Knowlpedia

Fundamental representation

An irreducible highest-weight representation whose highest weight is a fundamental weight.

Let g\mathfrak g be a complex semisimple with a fixed h\mathfrak h and a choice of positive roots (see ). Let {α1,…,αr}\{\alpha_1,\dots,\alpha_r\} be the set of , and let {ω1,…,ωr}⊂h∗\{\omega_1,\dots,\omega_r\}\subset \mathfrak h^* be the corresponding fundamental weights, characterized by

⟨ωi,αj∨⟩=δij,\langle \omega_i,\alpha_j^\vee\rangle=\delta_{ij},

where αj∨\alpha_j^\vee are the simple coroots.

Definition (Fundamental representation). A fundamental representation of g\mathfrak g is a finite-dimensional irreducible whose is one of the fundamental weights ωi\omega_i.

Remarks

Example (type An−1A_{n-1}). For sln(C)\mathfrak{sl}_n(\mathbb C) (see ), the fundamental representations are the exterior powers of the defining representation:

Λk(Cn),1≤k≤n−1.\Lambda^k(\mathbb C^n),\qquad 1\le k\le n-1.

With the standard choice of h\mathfrak h as diagonal trace-zero matrices, the highest weight of Λk(Cn)\Lambda^k(\mathbb C^n) is ωk\omega_k (in the usual convention where weights record the eigenvalues of h\mathfrak h on weight vectors; compare and ). In particular, Λ1(Cn)=Cn\Lambda^1(\mathbb C^n)=\mathbb C^n has highest weight ω1\omega_1, while Λn−1(Cn)≅(Cn)∗\Lambda^{n-1}(\mathbb C^n)\cong (\mathbb C^n)^* has highest weight ωn−1\omega_{n-1}.

Context. Fundamental representations correspond to the nodes of the and generate the representation ring in many settings; the classification of all irreducibles proceeds via the .