0

I would like to understand the proof of proposition $1.5$ from the book "Representation theory: a first course" of Fulton and Harris.

$\textbf{Proposition}$: If $W$ is a subrepresentation of a representation $V$ of a finite group $G$, then there is a complementary invariant subspace $W'$ of $V$, so that $V = W \oplus W'$.

$\textit{Proof:}\;$ You can introduce a Hermitian inner product $H$ on $V$ which is preserved by each $g \in G$. Indeed, if $H_0$ is any Hermitian product on $V$, one gets such an $H$ by averaging over $G$: $$ \bbox[yellow]{\;H(v,w) = \sum_{g \in G}^{} H_0(gv, gw).\;} $$

Then the perpendicular subspace $W^{\perp}$ is complementary to $W$ in $V$.

I don't understand why the highlighted equation is true and how this proves that the perpendicular subspace $W^{\perp}$ is complementary to $W$ in $V$.

Sammy Black
  • 25,273
danilocn94
  • 429

3 Answers3

3

  1. The highlighted equation is the definition of the $G$-invariant Hermitian product introduced in the first sentence.

  2. For an Hermitian product it is a general fact that $W$ and $W^\perp$ are complementary subspaces. This amounts to showing that $W \cap W^\perp = {0}$ (a quick check) and that $W + W^\perp = V$ (done using e.g. Gram-Schmidt).

  3. Rather, the content of the Proposition is that $W^\perp$ is invariant under the $G$-action. (The upshot being that we can break up the representation $V$ into smaller pieces that are easier to understand.) For this, observe that for all $w’ \in W^\perp$, for all $g \in G$, and for all $w \in W$, $$H(g w’, w) = H(g^{-1}(gw’), g^{-1} w) = H(w’, g^{-1} w) = 0, $$ so that $gw’ \in W^\perp$ and $W^\perp$ is invariant. (The first equality uses the $G$-invariance of $H$. The last equality follows from the fact that $w’ \in W^\perp$ and that $W$ is a subrepresentation, so that $g^{-1}w \in W$.)

Sam Freedman
  • 3,989
  • So, did the book introduce the definition in the proof ? Could you provide me others references with this definition, please. – danilocn94 Jul 25 '21 at 02:55
  • 1
    See this page on the unitary trick (aka the “averaging trick”): https://math.stackexchange.com/questions/1904449/weyls-unitarian-trick – Sam Freedman Jul 25 '21 at 03:23
2

The highlighted equation is merely the definition of $H.$ It is a standard fact of linear algebra that in a finite-dimensional inner product space, a subspace and its perpendicular subspace span the entire space, proven quickly by say Graham-Schmidt.

  • So, did the book introduce the definition in the proof ? Could you provide me others references with this definition, please. – danilocn94 Jul 25 '21 at 02:56
  • 1
    @danilocn94 There aren't really other definitions to cite, becaues it's a one off object. He is just defining this hermitian form, just as one might in a proof write "let $x = t^2$" –  Jul 25 '21 at 02:57
2

The equation is the definition of your Hermitian form $H$. It is a an Hermitian inner product, because it is a sum of Hermitian inner products: if $f$ is any element of the linear group and $H_0$ is Hermitian, then so is $H_0(f-, f-)$. To see that a sum of Hermitian forms is Hermitian: note that being sesquilinear and Hermite-symmetric is preserved by linear combination, and that a sum of positive definite forms is positive.

To see it is invariant under the action of the group note that from the defining formula it follows that $H(g-, g-) = H(-,-)$ for any $g \in G$.

Finally observe that if $H$ is invariant under the action of $G$ and $W$ is invariant under the action of $G$, then $W^{\bot}=\{v \in V: H(v, W)=0\}$ is invariant as well, just applying the definition.

The fact that $W^\bot$ is a complement to $W$ follows from the fact that $H$ is Hermitian.

Dunnò000
  • 683
  • 1
    So, did the book introduce the definition in the proof ? Could you provide me others references with this definition, please. – danilocn94 Jul 25 '21 at 02:55
  • 1
    @danilocn94 If you are wondering about what Hermitian inner products are and what are their properties: they are binary functions $H(x,y)$ from a complex vector space that are linear in one variable, conjugate-linear (or sesquilinear) in the other, Hermitian symmetric in the sense that $H(y,x)=\overline{H(y,x)}$, and positive definite in the sense that $H(x,x)\ge 0$ for every $x$, with equality holding only if $x=0$. You should find the definition and basic facts about them in any basic linear algebra book. Also the page https://en.wikipedia.org/wiki/Inner_product_space may be of help. – Dunnò000 Jul 26 '21 at 10:35