# 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 $$\fg$$:

• An inner product on $$\fg$$. We use Hilbert-Schmidt 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 finite-dimensional 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.