= Garnir relation
Take two adjacent columns of a numbered <Young tableau>, with a bottom segment $A$ of the left column and a top segment $B$ of the right column such that $|A|+|B|$ exceeds the height of the left column. Alternating over <permutations> of $A\cup B$, modulo <permutations> preserving $A$ and $B$ separately, gives a signed relation among <polytabloids>. The full alternation vanishes because each <tabloid> has two of these entries in one row. Solving for a nonstandard <polytabloid> replaces its offending row inversion by earlier column-standard tableaux. The relation is proved over the free integral <tabloid> lattice, where cancellation and division by nonzero integer factors are valid; reduction then makes it valid in every characteristic. Together with column antisymmetry these relations give the <standard polytabloid basis>.
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