In this short post, we will show two ways of combining existing representations to obtain new representations.

Recall

In the previous post, we saw two representations of $D_4$: the permutation representation, and the representation given in this Wikipedia example. Let’s first define these in Sage:

(The Sage cells in this post are linked, so things may not work if you don’t execute them in order.)

Direct Sums

If $(V_1,\rho_1), (V_2,\rho_2)$ are representations of $G$, the direct sum of these representations is $(V_1 \oplus V_2, \rho)$, where $\rho$ sends $g \in G$ to the block diagonal matrix

\[\begin{pmatrix} \rho_1(g) & 0 \\ 0 & \rho_2(g) \end{pmatrix}\]

Here $\rho_1(g), \rho_2(g)$ and the “zeros” are all matrices.

It’s best to illustrate with an example. We can define a function direct_sum in Sage that takes two representations and returns their direct sum.

Tensor products

We can also form the tensor product $(V_1 \otimes V_2,\rho)$, where $\rho$ sends $g \in G$ to the Kronecker product of the matrices $\rho_1(g)$ and $\rho_2(g)$.

We define a function tensor_prod that takes two representations and returns their tensor product.

Observe that

• $\dim V_1 \oplus V_2 = \dim V_1 + \dim V_2$,
• $\dim V_1 \otimes V_2 = \dim V_1 \times \dim V_2$,

which motivates the terms direct sum and tensor product.

We can keep taking direct sums and tensor products of existing representations to obtain new ones:

Decomposing representations

Now we know how to build new representations out of old ones. One might be interested in the inverse questions:

1. Is a given representation a direct sum of smaller representations?
2. Is a given representation a tensor product of smaller representations?

It turns out that Q1 is a much more interesting question to ask than Q2.

A (very poor) analogy of this situation is the problem of “building up” natural numbers. We have two ways of building up new integers from old: we can either add numbers, or multiply them. Given a number $n$, it’s easy (and not very interesting) to find smaller numbers that add up to $n$. However, finding numbers whose product is $n$ is much much harder (especially for large $n$) and much more rewarding. Prime numbers also play a special role in the latter case: every positive integer has a unique factorization into primes.

The analogy is a poor one (not least because the roles of “sum” and “product” are switched!). However, it motivates the question

• What are the analogues of “primes” for representations?

We’ll try to answer this last question and Q1 in the next few posts, and see what it means for us when working with representations in Sage.