Key polynomials, invariant factors and an action of the symmetric group on Young tableaux

Abstract

We give a combinatorial description of the invariant factors associated with certain sequences of product of matrices, over a local principal ideal domain, under the action of the symmetric group by place permutation. Lascoux and Sch¨utzenberger have defined a permutation on a Young tableau to associate to each Knuth class a right and left key which they have used to give a combinatorial description of a key polynomial. The action of the symmetric group on the sequence of invariant factors generalizes this action of the symmetric group, by Lascoux and Sch¨utzenberger, to Young tableaux of the same shape and weight. As a dual translation, we obtain an action of the symmetric group on words congruent with key-tableaux based on nonstandard pairing of parentheses.