The set-up is this: Let $G$ be a finite group, and $H$ a subgroup. We are given an irreducible representation of $H, \rho: H\rightarrow GL\_n(K)$ (I will notationally identify $\rho$ with its character). I want to decompose $Ind^G\_H(\rho)$ into irreducibles. I am given character tables of both $G$ and $H$.


If $K$ were $\mathbb{C}$, Frobenius reciprocity (<https://planetmath.org/FrobeniusReciprocity>) will do the trick. However, I am in the modular case; meaning: $char(K)||H|$. I still have all the character tables, except now they are Brauer character tables for the correct characteristic. 


Is there a method for decomposing $Ind^G\_H(\rho)$ into irreducible (Brauer) characters?


Edit: I wanted to make clear that since in the modular case we don't have Maschke's theorem, the ``decomposition'' into irreducibles would be in the Grothendieck group of Brauer characters of $G$. (representations of $G$ wouldn't nec. be direct sums of irreducible representations)



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 Φi corresponding to a Brauer character φi. I don't mean representation into projective groups, like PGL.


