Consider finite group G and its subgroup H, and representation of G in k[G/H] i.e. functions on G/H.


**Question:** What is known about the question: when k[G/H] is multiplicity free ? (Let us consider k - complex numbers for simplicity).


More generally consider $Ind^G\_H V$ some induced representation of G from $H$, same question. 
(In case V= trivial representation we get previous question).


Are there some general conditions ? What is know about the cases G = S\_n, A\_n, GL\_n(F\_q) ? 


**Example**: $S\_{n-1} \in S\_n$ is multiplicity free since G/H = C^n - standard representation.


**Particular question** $H=GL\_{n}(F\_q) \subset G= GL\_{n+1}(F\_q)$ is C[G/H] multiplicity free ?
(This is true from GL(C) and leads to Gelfand-Zeitlin basis theory, well , may be it is not correct). 
More generally what about induced representations in this case ? 


Morally multiplicity free means that $H$ is "big" subgroup of G, however I do not know any precise way to say what "big" means.


PS


I googled "Multiplicity-free permutation representations of the alternating groups", which promises survey on the topic, but but I have no access to this file:(


<https://www.ideals.illinois.edu/handle/2142/22828>


If someone can help ... would be great...



When k[G/H] is multiplicity free G module ? 
Here's a very simple elementary criterion: 



> 
> A $\mathbb CG$-module is multiplicity-free if and only if its endomorphism ring is commutative
> 
> 
> 


The reason is simple: if $W\oplus W$ (for some irreducible module $W$) is a direct summand of $V$, then the non-commutative matrix ring $\textrm{End}(W\oplus W) \cong M\_2(\mathbb C)$ embeds into $\textrm{End(V)}$).


You are dealing with permutation modules, and for those there is a very explicit description of the endomorphism ring in terms of double cosets:
$$
 \textrm{End}\_{\mathbb CG}(\mathbb C [G/H]) \cong \langle \sum\_{h\in HgH} h \mid \textrm{ $g$ runs over double coset representatives}\rangle\_{\mathbb C} 
$$


You just need to check whether this ring is commutative. One criterion is that $H\backslash G /H $ has cardinality $\leq 3$ (because any semisimple algebra of dimension $\leq 3$ must be commutative).


A reference is Curtis-Reiner: "Methods of representation theory vol. 1". In this book these endomorphism rings are called "Hecke algebras", defined on page 281 (the endomorphism ring above corresponds to the Hecke algebra $\mathcal H(G,H,1\_H)$). 


However, computations in GAP show that $\mathbb C [GL\_3(3)/GL\_2(3)]$ is not multiplicity-free, so I see little hope for you question about permutation modules for $GL\_n(q)$.


