Canonical decompostion of a representation- an example

Question

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 $$ \lambda = \frac{1}{n} \sum f(t)\chi_\rho (t) = \frac{\,|G|}{n}(f|\chi_\rho^*) $$

Answer 1

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).