Lecture notes for Introduction to Representation Theory
Lecture notes for Introduction to Representation Theory
Lecture notes for Introduction to Representation Theory
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is the rotation by 180 0 around an axis that is perpendicular <strong>to</strong> two opposite faces, (123) is the<br />
rotation around a main diagonal by 120 0 , and (1234) is the rotation by 90 0 around an axis that is<br />
perpendicular <strong>to</strong> two opposite faces; this allows us <strong>to</strong> compute the traces easily, using the fact that<br />
the trace of a rotation by the angle θ in R 3 is 1 + 2 cos θ. Now the character of C 3<br />
− is found by<br />
multiplying the character of C 3 by the character of the sign representation.<br />
+<br />
Finally, we explain how <strong>to</strong> obtain the character table of A 5 (even permutations of 5 items). The<br />
group A 5 is the group of rotations of the regular icosahedron. Thus it has a 3-dimensional “rotation<br />
representation” C 3 + , in which (12)(34) is the rotation by 180 0 around an axis perpendicular <strong>to</strong> two<br />
opposite edges, (123) is the rotation by 120 0 around an axis perpendicular <strong>to</strong> two opposite faces,<br />
and (12345), (13254) are the rotations by 72 0 , respectively 144 0 , around axes going through two<br />
opposite vertices. The character of this representation is computed from this description in a<br />
straight<strong>for</strong>ward way.<br />
Another representation of A 5 , which is also 3-dimensional, is C 3 + twisted by the au<strong>to</strong>morphism<br />
of A This representation is denoted by C 3<br />
5 given by conjugation by (12) inside S 5 .<br />
−. It has the<br />
same character as C 3 , except that the conjugacy classes (12345) and (13245) are interchanged.<br />
+<br />
There are two remaining irreducible representations, and by the sum of squares <strong>for</strong>mula their<br />
dimensions are 4 and 5. So we call them C 4 and C 5 .<br />
The representation C 4 is realized on the space of functions on the set {1, 2, 3, 4, 5} with zero<br />
sum of values, where A 5 acts by permutations (check that it is irreducible!). The character of<br />
this representation is equal <strong>to</strong> the character of the 5-dimensional permutation representation minus<br />
the character of the 1-dimensional trivial representation (constant functions). The <strong>for</strong>mer at an<br />
element g equals <strong>to</strong> the number of items among 1,2,3,4,5 which are fixed by g.<br />
The representation C 5 is realized on the space of functions on pairs of opposite vertices of the<br />
icosahedron which has zero sum of values (check that it is irreducible!). The character of this<br />
representation is computed similarly <strong>to</strong> the character of C 4 , or from the orthogonality <strong>for</strong>mula.<br />
3.9 Computing tensor product multiplicities using character tables<br />
Character tables allow us <strong>to</strong> compute the tensor product multiplicities N ij<br />
k<br />
using<br />
V i V j = N ij<br />
k<br />
V k , N ij<br />
k<br />
= (ν i ν j , ν k )<br />
Example 3.16. The following tables represent computed tensor product multiplicities of irre<br />
S C 2<br />
3 C + C −<br />
C − C 2<br />
+ C + C<br />
ducible representations of S 3 , S 4 , and A 5 respectively.<br />
C−<br />
C + C 2<br />
C 2 C + C − C 2<br />
S C 2 C 3<br />
C3<br />
4 C + C − +<br />
−<br />
C 2 C 3<br />
C +<br />
C 3<br />
+ C + C −<br />
−<br />
C C + C 2 3<br />
C 3<br />
− C −<br />
+<br />
C 2<br />
C C2 C3 C3<br />
− + −<br />
C<br />
C 3<br />
C C3<br />
+<br />
+ 3<br />
+ 3 C 3<br />
+ C<br />
−<br />
C<br />
3<br />
−<br />
C C<br />
C−<br />
3 + 3<br />
+ C 2 C− C 2<br />
−<br />
C + C−<br />
3<br />
+ C 2 C 3<br />
42