Irreducible representations

1. Assume that the field is algebraically closed and has zero characteristic, G is finite and representations are finite-dimensional. Show that this statement is true under the above assumptions: "Let p be an irreducible representation of G, and q be an irreducible representation of H. Is it always true that the exterior t ...continues

irreducible representations of a quaterion subgroup

Let G be the subgroup of quaternions of 8 elements, that contains ±1, ±i, ±j, ±k with relations i^2=j^2=k^2= −1, ij=k, jk=i, ki=j, ij=−ji, ik=−ki, jk=−kj. Classify irreducible representations of G over C.

Dodecahedron problem A_5 irreducibles

Consider the action of the group A_5 on the faces of a dodecahedron. Decompose the corresponding representation of A_5 into a sum of irreducibles and solve the problem by diagonilizing the interwining operator.

Matrix irreducible representations

Let G be the group of matrices 1 x y 0 1 z 0 0 1 where x, y, z are elements of the finite field F_5 . Classify irreducible representations of G over C.

Transitivity of induced representations

PLEASE SEE ATTACHMENT. A very standard proof in induced representation theory. can be found in any relevant book yet I have trouble unerstanding it. Please write the complete proof with details and everything. Thanks a lot. This will help me prepare for my exams.

Irreducible representation of dihedral group

Let D_n be the dihedral group. Classify the irreducible representations of D_n over C (complex).

SO(4) isomorphic

Show that SO(4) is isomorphic to the quotient of SU(2) X SU(2) by the subgroup generated by (-1,1)

Mathematical system

I am having a problem drawing the table for the following system: Define a universal set U as the set of counting numbers. Form a new set that contains all possible subsets of U. This new set of subsets together with the operation of set intersection forms a mathematical system. Then I have to tell which properties that we d ...continues

Indecomposable representations of quivers

Classsify the indecomposable representations of the following quivers: 1. o -> o <- o 2. o -> o <- o ^ l o

Modern Algebra, Group Theory (I): G contains all symbols ai, i = 0,1,2, …….,n-1 where we insist that a0 = an = e, ai.aj = ai+j if i+j ≤ n and ai.aj = ai+j-n if i+j > n .To prove that G is a cyclic group of order n.

Modern Algebra Group Theory (I) G contains all symbols a^i, i = 0,1,2, …….,n-1 where we insist that a^0 = a^n = e, a^i.a^j = a^(i+j) if i+j ≤ n and a^i.a^j = a^(i+j-n) if i+j > n . T ...continues