be the corresponding character. Then

\begin{align} \chi_\mu (H) &= {\sum_{w \in W} \det(w) e^{\braket{w \cdot (\mu + \delta), H}} \over \sum_{w \in W} \det(w) e^{\braket{w \cdot \delta, H}}}. \\ \dim V_\mu &= {\prod_{\alpha \in R^+} \braket{\alpha, \mu + \delta} \over \prod_{\alpha \in R^+} \braket{\alpha, \delta}}. \end{align}

IFDR of \(\fsl_n\)