Set of self-conjugate unordered integer partitions

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Definition

Let n be a nonnegative integer. A self-conjugate unordered integer partition of n is an unordered integer partition of n whose Ferrers diagram (or Young diagram) is self-conjugate. In other words, it is an unordered integer partition that is equal to its conjugate partition.

The set of self-conjugate unordered integer partitions is in canonical bijection with the following sets:

No. Set description Explanation of bijection
1 The set of unordered integer partitions of n into distinct odd parts. For any self-conjugate partition, count the total number of elements in the first row and first column. That's the first odd part. Now delete the first row and first column and get a smaller Young diagram. Repeat the process with the new Ferrers diagram to get the second part. The parts are odd because of the self-conjugate property. Further, we can use the self-conjugate property to reconstruct the partition uniquely.
2 The set of conjugacy classes of even permutations in the symmetric group S_n that get split in the alternating group A_n. Via splitting criterion for conjugacy classes in the alternating group, a conjugacy class gets split if its cycle type involves distinct odd pieces. The bijection thus goes via (1).
3 The set of irreducible representations of the symmetric group S_n that get split into two irreducible representations for the alternating group A_n. Via the Specht module description of a representation by a partition. The key is that since the representation is self-conjugate, it equals its tensor product with the sign representation and hence its character value is zero outside the alternating group. From this, we can work out that the Hermitian inner product of the character with itself over the alternating group is 2, forcing it to split.