SL2equivariant

SL2equivariant refers to objects or maps that are compatible with the action of the special linear group SL(2). Over a field k, SL2 denotes the group of 2-by-2 matrices with determinant 1, and an SL2-action on a vector space V is a group homomorphism ρ: SL2 → GL(V). A linear map f: V → W between SL2-modules is SL2-equivariant if f(ρ(g)v) = ρ(g)f(v) for all g in SL2 and v in V. Equivalently, f commutes with the SL2-action and is a morphism in the category of SL2-representations.

If V and W are SL2-representations, the set Hom_SL2(V,W) consists of SL2-equivariant linear maps, forming the intertwiner space between the representations. Over fields of characteristic zero, finite-dimensional SL2-representations are completely reducible, and every representation decomposes into direct sums of symmetric powers of the standard 2-dimensional representation, denoted Sym^n(k^2). Schur’s lemma implies that Hom_SL2(Sym^m(k^2), Sym^n(k^2)) is zero unless m = n, in which case it is one-dimensional.

Equivariance extends beyond linear maps to other mathematical objects. An SL2-equivariant morphism between SL2-varieties, or between SL2-equivariant sheaves, is one that commutes with the group action. In algebraic geometry and representation theory, SL2-equivariant structures play a central role in the study of representations, invariants, and geometric constructions, including the formulation of morphisms, decompositions, and functorial relations that respect the SL2-action.