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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)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...

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# When k[G/H] is multiplicity free G module ?

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). 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...