The first proof I ever saw of the orthogonality relations for characters of finite groups was computational: it did a lot of matrix computations and manipulations of sums, which I didn't like at all. There is a much more conceptual proof which begins by observing that Schur's lemma is equivalent to the claim that

$$\text{dim Hom}(A, B) = \delta_{ab}$$

for irreducible representations $A, B$, where $\text{Hom}$ denotes the set of $G$-module homomorphisms. One then observes that $\textbf{Hom}(A, B) = A^{*} \otimes B$ is itself a $G$-module and $\text{Hom}$ is precisely the submodule consisting of the copies of the trivial representation. Finally, the projection from $\textbf{Hom}$ to $\text{Hom}$ can be written

$$v \mapsto \frac{1}{|G|} \sum_{g \in G} gv$$

and the trace of a projection is the dimension of its image.

I particularly like this proof because the statement of the orthogonality relations is concrete and not abstract, but this proof shows exactly where the abstract content (Schur's lemma, Maschke's theorem) is made concrete (the trace computation). It also highlights the value of viewing the category of $G$-modules as an algebraic object in and of itself: a symmetric monoidal category with duals.

In addition, this interpretation of Schur's lemma suggests that $\text{Hom}(A, B)$ behaves like a categorification of the inner product in a Hilbert space, where the contravariant/covariant distinction between $A, B$ corresponds to the conjugate-linear/linear distinction between the first and second entries of an inner product. This leads to 2-Hilbert spaces and is a basic motivation for the term "adjoint" in category theory, as explained for example by John Baez here. It is also related to quantum mechanics, where one thinks of the inner product as describing the amplitude of a transition between two states occurs and of $\text{Hom}(A, B)$ as describing the way those transitions occur. John Baez explains related ideas here.

The first proof I ever saw of the orthogonality relations for characters of finite groups was computational: it did a lot of matrix computations and manipulations of sums, which I didn't like at all. There is a much more conceptual proof which begins by observing that Schur's lemma is equivalent to the claim that

$$\text{dim Hom}(A, B) = \delta_{ab}$$

for irreducible representations $A, B$, where $\text{Hom}$ denotes the set of $G$-module homomorphisms. One then observes that $\textbf{Hom}(A, B) = A^{*} \otimes B$ is itself a $G$-module and $\text{Hom}$ is precisely the submodule consisting of the copies of the trivial representation. Finally, the projection from $\textbf{Hom}$ to $\text{Hom}$ can be written

$$v \mapsto \frac{1}{|G|} \sum_{g \in G} gv$$

and the trace of a projection is the dimension of its image.

I particularly like this proof because the statement of the orthogonality relations is concrete and not abstract, but this proof shows exactly where the abstract content (Schur's lemma, Maschke's theorem) is made concrete (the trace computation). It also highlights the value of viewing the category of $G$-modules as an algebraic object in and of itself: a symmetric monoidal category with duals.

In addition, this interpretation of Schur's lemma suggests that $\text{Hom}(A, B)$ behaves like a categorification of the inner product in a Hilbert space, where the contravariant/covariant distinction between $A, B$ corresponds to the conjugate-linear/linear distinction between the first and second entries of an inner product. This leads to 2-Hilbert spaces and is a basic motivation for the term "adjoint" in category theory, as explained for example by John Baez here. It is also related to quantum mechanics, where one thinks of the inner product as describing the amplitude of a transition between two states occurs and of $\text{Hom}(A, B)$ as describing the way those transitions occur. John Baez explains related ideas here.

The first proof I ever saw of the orthogonality relations for characters of finite groups was computational: it did a lot of matrix computations and manipulations of sums, which I didn't like at all. There is a much more conceptual proof which begins by observing that Schur's lemma is equivalent to the claim that

$$\text{dim Hom}(A, B) = \delta_{ab}$$

for irreducible representations $A, B$, where $\text{Hom}$ denotes the set of $G$-module homomorphisms. One then observes that $\textbf{Hom}(A, B) = A^{*} \otimes B$ is itself a $G$-module and $\text{Hom}$ is precisely the submodule consisting of the copies of the trivial representation. Finally, the projection from $\textbf{Hom}$ to $\text{Hom}$ can be written

$$v \mapsto \frac{1}{|G|} \sum_{g \in G} gv$$

and the trace of a projection is the dimension of its image.

I particularly like this proof because the statement of the orthogonality relations is concrete and not abstract, but this proof shows exactly where the abstract content (Schur's lemma, Maschke's theorem) is made concrete (the trace computation). It also highlights the value of viewing the category of $G$-modules as an algebraic object in and of itself: a symmetric monoidal category with duals.

In addition, this interpretation of Schur's lemma suggests that $\text{Hom}(A, B)$ behaves like a categorification of the inner product in a Hilbert space, where the contravariant/covariant distinction between $A, B$ corresponds to the conjugate-linear/linear distinction between the first and second entries of an inner product. This leads to 2-Hilbert spaces and is a basic motivation for the term "adjoint" in category theory, as explained for example by John Baez here. It is also related to quantum mechanics, where one thinks of the inner product as describing the probability that a transition between two states occurs and of $\text{Hom}(A, B)$ as describing the way those transitions occur. John Baez explains related ideas here.

The first proof I ever saw of the orthogonality relations for characters of finite groups was computational: it did a lot of matrix computations and manipulations of sums, which I didn't like at all. There is a much more conceptual proof which begins by observing that Schur's lemma is equivalent to the claim that

$$\text{dim Hom}(A, B) = \delta_{ab}$$

for irreducible representations $A, B$, where $\text{Hom}$ denotes the set of $G$-module homomorphisms. One then observes that $\textbf{Hom}(A, B) = A^{*} \otimes B$ is itself a $G$-module and $\text{Hom}$ is precisely the submodule consisting of the copies of the trivial representation. Finally, the projection from $\textbf{Hom}$ to $\text{Hom}$ can be written

$$v \mapsto \frac{1}{|G|} \sum_{g \in G} gv$$

and the trace of a projection is the dimension of its image.

I particularly like this proof because the statement of the orthogonality relations is concrete and not abstract, but this proof shows exactly where the abstract content (Schur's lemma, Maschke's theorem) is made concrete (the trace computation). It also highlights the value of viewing the category of $G$-modules as an algebraic object in and of itself: a symmetric monoidal category with duals.

In addition, this interpretation of Schur's lemma suggests that $\text{Hom}(A, B)$ behaves like a categorification of the inner product in a Hilbert space, where the contravariant/covariant distinction between $A, B$ corresponds to the conjugate-linear/linear distinction between the first and second entries of an inner product. This leads to 2-Hilbert spaces and is a basic motivation for the term "adjoint" in category theory, as explained for example by John Baez here. It is also related to quantum mechanics, where one thinks of the inner product as describing the amplitude of a transition between two states occurs and of $\text{Hom}(A, B)$ as describing the way those transitions occur. John Baez explains related ideas here.