Tuesdays 2pm - Strand S2.29
Tuesdays 4pm - Bush House (North East wing) 1.03
Tutorial I (Link here)
Symmetric groups S_n : Signs and Inverses,
Dihedral groups D_2n : Centre, Conjugacy classes
Linear algebra: Commutating matrices have simultaneous eigenvectors
Tutorial II (Link here)
Group actions: orbits and stabilisers,
Conjugacy action of S_3,
Group actions from geometry
Tutorial III (Link here)
Homogeneous G-sets and their correspondence with conjugate subgroups,
Representations and their Invariant Subspaces,
Maschke's Theoerem : orthogonal complement with respect to <- , ->_{new}
Tutorial IV (Link here)
Conjugacy classes of S_n determined by cycle-type,
Permutation representations,
Representations of Z/nZ
Tutorial V (Link here)
New representations from olds: Hom(V, W),
Characters I : Character formulas for trivial and permutation representations
Average of a character : Cauchy-Frobenius orbit counting lemma
Tutorial VI (Link here)
Characters II : Irreducible criterion, multiplicities and two miracles
Class functions and their convolution
Sum and difference of characters
Character table for abelian groups
Tutorial VII (Link here)
Character III : row orthogonality & column analysis ,
Character tables for C_2 x C_2 and Q_8 ,
Projection map : averaging a group action
Tutorial VIII (Link here)
Dual space : definition, representation and character formula,
Tensor product : definition, representation and character formula,
Explicit vector space isomorphism between V*\otimes W and Hom(V,W)
Tutorial IX (Link here)
Proof : V*\otimes W isomorphic to Hom(V,W) as representations
Schur's lemma : Hom(V,V) contains the trivial rep,
Symmetric and exterior powers : their representations and character formulas
Tutorial X (Link here)
Restricted and induced representations : their representations and character formulas
Examples : (1) representations of A_4 and S_4, (2) inducing the trivial representation
Frobenius reciprocity and applications