[Math] Irreducibility of Induced Representation

rt.representation-theory

Mackey's test for irreducibility of induced representation over $\mathbb{C}$ is:

Let $G$ be a finite group, $H\leq G$, $W$ be a representation of $H$, and $W^x$ be conjugate representation of $H^x=xHx^{-1}$. Then following are equivalent:

(i) $Ind^G_H(W)$ is irreducible.

(ii) $W$ is irreducible and for each $x\in G\setminus H$, $W$ and $W^x$ have no common irreducible component, when restricted to $H\cap H^x$.

There is question posted, about changing the ground field, and the answer posted is (are) "Yes".

But, $(ii)\Rightarrow (i)$ is true for any field of characteristic zero or prime to $|G|$; how does $(i)\Rightarrow (ii)$ for such fields?

Best Answer

No. Over $\mathbb R$ let $G$ be the quaternion group of order $8$, $H$ the subgroup of order $2$, $W$ the nontrivial one-dimensional representation.

EDIT For an even simpler example, see Kevin Ventullo's comment!

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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] Irreducibility of Induced Representation over arbitrary field

I think so. The key property used in the proof is $$ Res^G_H Ind^G_H V=\bigoplus_{s\in H\setminus G/H} Ind^H_{H_s}V_s, $$ where $V_s$ is the twist of $V$ by $s$, and $H_s=sHs^{-1}\cap H$, and its proof is more or less manifestly characteristic-free. Other than that, instead of characters and scalar products $\langle \chi_U,\chi_W\rangle$ you can use the actual spaces of intertwiners $Hom_G(U,W)$, and finally instead of saying that irreducibility of $W$ means the intertwining number $\dim Hom_G(W,W)$ is equal to 1 (which is true over an algebraically closed field of characteristic zero), you can say that irreducibility of $W$ means that $Hom_G(W,W)$ is a division algebra (which is true in the semisimple case, so under the assumption you want to make).

Let me, following a comment of Darij Grinberg, make the last step a bit more clear. There is a natural homomorphism of algebras $Hom_H(V,V)\to Hom_G(Ind^G_HV, Ind^G_HV)$. The induced representation $Ind^G_HV$ is irreducible if and only if this map is an isomorphism of algebras. However, to conclude that it is sufficient to show that it is an isomorphism of vector spaces, which we do without trouble via the Frobenius reciprocity.

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

[Math] Irreducibility of Induced Representation over arbitrary field

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