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Description: |
In this talk, we consider the problem of the existence of a matrix
realization, over a local principal ideal domain, for a pair
$(T,K(\sigma))$, where $T$ is a skew tableau over the alphabet
$\{1,\ldots,t\}$ and $K(\sigma)$ is the key associated with the
permutation $\sigma\in S_t$ with the same evaluation as $T$. Using a
variant of the dual Robinson-Schensted-Knuth correspondence, we
determine a necessary condition for the existence of a matrix
realization for $(T,K(\sigma))$, generalizing the existing results
for the permutations identity, reverse and transpositions of two
consecutive integers. This condition is also sufficient when the
word of $T$ is a shuffle of all columns of $K(\sigma)$. The shuffle
of all columns of $K(\sigma)$ is always a subset of the plactic
class of $K(\sigma)$. The problem is, therefore, completely solved
for the permutations whose plactic classes of associated keys are
characterized by the shuffle of their columns. This family of
permutations is identified. We also solve the problem for the keys
associated with the symmetric group $S_4$, where there are keys
whose plactic classes cannot be described by the shuffle of their
columns. In particular, the analysis of the case $\sigma\in S_3$
allows us to generalize the action of the symmetric group on frank
words and words congruent with keys described by Lascoux and
Sch\"utzenberger.
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