# Rep tensor product of GL(2,C)

1. Dec 28, 2008

### swallowtail

Hi PF bloggers,
I'm trying to decompose a representation of $$GL(2,C)$$ on $$C^2\otimes Sym^{N-2}(C^2)$$ into IRREPS and I'm wondering if there's anything similar to Clebsh-Gordan coefficients which could assist one in this task?
Any good references one could point out?
Happy holidays!

P.S.: action is described as $$g(v\otimes w) := g(v)\otimes g(w)$$, and '$$Sym^{N-2}(C^2)$$ ' is thought as homogeneous polynomials of degree (N-2) in two variables.

2. Jan 1, 2009

### masnevets

Hi,

To decompose a tensor product of representations of GL(n,C) into a direct sum of irreps, use the Littlewood-Richardson rule:
http://en.wikipedia.org/wiki/Littlewoodâ€“Richardson_rule

In your case, C^2 is the standard representation represented by the partition (1), and Sym^{N-2}(C^2) is the representation represented by the partition (N-2), so the decomposition is

Sym^{N-1}(C^2) \oplus S_{(N-2,1)}(C^2)

where the second thing is the irrep corresponding to the partition (N-2,1). See this for one possible construction:
http://en.wikipedia.org/wiki/Schur_functor
A more combinatorial description can be found in Section 2 of this paper:
http://arxiv.org/abs/0810.4666