1. 18.715 takehome assignment, December 3-9, 2014
Problem 1. Let p be a prime, E a 2-dimensional vector space over Fp ,
and A : E → E an invertible linear operator. Assume that m is the smallest
positive number such that Am = 1, and GCD(m, p) = 1. Also assume that
det(A) = 1. Let G = Zm n E be the semidirect product group (of order
mp2 ), where 1 ∈ Zm acts on E by the operator A. Describe the irreducible
complex representations of G, and find their characters.
Problem 2. Let V be a finite dimensional complex vector space of
dimension n, and G = SL(V ) be the group of linear transformations of
V with determinant 1.
(a) Show that V ⊗N contains a nonzero G-invariant if and only if N is a
multiple of n.
(b) If N = mn where m is a positive integer, compute the dimension of
the space of invariants (V ⊗N )G .
Hint. Use Schur-Weyl duality.
(c) Show that if GCD(N, n) = 1 then the dimension of each subrepresentation Y of V ⊗N is divisible by n.
Hint. View Y as a GL(V )-module (where V = Cn ), and restrict it to
the subgroup H ⊂ GL(V ) generated by the cyclic permutation A and the
diagonal matrix B = diag(1, q, q 2 , ..., q n−1 ), where q is a primitive n-th root
of unity. Then use the representation theory of this finite subgroup.
Problem 3. Let λ be a partition of n, and λ∗ the dual partition (i.e.
∗
having the transposed Young diagram). Let zi be vectors in Cλi , and
Y
Fλ =
∆λ∗i (zi ),
i
Q
where ∆m (x) = 1≤i<j≤m (xi − xj ) is the Vandermonde determinant. Then
Fλ is a polynomial in n variables (say, x1 , ..., xn ). Let Wλ is the linear span
of the Sn -translates of Fλ . Show that Wλ is isomorphic to the Specht module
Vλ of Sn .
Hint. For partitions λ = (λ1 , ..., λn ) and µ = (µ1 , ..., µn ) of n, say that
e if λ − µ is a sum of vectors ei − ej , i < j (this is a partial order).
µ<λ
Let Uλ− = IndSQnλ (C− ), where C− is the sign representation. Show that
−
Uλ− = Vλ ⊕ ⊕µ<λ
e Mλµ Vµ , where Mλµ ∈ Z (show that HomSn (Uλ , Vµ ) = 0
e or µ = λ by using the character formulas). Then show that the
unless µ<λ
map g 7→ gFλ defines a surjective homomorphism Uλ− → Wλ . Finally, show
e then bµ Wλ = 0 (look at the smallest degree of a polynomial not
that if µ<λ
annihilated by bµ ). Deduce that HomSn (Vµ , Wλ ) = 0.
Problem 4. Let V = Cn−1 be the reflection representation of Sn , and
let Em,n := (V ⊗m )Sn , where m ≤ n. Show that the dimension dm of Em,n
is independent of n. Compute the exponential generating function
X dm
zm.
f (z) :=
m!
m≥0
1
Hint. First solve this problem for the permutation representation P = Cn
instead of V , then use that P = V ⊕ C.
Problem 5. Compute the dimensions of the radicals of the group algebras of S3 and S4 over algebraically closed fields of characteristic 2 and
3.
2