GRAPH ISOMORPHISM AND REPRESENTATION THEORY

✨ Summary

The paper proves that, for fixed n, the number g_{n,k} of isomorphism classes of unlabeled graphs with n vertices and k edges forms a unimodal sequence as k varies. It first identifies g_{n,k} with the dimension of the S_n-invariant subspace of the vector space spanned by labeled graphs. Operators that add and remove edges are then combined to define an sl_2(C)-representation, with the operator H acting on the k-edge subspace by the scalar 2k - binom(n,2). The dimensions of the corresponding H-eigenspaces are therefore the values g_{n,k}; finite-dimensional sl_2-representation theory implies that these multiplicities are symmetric and unimodal. The same construction also yields unimodality for multiplicities associated with every irreducible representation of S_n. (daniel-litt.squarespace.com)

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