# Canonical decompostion of a representation- an example

Can someone create an example using these notes:

Let $$\rho : G \rightarrow GL(V)$$ be a linear representation of $$G$$. We define below a direct sum decomposition of $$V$$ which is ''weaker'' than the decomposition into irreducible representations since the latter has the advantage of being unique. Consider the following:

Definition

Let $$\chi_i$$ be the distinct characters of the irreducible representations $$W_i$$ of $$G$$ and $$n_i$$ be their degrees. Let $$V=U_1 \oplus \cdots \oplus U_m$$ be a decomposition of $$V$$ into a direct sum of irreducible representations. Let $$V_i$$ for $$i=1, \cdots, h$$ be the direct sum of those the $$U_1, \cdots, U_m$$ which are isomorphic to $$W_i$$. So $$V=V_1 \oplus \cdots \oplus V_h$$ This is the canonical decomposition.

That is, we have decomposed $$V$$ into a direct sum of irreducible representations and collected together'' the isomorphic representations with the following properties

Theorem

(i) The decomposition $$V=V_1 \oplus \cdots \oplus V_h$$ does not depend on the initially chosen decomposition of $$V$$ into irreducible representations.

(ii) The projection $$P_i$$ of $$V$$ onto $$V_i$$ associated with this decomposition is $$P_i = \frac{n_i}{\,|G|}\sum_{t\in G} \chi_i(t)^* \rho_t$$

Proof

We shall prove claim $$(ii)$$ since this will prove $$(i)$$ as the projections $$P_i$$ dictate the $$V_i$$. Let $$q_i = \frac{n_i}{\,|G|} \sum_{t\in G} \chi_i(t)^*\rho_t$$

Proposition 2.5.1* tells us that the restriction of $$q_i$$ to an irreducible representation $$W$$ with character $$\chi$$ and of degree $$n$$ is a multiple in the ratio $$\frac{n_i}{n}(\chi_i|\chi)$$, hence it is $$1$$ if $$\chi = \chi_i$$ and $$0$$ if $$\chi \neq \chi_i$$ namely $$q_i$$ is the identity on an irreducible representation isomorphic to $$W_i$$ and is zero on the others. From the definition of $$V_i$$, it follows that $$q_i$$ is the identity on $$V_i$$ and is zero on $$V_k$$ for $$k\neq i$$. Now if we decompose an element $$v \in V$$ into its components $$v_i \in V_i$$ we have $$v= v_1 + \cdots+ v_h$$ so $$q_i(v) = q_i(v_1) + \cdots + q_i(v_h) = v_i$$. This means $$q_i$$ is equal to the projection $$p_i$$ of $$V$$ onto $$V_i$$.

We see that the decomposition of a representation can be done via two phases. Initially by deducing the canonical decomposition $$V_1 \oplus \cdots \oplus V_n$$ using the formulas for $$p_i$$. Next we decompose $$V_i$$ into a direct sum of irreducible representations each isomorphic to $$W_i$$ $$V_i = W_i \oplus \cdots \oplus W_i$$

*Proposition 2.5.1 Let $$f$$ be a class function on $$G$$. Let $$\rho: G\rightarrow GL(V)$$ be an irreducible representation of $$G$$. Then $$\hat{f}(\rho)=\lambda I$$ with $$\lambda = \frac{1}{n} \sum f(t)\chi_\rho (t) = \frac{\,|G|}{n}(f|\chi_\rho^*)$$

Since I've been working with this example lately, thought I would show you a slightly esoteric example.

Let $$V$$ be an $$n$$-dimensional vector space over a field field $$\mathsf{k}$$, and let $$\mathcal{F} = \mathcal{F}(V)$$ be the set of all complete affine flags in $$V$$, where a complete linear affine $$F$$ in $$V$$ is a filtration of $$V$$ into affine subspaces:

$$F_0 \subsetneq F_1 \subsetneq F_2 \subsetneq \dots \subsetneq F_{n-1} = V,$$

where $$F_k$$ is a $$k$$-dimensional affine subspace of $$V$$. An affine subspace of a vector space is just a translation of a linear subspace, so think maybe a line, or a plane or something like that. We just drop the assumption that it must contain zero.

