Number of copies of irreducible representations for transitive actions

Question

Let a finite group $G$ act on a set $X$ transitively by permuting its elements. Then $|X| \le |G|$ since $|X| \big| |G|$ by the orbit-stabilizer theorem. Let $\pi_i;\ \ 1\le i \le m$ be irreducible representations of $G$ with degree $n_i$ over complex numbers.

This action induces a permutation representation $\rho: G \to GL(V)$, where $V$ is a free vector space on $X$ over $\mathbb{C}$. Moreover, $\rho$ decomposes to direct sum of some copies of $\pi_i$: $$\rho=c_1\pi_1 \oplus \cdots \oplus c_m \pi_m$$

If $|X|=|G|$, then $\rho$ is the regular representation, and $c_i=n_i$.

Question: Is it true that $c_i \le n_i$ when $|X| < |G|$?

For example, this is true for $c_1$. Since the action is transitive, the number of different orbits is always one, which is equal to $c_1$.

Answer 1

A $G$-set is a set $X$ equipped with a group action. A morphism $f:X\to Y$ of $G$-sets is an intertwining a.k.a. equivariant function, i.e. satisfies $f(gx)=gf(x)$ for all $x\in X$. Given any $G$-set $X$, the free vector space $\mathbb{C}X$ with basis $X$ becomes a complex representation of $G$. Any morphism of $G$-sets then extends to a linear transformation which is a morphism of representations. (IOW: linearization is a functor from $\mathsf{Set}_G$ to $\mathsf{Rep}_G$.)

Given $x\in X$, the map $g\mapsto gx$ is a morphism $G\to X$. The range is the orbit $\mathrm{Orb}(x)$ and the fibers are the cosets of the stabilizer $\mathrm{Stab}(x)$ (orbit-stabilizer theorem). This extends to a morphism of representation $\mathbb{C}G\to\mathbb{C}X$, where $\mathbb{C}G$ is the regular representation. If $G\curvearrowright X$ transitively then this is onto.

Any complex finite-dimensional representation $V$ of a finite group $G$ is unitarisable (via the so-called unitary trick), in which case the orthogonal complement of any subrepresentation is another subrepresentation. (This is used to show semisimplicity in the first place.) Given any morphism $V\to W$, the orthogonal complement of the kernel is equivalent (as a $G$-rep) to the image in $W$ (the $1$st iso thm). Thus for $\mathbb{C}G\to\mathbb{C}X$, we conclude $\mathbb{C}G$ has a subrep equivalent to $\mathbb{C}X$.

Say $V_1,V_2,\cdots$ are the $G$-irreps (up to iso) and denote $nV=V\oplus\cdots\oplus V$ ($n$ times). Write $\mathbb{C}G\cong A\oplus B$ where $A\cong\mathbb{C}X\cong\bigoplus a_iV_i$ and $B\cong\bigoplus b_iV_i$. Then $a_i+b_i=n_i$ implies $a_i\le n_i$ for each $i$.