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 finitedimensional 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 \(\fg\):
 An inner product on \(\fg\). We use HilbertSchmidt inner products \(\braket{X, Y} = \Tr (X^* Y)\) for matrix Lie algebras.
 Cartan subalgebra \(\fh\): A maximal commutative algebra \(\fh \subset \fg\) where \(\Ad_H\) is diagonalisable for each \(H \in \fh\).
 Root \(\alpha \in \fh\) and corresponding root vector \(0 \neq X \in \fg\), defined by the relation \[ [H, X] = \braket{\alpha, H} X, \qquad \forall H \in \fh \] The set of roots is denoted by \(R\).
 Given a representation \((\pi, V)\), define weight \(\mu \in \fh\) and corresponding weight vector \(0 \neq v \in V\) by the relation \[ \pi(H) v = \braket{\mu, H} v, \qquad \forall H \in \fh \] so that the roots and root vectors are weights and weight vectors where \(\pi: X \mapsto \Ad_X\) is the adjoint representation.

The roots span a real space \(E\).
We estblish a partial ordering on \(E\).
 Base \(\Delta \subset R\): a subset of \(R\) and a basis of \(E\), such that \(\forall \alpha \in R\), \(\alpha\) expressed as a linear combination of \(\Delta\) have all nonnegative or all nonpositive coefficients.
 For \(x \in E\), \(x\) is dominant if \(x\) expressed as a linear combination of \(\Delta\) have all nonnegative coefficients.
 For \(x, y \in E\), \(x\) is said to be lower than \(y\) (denoted as \(x \preceq y\)) if \(y  x\) is dominant.
 Let \(R^+ = \{\alpha \in R: \alpha \text{ is dominant}\}\) be the set of positive roots.
 Let \(\delta = {1 \over 2}\sum_{\alpha \in R^+} \alpha\) be the half sum of the positive roots.
 \(x \in E\) is called integral if for any \(\alpha \in R\), \(2 {\braket{x, \alpha} \over \braket{\alpha, \alpha}} \in \intg\). Any weight of a finite representation of \(\fg\) is integral.
 Weyl group \(W\) is the group generated by the reflections \(s_\alpha\) for \(\alpha \in R\), where \(s_\alpha\) is defined by its action on \(E\) as \(I  2 P_\alpha\) as follows: \[ s_\alpha \beta = \beta  2 {\braket{\alpha, \beta} \over \braket{\alpha, \alpha}} \alpha. \]
 Given a representation \((\pi, V)\) of \(\fg\), the character is \[ \chi_\pi (X) := \Tr e^{\pi (X)} \] and so the dimension of \(V\) is \(\chi_\pi (0)\).
The Big Theorem. Irreducible finitedimensional representation (IFDR) of \(\fg\) are characterised by highest weights:
 Every IFDR of \(\fg\) has a highest weight, which is dominant integral, and for every dominant integral \(\mu\) there is an IFDR of \(\fg\) with highest weight \(\mu\).
 Two IFDRs of \(\fg\) with the same highest weight are isomorphic.
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 \(\fsl_n\)
 \(\fh = \{\diag(\lambda_{1 : n}) : \lambda_i \in \cplx, \lambda_1 + ... + \lambda_n = 0 \}\) are diagonal matrices with trace \(0\).
 The roots are \(E_{ii}  E_{jj}\) for \(i \neq j\) with root vectors \(E_{ij}\): \[ [H, E_{ij}] = (\lambda_i  \lambda_j) E_{ij} = \braket{E_{ii}  E_{jj}, H} E_{ij} \]
 Hence \(E = \{(x_1, ..., x_n) \in \real^n: \sum_i x_i = 0\}\) with the standard product of \(\real^n\), and the root \(E_{ii}  E_{jj}\) is identified as \(e_i  e_j\) in \(E\).
 The positive simple roots are \(\Delta = \{e_i  e_{i + 1}: i = 1 : n  1\}\).
 The positive roots are of the form \(e_i  e_j\) for \(1 \le i < j \le n\).
 \(\delta = {1 \over 2} \sum_{1 \le i < j \le n} (e_i  e_j) = ((n  1) / 2, (n  1) / 2  1, ..., 1  (n  1) / 2,  (n  1) / 2)\).
 The Weyl group \(W = S_n\) is the \(n\)th symmetric group, thus \(\det \sigma = \sgn \sigma\) for \(\sigma \in S_n\).
 The highest weights / dominant integral elements are of the form \(\mu = (\mu_1, \mu_2, ..., \mu_n) \in \intg^n\) such that \(\mu_i \ge \mu_{i + 1}\) for all \(i = 1 : n  1\).
 By the Weyl dimensional formula \[ \dim V_{\mu} = {\prod_{1 \le i < j \le n} (\mu_i  \mu_j + \delta_i  \delta_j) \over \prod_{1 \le i < j \le n} (\delta_i  \delta_j)} = {\prod_{1 \le i < j \le n} (\mu_i  \mu_j + j  i) \over \prod_{1 \le i < j \le n} (j  i)}. \] And the right hand side is the number of semistandard Young tableaux of shape \((\mu_1 + N, \mu_2 + N, ..., \mu_n + N)\) for \(N\) sufficiently large with entries in \([n]\).
 For \(H = \diag(\lambda_{1 : n})\), identify \(H\) with \((\lambda_1, ..., \lambda_n) \in E\). By the Weyl character formula and using the fact \(\sum_i\lambda_i = 0\), \begin{align} \chi_\mu (H) &= {\sum_{\sigma \in S_n} \sgn(\sigma) e^{\sum_i (\mu_{\sigma_i} + {n + 1 \over 2}  \sigma_i) \lambda_i} \over \sum_{\sigma \in S_n} \sgn(\sigma) e^{\sum_i ({n + 1 \over 2}  \sigma_i) \lambda_i}}\\ &= {\sum_{\sigma \in S_n} \sgn(\sigma) e^{\sum_i (\mu_{\sigma_i} + n  \sigma_i) \lambda_i} \over \sum_{\sigma \in S_n} \sgn(\sigma) e^{\sum_i (n  \sigma_i) \lambda_i}}\\ &= {\det (e^{\lambda_i})^{\mu_j + n  j} \over \det (e^{\lambda_i})^{n  j}} = s_\mu(x) \end{align} where \(x_i = e^{\lambda_i}\).
References
 [hall03] Lie groups, Lie algebras, and representations: an elementary introduction, , , Vol. 222 2003.