Degrees of irreducible representations of direct product are pairwise products of degrees of irreducible representations of direct factors

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Statement for two groups

Over any field

Suppose G_1 and G_2 are finite groups. Suppose K is a field. Suppose d_{11},d_{12},\dots,d_{1r} are the degrees of irreducible representations (possibly with repetitions) of G_1 over K and d_{21},d_{22},\dots,d_{2s} are the degrees of irreducible representations of G_2.

Then, the degrees of irreducible representations of G_1 \times G_2 over K are:

d_{11}d_{21}, d_{11}d_{22}, \dots,d_{11}d_{2s},d_{12}d_{21},d_{12}d_{22},\dots,d_{12}d_{2s},d_{1r}d_{21},\dots,d_{1r}d_{2s}

Over a splitting field

If we take K to be a splitting field for both G_1 and G_2, then the above relates the degrees of irreducible representations for G_1,G_2,G_1 \times G_2 over a splitting field. For instance, we could take K to be \mathbb{C} or \overline{\mathbb{Q}}. Note that degrees of irreducible representations are the same for all splitting fields (as long as the characteristic does not divide the group order), so this relates universal lists of numbers for G_1,G_2,G_1 \times G_2.

Statement for multiple groups

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Related facts

Facts used

  1. Tensor product establishes bijection between irreducible representations of direct factors and direct product