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Conjugating $\mathcal{C}$ by the automorphism that interchanges the elementary and homogeneous symmetric functions may be helpful. Doing this in your example yields a triangular matrix (after deleting the first column).
Have your read the first chapter of Kleshchev's "Linear and projective representations of symmetric groups"? This reminds me of ideas of Okounkov & Vershik front.math.ucdavis.edu/0503.5040.