Exercise set 4

The symbol G denotes a finite group. The character of a finite dimensional (complex) representation V is denoted by  _V .

(1)

(Difficulty level: 1) Convince yourself of the following:

• Hom(V,W) ≃ V ^*⊗ W
• _{V ⊕W} = _V + _W
• _{V ⊗W} = _{V W}
• _{V ^*} = _V
• _Hom(V,W) = _{V W}

(2)

(Difficulty level: 3) Calculate character tables of the symmetric groups on 3, 4, and 5 letters; of cyclic groups; of the dihedral groups; of the non-cyclic group of order 4; of all groups of order 8; of all groups up to order 12; etc.

(3)

(Difficulty level: 2) Deduce the following from Wedderburn’s structure theorem, where R denotes a finite dimensional semisimple algebra over an algebraically closed field k:

• (a criterion for simplicity) R is simple if and only if it admits a unique simple module.
• (density) If V ₁, …, V _k are pairwise non-isomorphic simple R-modules, and φ₁, …, φ_k are arbitrarily specified k-linear transformations, then there exists r in R which acts on V _i like φ_i for all 1 ≤ i ≤ k.
• (”left semisimplicity and right semisimplicity are equivalent”) R is semisimple as a right module.

Additional problems (Optional)

(1)

(Burnside’s lemma) Let k be an algebraically closed field. Let R be a k-algebra (not necessarily finite dimensional) and let V be a finite dimensional simple R-module. Show that R → End_kV (the map defining V as an R-module) is surjective. (Hint: Let the image of R in End_kV be denoted by S. Note that V is a simple module for S. Since End_kV = V ⊕⊕ V (dim_kV times) as a End_kV module, it follows that it is semisimple as an S-module. Thus S is semisimple as a module over itself (being a submodule of a semisimple module), which means that S is a semisimple algebra. Since V is a simple module for S, it follows from Wedderburn that dim_kS ≥ (dim_kV )². Thus S = End_kV . □)

(2)

Let k be an algebraically closed field. Show that any finite dimensional simple k-algebra is isomorphic to the matrix ring M_n(k) for some n. (Hint: Let R be such an algebra. By definition, R≠0. Choose a simple module V for R and consider R → End_kV defining V as an R-module (note that V exists). This map is surjective by Burnside’s lemma (previous item) and injective because R is simple. □)