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Séminaire Lotharingien de Combinatoire, B59g (2010), 27 pp.

# Olga Azenhas and Ricardo Mamede

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

**Abstract.**
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 two-column 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 non-standard 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.

Received: October 19, 2007.
Revised: April 15, 2010.
Accepted: April 18, 2010.
Final Version: April 19, 2010.

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