On a non-basis of the coinvariant algebra
A conjecture arising from a question of Procesi proposes a basis of the coinvariant algebra of $S_n$ consisting of column-antisymmetrized monomials indexed by pairs of standard Young tableaux. We show that the proposed family need not even span an $S_n$-subrepresentation: for $n=8$, a generator of degree $15$ is sent outside the span by the adjacent transposition $(4,5)$. The proof is an exact finite computation with an explicit separating functional, requiring neither a rank computation nor Gr\"obner reduction. We also retain an explicit relation for $n=7$, where the basis assertion first…
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