Highest-weight representation

A highest-weight representation is generated by a weight vector killed by every positive-root subspace.

Every finite-dimensional representation of a complex semisimple Lie algebra is a direct sum of irreducible representations.

If $\lambda ={\sum }_i{m}_i{\omega }_i$ is dominant integral, the tensor product ${⨂}_{i}V({\omega }_i{)}^{\otimes m_i}$ contains $V(\lambda )$ as the irreducible summand generated by the tensor product of highest-weight vectors.

If $\mu$ and $\lambda$ are weights of a finite-dimensional representation, $\mu$ is dominant, and $\mu ⪯\lambda$, thenSuccessively subtract a simple root ${\alpha }_i$ for which $⟨\lambda ,{\alpha }_i^{\vee }⟩>0$ and use injectivity of the corresponding ${\mathfrak{sl}}_2$ lowering operator.

A vector $v$ has weight $\mu \in {\mathfrak{t}}^{\ast }$ when $hv=\mu (h)v$ for every $h$ in the Cartan subalgebra.

A weight of a representation is a functional $\mu$ whose weight space is nonzero.

The weight space $V_{\mu }$ consists of all weight vectors of weight $\mu$, together with zero.

The multiplicity of a weight $\mu$ is the dimension of its weight space.

A singular vector is a nonzero weight vector annihilated by the positive nilpotent subalgebra ${\mathfrak{n}}^{+}$.

For a Borel subalgebra $\mathfrak{b}=\mathfrak{t}\oplus {\mathfrak{n}}^{+}$ and a weight $\lambda$, the Verma module iswhere ${\mathfrak{n}}^{+}$ kills ${\mathbb{C}}_{\lambda }$ and $h\in \mathfrak{t}$ acts by $\lambda (h)$.

For a module with finite-dimensional weight spaces, its formal character is $\mathrm{ch}V={\sum }_{\mu }(\dim V_{\mu })e^{\mu }$ in the completed group algebra of the weight lattice.

For a dominant integral weight $\lambda$, the irreducible character is

The dimension obtained by evaluating the Weyl character formula at the identity is

For the principal specialization, let $h_{\mathrm{pr}}=2{\rho }^{\vee }$, so every simple root takes value two on $h_{\mathrm{pr}}$. The q-character isIt is obtained from the Weyl character formula by substituting $e^{\mu }=q^{\mu (h_{\mathrm{pr}})}$.

A crystal basis is the combinatorial limit of a basis of a quantum-group representation. Its colored directed graph records the actions of the Kashiwara lowering operators.

On a crystal, ${\tilde{f}}_i$ follows an edge of color $i$ and ${\tilde{e}}_i$ follows that edge backwards.

The tensor product crystal uses the $i$-string data ${\epsilon }_i$ and ${\phi }_i$ to decide on which tensor factor a Kashiwara operator acts. Its connected components are highest-weight crystals and encode the decomposition of the tensor product representation.