The regular representation of $G$ is obtained by inducing the trivial representation from the trivial subgroup: $\mathbb{F}[G] = \text{Ind}_{\{e\}}^G(\mathbb{F})$

This shows the regular representation decomposes as: $\mathbb{F}[G] \cong \bigoplus_{V \text{ irreducible}} V^{\oplus \dim V}$

Let $G$ act transitively on a set $X$, and let $H = \text{Stab}(x)$ be the stabilizer of a point. Then the permutation representation $\mathbb{F}[X]$ is isomorphic to: $\mathbb{F}[X] \cong \text{Ind}_H^G(\mathbb{F})$ where $\mathbb{F}$ is the trivial representation of $H$.

For $S_n$ acting on $n$ points: $\text{Ind}_{S_{n-1}}^{S_n}(\mathbb{F}) \cong \mathbb{F}^n$ with the standard permutation action.

Let $H = K = \langle (12) \rangle$ (a subgroup of order 2 in $S_3$). The double cosets $H \backslash S_3 / K$ are:

• $H \cdot e \cdot K = H$
• $H \cdot (23) \cdot K$ (distinct double coset)

For $W = \mathbb{C}$ (trivial representation of $H$): $\text{Res}_K^{S_3}(\text{Ind}_H^{S_3}(\mathbb{C})) \cong \text{Ind}_{K}^K(\mathbb{C}) \oplus \text{Ind}_{K \cap (23)H(23)^{-1}}^K(\text{something})$

The first term is just $\mathbb{C}$, and computing the second term involves the intersection $K \cap \langle (13) \rangle = \{e\}$.