Homework 10. Due Tues. May 2 (One Week Later!)

Let k be an algebraically closed field. Let V be a 3-dimensional k-vector space and let V^* be the
dual vector space. Let m,n be in {0,1,2,...}.
Let P_m(V) be the vector space of all functions V\to k  which are homogeneous
polynomials of degree m. Let P_n(V^*) be the vector space of all functions V^*\to k which are homogeneous
polynomials of degree n. Show that P_m(V), P_n(V^*) are naturally representations of GL(V)
hence P_m(V)\otimes P_n(V^*) is naturally a representations of GL(V). (Here \otimes=tensor product over k.)
Assume that m,n are in {1,2,3,...}. Show:

(i) that P_{m-1}(V)\otimes P_{n-1}(V^*) is naturally a subspace of
 P_m(V)\otimes P_n(V^*) which is stable under the GL(V)-action  and

(ii) that (assuming k of characteristic 0) the quotient
( P_m(V)\otimes P_n(V^*))/(P_{m-1}(V)\otimes P_{n-1}(V^*)) is an irreducible representation of GL(V)
of dimension (m+1)(n+1)(m+n+2)/2.  (You may assume known the Weyl character formula.)

Hint for (i).
Consider the composition
P_{m-1}(V)\otimes P_{n-1}(V^*)\to P_{m-1}(V)\otimes P_{n-1}(V^*)\otimes P_1(V)\otimes P_1(V^*)\to
=P_{m-1}(V)\otime P_1(V)\otimes P_{n-1}(V^*)\otimes \to  P_m(V)\otimes P_n(V^*)
where the first map is 1\otimes A with A:\kk\to P_1(V)\otimes P_1(V^*) an imbedding of GL(V) representations
and the second map is (x\otimes y\otimes z\otimes u)\to (xy)\otimes(zu).