Representation theory of sl(n; C)

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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 \(\fg = \fsl_n\).

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}

• \(\fh = \{\diag(\lambda_{1 : n}) : \lambda_i \in \cplx, \lambda_1 + ... + \lambda_n = 0 \}\) 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, Brian Hall, , Vol. 222 2003.