[Math] Decomposition into irreducible of induced representations

gr.group-theoryrt.representation-theory

If we consider an induced representation $Ind_{H}^{G}1_{H}$ of finite groups, where $1_{H}$ is the trivial caracter of $H$, how we decompose this representation into irreducible ? There is a decomposition $Ind_{H}^{G}1_{H}=V\oplus W$, where $V$ is formed of canstant functions on $G$ and $W$ is formed of functions $f:G\longrightarrow\mathbb{C}$, $f\in Ind_{H}^{G}1_{H}$, such taht $$\sum_{x\in G/H}f(x)=0$$
but I don't know if $ W $ is irreducible and if it is not irreducible how decomposed there into irreducible ?

The question arises also for profinite groups : If $G$ is totaly disconnected compact group and $H$ is an open subgroup, how to decompose the representation $Ind_{H}^{G}1_{H}$ into irreducibles ?

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Since it has not really been made completely explicit in the answers above, I point out that the specific question of whether your invariant subspace $W$ is irreducible or not is easier to check, at least in the case of finite groups. It rarely is. As is implicit in the earler answers, there is a formula due to Mackey for the norm (squared) of the induced character (and Mackey's formula works in more general contexts. In the case of finite groups, it is a double application of Frobenius reciprocity). We have $\langle {\rm Ind}_{H}^{G}(1), {\rm Ind}_{H}^{G}(1) \rangle = |T|$ , where $T$ is a full set of representatives for the distinct $(H,H)$-double cosets (so $G$ is the disjoint union $\cup_{t \in T} HtH ).$ The space $W$ is irreducible if and only if this norm is $2,$ which happens if and only if there are just two $(H,H)$-double cosets. This in turn happens precisely when $G$ is doubly transitive in the usual action on the ( say right) cosets of $H.$

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[Math] How does one compute induced representations for modular representations

Frobenius reciprocity for Brauer characters is a little more complicated (a lot more complicated if you don't have complete tables). You need the projective characters to compute the multiplicities. I don't use anything special about the character being induced, though sometimes you can leverage that information (especially if you don't have complete tables).

In GAP, this is easily done:

g:=CharacterTable("ON");
# Let g be the O'Nan simple group
chi:=Sum([1..10],i->Random(Irr(g mod 2)));;
# chi is a random reducible Brauer character
ipr:=Irr(g)*DecompositionMatrix(g mod 2);;
# ipr is the list of projective Brauer characters
vec:=MatScalarProducts(ipr,[InducedClassFunction(chi,g)])[1];
# vec is the multiplicity of each Brauer character in chi
chi = vec * Irr( g mod 2 );
# should be true: we have decomposed it correctly.

To get the decomposition matrix, you must not only have the Brauer table but also the ordinary table. You don't need much from the subgroup H, just a character to induce and the element fusion from H into G.

This is theorem 2.13 on page 25 of Navarro's textbook on Characters and Blocks of Finite Groups.

By projective, I mean in the module theory sense, an indecomposable direct summand of the regular representation, usually these are denoted Φ_i corresponding to a Brauer character φ_i. I don't mean representation into projective groups, like PGL.

[Math] Decomposition of a group into a Product

In every finite $p$-group $G$ of order $p^n$ there is a normal subgroup $N_k$ of order $p^k$ for any $k\le n$. The proof is by induction on $k$. For $k=1$ take a cyclic subgroup of order $p$ in the center, for the step, take a cyclic central subgroup in $G/N_k$ and its preimage in $G$. So if you do not assume that $A\cap B=\{1\}$, one can't say anything specific about the orders. If you assume that $A\cap B=\{1\}$, i.e. your product is semi-direct, then some information (not much, though) can be deduced.

If $|A|=p$ and $A$ is normal, then $A$ is central because in a nilpotent group every non-trivial normal subgroup intersects the center non-trivially. In that case either $B=G$ or $A\cap B=\{1\}$ and $G=A\times B$.

If $|A|=p^k, k\ge 2$, then again you can take a central cyclic of order $p$ subgroup $T$ of $A$, and you will have $G/T=(A/T)(BT/T)$. This reduces your problem to the same problem for a smaller group $G/T$.

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[Math] How does one compute induced representations for modular representations

[Math] Decomposition of a group into a Product

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