Actions of the Symmetric Group Generated by Comparable Sets of Integers and Smith Invariants

Lascoux and Schützenberger have shown that there exists a unique action of the symmetric group generated by the commutation of the column lengths of a twocolumn tableau and preserving the plactic class. We describe more general operators on pairs of comparable subsets of {1, . . . , n} which commute their cardinalities, and we prove that those operators define an action of the symmetric group by checking the braid relations on triples of sets of integers. The action of the symmetric group by Lascoux and Schützenberger appears in our construction as an extreme case as we only require the invariance of the shape and the weight of the insertion tableau. Instead of sets of positive integers one may take other equivalent objects as words in a two letter alphabet, and describe an action of the symmetric group on words congruent to key-tableaux defined by reflection crystal operators type based on nonstandard pairing of parentheses. This construction arises naturally as a combinatorial description of the Smith invariants of certain sequences of products of matrices, over a local principal ideal domain, under a natural action of the symmetric group.