= Standard polytabloid
{title2=$e_t\quad(t\text{ a standard Young tableau})$}
A standard polytabloid is the <polytabloid> associated with a <standard Young tableau>. The standard polytabloids of one shape form a basis of the corresponding <Specht module>. For linear independence, sort each row of a <tabloid> increasingly and compare row-reading words lexicographically. In the column antisymmetrization of a standard tableau, the identity term is its unique least row word: in the first row affected by a nontrivial column permutation, each moved-in entry is larger than the entry it replaces. Hence its coefficient is one, and all other terms are larger. Distinct standard tableaux have distinct least tabloids, so a linear relation is impossible by taking its least nonzero term. Spanning is the straightening part of the standard polytabloid basis theorem.
Back to article page
