Isoclinic groups have same proportions of degrees of irreducible representations

Statement
Suppose $$G_1$$ and $$G_2$$ are finite groups that are isoclinic groups. Suppose $$d$$ is a positive integer. Denote by $$m_1$$ the number of equivalence classes of irreducible linear representations of $$G_1$$ of degree $$d$$ and denote by $$m_2$$ the number of equivalence classes of irreducible linear representations of $$G_2$$ of degree $$d$$. Then, $$m_1$$ is nonzero if and only if $$m_2$$ is nonzero, and if so, we have:

$$\frac{m_1}{m_2} = \frac{|G_1|}{|G_2|}$$

All this is over a field that is a splitting field for both $$G_1$$ and $$G_2$$. Note that the minimal splitting field for $$G_1$$ may be different from that of $$G_2$$.

In other words, given the degrees of irreducible representations of $$G_1$$, we can obtain the degrees of irreducible representations of $$G_2$$ by scaling the number of occurrences of each degree by a factor of $$|G_2|/|G_1|$$.

In particular, if $$G_1$$ and $$G_2$$ also have the same order, then they have precisely the same degrees of irreducible representations.

Relation with Schur covering groups
All the Schur covering groups of a given finite group have the same collection of degrees of irreducible representations and these correspond, with some adjustment for multiplicity, to the degrees of the irreducible projective representations of the original finite group.

Related facts

 * Isoclinic groups have same proportions of conjugacy class sizes

Facts used

 * 1) uses::Criterion for projective representation to lift to linear representation
 * 2) uses::Character determines representation in characteristic zero
 * 3) uses::Third isomorphism theorem
 * 4) uses::Fundamental theorem of group actions

Proof idea
The idea will be to show the following two facts:


 * For any irreducible projective representation of the inner automorphism lift, there exists a lift to a linear representation of $$G_1$$ if and only if there exists a lift to a linear representation of $$G_2$$.
 * The number of such lifts are in the ratio $$|G_1|/|G_2|$$.

Showing the result from that point onwards is straightforward.

Proof details
Given: Two isoclinic groups $$G_1$$ and $$G_2$$, a positive integer $$d$$. $$m_1$$ and $$m_2$$ are respectively the number of irreducible representations of $$G_1$$ and $$G_2$$ of degree $$d$$ over $$\mathbb{C}$$.

To prove: $$m_1$$ is nonzero if and only if $$m_2$$ is nonzero, and if so, $$m_1/m_2 = |G_1|/|G_2|$$.

Proof: We prove some subclaims. Let $$W$$ be the group $$\operatorname{Inn}(G_1) \cong \operatorname{Inn}(G_2)$$ and $$T$$ be the group $$G_1' \cong G_2'$$. Denote by $$\gamma: W \times W \to T$$ the map obtained from the commutator map in either group (we know both maps are equivalent via the isoclinism). Denote by $$\alpha_1:G_1 \to W$$ and $$\alpha_2:G_2 \to W$$ the quotient maps modulo the respective centers. In other words, we have the following short exact sequences:

$$\! 1 \to Z(G_1) \to G_1 \stackrel{\alpha_1}{\to} W \to 1$$

$$\! 1 \to Z(G_2) \to G_2 \stackrel{\alpha_2}{\to} W \to 1$$

Existence of lift of projective representation
Given: A projective representation $$\rho: W \to PGL_d(\mathbb{C})$$. Let $$\pi_d: GL_d(\mathbb{C}) \to PGL_d(\mathbb{C})$$ be the natural quotient map.

To prove: There exists a linear representation $$\theta_1: G_1 \to GL_d(\mathbb{C})$$ that is a lift of $$\rho$$ (i.e., $$\pi_d \circ \theta_1 = \rho \circ \alpha_1$$) if and only if there exists a linear representation $$\theta_2: G_2 \to GL_d(\mathbb{C})$$ that is a lift of $$\rho$$ (i.e., $$\pi_d \circ \theta_2 = \rho \circ \alpha_2$$).

Proof: By Fact (1), checking that the projective representation lifts is equivalent to checking a certain condition, that turns out to be invariant under isoclinism.

Proportionality of number of lifts
Given: A projective representation $$\rho: W \to PGL_d(\mathbb{C})$$ that lifts to linear representations for both $$G_1$$ and $$G_2$$. Let $$\pi_d: GL_d(\mathbb{C}) \to PGL_d(\mathbb{C})$$ be the natural quotient map.

To prove: The number of lifts to $$G_1$$ and the number of lifts to $$G_2$$ are in the ratio $$|G_1|/|G_2|$$.

Proof:

Proportionality of the overall number of representations
We can now finish off the proof: first, list the set of all irreducible projective representations of $$W$$ of degree $$d$$. Next, filter this to the set of those that have linear lifts to $$G_1$$ (and equivalently, to $$G_2$$). For each such, the number of lifts to $$G_1$$ and to $$G_2$$ are in the ratio $$|G_1|/|G_2|$$. Hence, the overall number of irreducible linear representations of $$G_1$$ and $$G_2$$ of degree $$d$$ are in the ratio $$|G_1|/|G_2|$$.