Lecture notes for Introduction to Representation Theory

y
θ(a 1 , . . . , a n ) = a 1 y 1 + · · · + a n y n
where {y i } is a basis of X ⊕ . θ is clearly surjective, as k → A. Thus, the dual map θ ⊕ : X −⊃ A n⊕
is injective. But A n⊕ ∪ = A
n
as representations of A (check it!). Hence, Im θ ⊕ ∪ = X is a subrepresentation
of A n . Next, Mat di (k) = d i V i , so A = r i=1 d iV i , A n = r i=1 nd iV i , as a representation of A.
Hence by Proposition 2.2, X = r i=1 m i V i , as desired.
Exercise. The goal of this exercise is to give an alternative proof of Theorem 2.6, not using
any of the previous results of Chapter 2.
Let A 1 , A 2 , ..., A n be n algebras with units 1 1 , 1 2 , ..., 1 n , respectively. Let A = A 1 A 2 ...A n .
Clearly, 1 i 1 j = ζ ij 1 i , and the unit of A is 1 = 1 1 + 1 2 + ... + 1 n .
For every representation V of A, it is easy to see that 1 i V is a representation of A i for every
i {1, 2, ..., n}. Conversely, if V 1 , V 2 , ..., V n are representations of A 1 , A 2 , ..., A n , respectively,
then V 1 V 2 ... V n canonically becomes a representation of A (with (a 1 , a 2 , ..., a n ) A acting
on V 1 V 2 ... V n as (v 1 , v 2 , ..., v n ) ⊃ (a 1 v 1 , a 2 v 2 , ..., a n v n )).
(a) Show that a representation V of A is irreducible if and only if 1 i V is an irreducible representation
of A i for exactly one i {1, 2, ..., n}, while 1 i V = 0 for all the other i. Thus, classify the
irreducible representations of A in terms of those of A 1 , A 2 , ..., A n .
(b) Let d N. Show that the only irreducible representation of Mat d (k) is k d , and every finite
dimensional representation of Mat d (k) is a direct sum of copies of k d .
Hint: For every (i, j) {1, 2, ..., d} 2 , let E ij Mat d (k) be the matrix with 1 in the ith row of the
jth column and 0’s everywhere else. Let V be a finite dimensional representation of Mat d (k). Show
that V = E 11 V E 22 V ... E dd V , and that i : E 11 V ⊃ E ii V , v ⊃ E i1 v is an isomorphism for
every i {1, 2, ..., d}. For every v E 11 V , denote S (v) = ◦E 11 v, E 21 v, ..., E d1 v. Prove that S (v)
is a subrepresentation of V isomorphic to k d (as a representation of Mat d (k)), and that v S (v).
Conclude that V = S (v 1 ) S (v 2 ) ... S (v k ), where {v 1 , v 2 , ..., v k } is a basis of E 11 V .
(c) Conclude Theorem 2.6.
2.4 Filtrations
Let A be an algebra. Let V be a representation of A. A (finite) filtration of V is a sequence of
subrepresentations 0 = V 0 → V 1 → ... → V n = V .
Lemma 2.8. Any finite dimensional representation V of an algebra A admits a finite filtration
0 = V 0 → V 1 → ... → V n = V such that the successive quotients V i /V i−1 are irreducible.
Proof. The proof is by induction in dim(V ). The base is clear, and only the induction step needs
to be justified. Pick an irreducible subrepresentation V 1 → V , and consider the representation
U = V/V 1 . Then by the induction assumption U has a filtration 0 = U 0 → U 1 → ... → U n−1 = U
such that U i /U i−1 are irreducible. Define V i for i ⊂ 2 to be the preimages of U i−1 under the
tautological projection V ⊃ V/V 1 = U. Then 0 = V 0 → V 1 → V 2 → ... → V n = V is a filtration of V
with the desired property.
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