Note that $$\mathcal{F}$$ is a finite set, since $$V$$ is finite as a set. Now we define $$C = C(\mathcal{F}, \mathsf{k})$$ to be the set of functions from $$\mathcal{F}$$ to $$\mathsf{k}$$. Note that we want $$C$$ to be all functions, we are not supposing any additional structure on $$C$$. However $$C$$ is a $$\mathsf{k}$$ vector space under point wise addition and scalar multiplication. I.e, if $$f,g \in C$$, $$F \in \mathcal{F}$$, $$\lambda \in \mathsf{k}$$ then $$(f + \lambda g)(F) = f(F) + \lambda g(F)$$.

We are now going to define a vector subspace of $$C$$ as follows. For each $$0 < k < n$$, and $$F \in \mathcal{F}$$, let $$\mathscr{L}_k(F) = \left\{ U < V \mid \operatorname{dim}U = k, F_i < U < F_j \ \text{for all} \ i < k < j \right\}.$$ Essentially, $$\mathscr{L}_k(F)$$ is the set of $$k$$-dimensional affine subspaces of $$V$$ that would fit into the $$k^{\text{th}}$$-dimensional spot of $$F$$. Then for $$F \in \cal{F}$$, $$U \in \mathscr{L}_k(F)$$ we define $$\tau(F,U) \in \cal{F}$$ be the complete affine flag in $$V$$ that agrees with $$F$$ at every dimension other than the $$k^{\text{th}}$$-dimension, and at the $$k^{\text{th}}$$-dimensional we swap out $$F_k$$ for $$U$$. Then for each $$0 < k < n$$ we define the $$\mathsf{k}$$-linear endomorphism of $$C$$ as follows, for $$f \in C$$, $$F \in \cal{F}$$, define

$$\sigma_k(f)(F) = \sum_{U \in \mathscr{L}_k(F)} f(\tau(F,U)).$$

Then we define $$\Delta = \bigcap_{0 < k < n} \operatorname{ker} \sigma_k$$. This is a vector subspace of $$C$$.

Now let $$G$$ be the multiplicative group $$\mathsf{k}^{\times}$$ of $$\mathsf{k}$$ (i.e the non-zero elements). We will show that $$\Delta$$ has the structure of a $$G$$-representation. For $$f \in \Delta$$, $$\lambda \in G$$, $$F \in \mathcal{F}$$ define $$(\lambda \cdot f)(F) = f(\lambda F)$$ where $$\lambda F$$ is the complete affine flag with $$k^{\text{th}}$$-dimensional element equal to $$\lambda F_k$$. It is clear that this indeed defines a $$G$$-representation with representation space equal to $$\Delta$$ (note that $$G$$ is Abelian). Note that $$\Delta$$ is finite dimensional.

Now note that $$G$$ is cyclic of order $$q-1$$, where $$q = \left| \mathsf{k} \right|$$. Suppose now that $$p = \operatorname{char}(\mathsf{k})$$ is the characteristic of $$\mathsf{k}$$. Then $$\left| G \right| \equiv -1 \not \equiv 0 \ (\operatorname{mod} \ p)$$, and since $$\mathsf{k}$$ is the ground field of $$\Delta$$, we have that the representation $$\Delta$$ is finite dimensional over a field whose characteristic does not divide the order of $$G$$, and so is semisimple by Maschke's theorem.

Now, it can be shown with not too much difficulty that the irreducible representation of $$\mathsf{k}^{\times}$$ over $$\mathsf{k}$$ are all one-dimensional and given by $$a \mapsto a^{k}$$ for some $$k$$. Thus, since $$\Delta$$ is semisimple, it decomposes as a direct sum $$\Delta = \bigoplus_{0 < k < q-1} \Delta^{(k)}$$, where

$$\Delta^{(k)} = \left\{ f \in \Delta \mid f(\lambda F) = \lambda^{k}f(F) \ \text{for every} \ F \in \mathcal{F} \right\}.$$

This is exactly the decomposition that you call the "canonical decomposition". I think it is more widely known as the isotypic decomposition.

We can show this direction directly by calculating the formulae for $$f_k \in \Delta^{(k)}$$ such that $$f = \sum_k f_k$$ for any given $$f \in \Delta$$ by using that $$G$$ is cyclic (see if you can do this).

• Once again, what in particular do you take issue with? Commented May 9, 2019 at 12:02