Using elementary techniques, an algorithmic procedure to construct skew-symmetric matrices realizing the real irreducible representations of so(3) is developed. We further give a simple criterion that enables one to deduce the decomposition of an arbitrary real representation R of so(3) into real irreducible components from the characteristic polynomial of an arbitrary representation matrix.

Representation theory. Let us try to study a given object X through vector spaces V associated naturally to it. The symmetries S of X can form a group or more generally some kind of an algebra usually related to a group (Lie algebra, quantum group,...). The Vector space V inherits the symmetries of X, and they act on V by linear operators. This is what one means by “V is a representation of the...

We establish the existence of an irreducible representation of An whose dimension does not occur as the dimension of an irreducible representation of Sn, and vice versa. This proves a conjecture by Tong-Viet. The main ingredient in the proof is a result on large prime factors in short intervals.

Suppose G is a real reductive Lie group, with maximal compact subgroup K. The representation theory of K is well-understood and well-behaved: K̂ is a countable discrete set consisting of finite-dimensional representations (δ, Eδ). If (π, V ) is a quasisimple irreducible representation of G, HarishChandra proved that each irreducible representation of K appears at most finitely often in V ; so th...

Let G be a p-adic connected reductive group with Lie algebra g. For a parabolic subgroup P ⊂ G and a finite-dimensional locally analytic representation V of a Levi subgroup of P, we study the induced locally analytic G-representation W = IndP (V ). Our result is the following criterion concerning the topological irreducibility of W : If the Verma module U(g) U(p) V ′ associated to the dual repr...

We discuss irreducible highest weight representations of the sl2 loop algebra and reducible indecomposable ones in association with the sl2 loop algebra symmetry of the six-vertex model at roots of unity. We formulate an elementary proof that every highest weight representation with distinct evaluation parameters is irreducible. We present a general criteria for a highest weight representation ...

If F is a local non-Archimedean field, then every irreducible admissible representation π of GL(r, F) is a quotient of a representation ξ induced by tempered ones. We show that £ has a Whittaker model, even though it may fail to be irreducible.

The Weil representation of the symplectic group associated to a finite abelian group of odd order is shown to have a multiplicity-free decomposition. When the abelian group is p-primary, the irreducible representations occurring in the Weil representation are parametrized by a partially ordered set which is independent of p. As p varies, the dimension of the irreducible representation correspon...

We classify the irreducible, admissible, smooth, genuine mod p representations of the metaplectic double cover of SL2(F ), where F is a p-adic field and p 6= 2. We show, using a generalized Satake transform, that each such representation is isomorphic to a certain explicit quotient of a compact induction from a maximal compact subgroup by an action of a spherical Hecke operator, and we define a...