Algebra and Combinatorics

 

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Author(s)
Olga Azenhas; Ricardo Mamede;

Title
Matrix realizations of pairs of Young tableaux, keys and shuffles

Abstract
Let $\sigma\in{\mathcal S}_t$ and ${\mathcal H}_{\sigma}$ an associated key [6, 7, 12]. Given the pair of Young tableaux $({\mathcal T},{\mathcal H}_{\sigma})$, where ${\mathcal T}$ is a skew-tableau with the same evaluation as ${\mathcal H}_{\sigma}$, we consider the problem of a matrix realization of the pair $({\mathcal T},{\mathcal H}_{\sigma})$, over a local principal ideal domain [2, 3, 4]. It has been shown that, when $\sigma$ is the identity [2], the reverse permutation in ${\mathcal S}_t$ [4], or any permutation in ${\mathcal S}_3$ [3], $({\mathcal T},{\mathcal H}_{\sigma})$ has a matrix realization if and only if the word of $\mathcal T$ is in the plactic class of ${\mathcal H}_{\sigma}$. Here, we extend the only if condition of this result to any $\sigma\in{\mathcal S}_t$, $t\geq 1$. For a sequence of nonnegative integers $(m_1,\ldots,m_t)$ and $1\leq i\leq t$, this amounts to give a criterion which extends a word congruent with the key of evaluation $(m_1,\ldots,m_{i-1},m_{i+1},\ldots,m_t)$, to one congruent with the key of evaluation $(m_1,\ldots,m_t)$.

For the identity, the reverse permutation in ${\mathcal S}_t$, or an arbitrary permutation in ${\mathcal S}_3$, the plactic class of an associated key may be described as shuffles of their rows [3]. For $t\geq 4$, this is no longer true for an arbitrary permutation $\sigma\in{\mathcal S}_t$. Instead, we find that shuffling together the rows of a key always leads to a congruent word. The permutations $\sigma\in{\mathcal S}_t$ whose plactic classes of associated keys are described by shuffling together their rows, are identified. For $\sigma\in{\mathcal S}_4$, we show that we may describe the plactic class of any associated key, in terms of shuffling, by adding, in those cases where the rows of the key are not enough, just one single word.

Journal
Séminaire Lotharingien de Combinatoire

Volume
53

Year
2006

Issue
B53h

Page(s)
22pp.

 
     
 

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