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Description: |
Let $(m_1,...,m_t)$ be a nonnegative integral vector, and
$\sigma\in{\cal S} _t$ such that
$(m_{\sigma(1)},...,m_{\sigma(t)})$ is a partition. A Young
tableau ${\cal T}$ of type $(a^0,(m_1,...,m_t),a^t)$ is a sequence
of partitions $(a^0,a^1,...,a^t)$ where $a^0\subseteq
a^1\subseteq...\subseteq a^t$, such that for each $i=1,...,t$,
the skew tableau $a^i/a^{i-1}$ is a vertical strip with $m_i$
entries labelled with $i$. Associated with ${\cal T}$ there is one
and only one Young tableau ${\cal H}_ {\sigma}$ of type
$(0,(m_1,...,m_t),\sum_{i=1}^t(1^{m_i}))$.
Given the pair $({\cal T},{\cal H}_{\sigma})$, we consider the
problem of the existence of a sequence of $n\times n$ nonsingular
matrices $A_0,B_1,...,B_t$, over a local principal ideal domain,
with invariant partitions $a^0,(1^{m_1}),...,(1^{m_t})$,
respectively, such that $a^i$ is the invariant partition of
$A_0B_1...B_i$, for $i=1,...,t$, and $\sum_{i=1}^t(1^ {m_i})$ is
the invariant partition of $B_1...B_t$. The sequence
$A_0,B_1,...,B_t$ is called a matrix realization for the pair
$({\cal T},{\cal H}_{\sigma})$.
This problem has been solved when $\sigma$ is the identity [1],
the reverse permutation [2], and for any $\sigma\in{\cal S}_3$
[3]. In the first two cases the answer is given by Yamanouchi and
dual Yamanouchi words, respectively. In the last case, the answer
is given by $\sigma$-Yamanouchi words [3] over a three letters
alphabet. The concept of $\sigma$-Yamanouchi words, with
$\sigma\in{\cal S}_t$, has been introduced in [3] using the usual
matching of parentheses on words, described by A. Lascoux and M.
P. Schutzenberger in the plactic monoid [4].
In this talk, we extend the necessary condition of the previous
cases to any $t\geq 1$:
There exists a matrix realization of the pair $({\cal T},{\cal
H}_{\sigma})$ only if $w({\cal T})$, the word of the tableau, is a
$\sigma$-Yamanouchi word.
When $t=2,3$, $\sigma$-Yamanouchi words may be characterized as
shuffles of the rows of ${\cal H}_{\sigma}$. Although this
property does not remain true, in general, for $t\geq 4$, a
shuffle of the rows of
${\cal H}_{\sigma}$ is always a $\sigma$-Yamanouchi word.
Considering
$\sigma$-Yamanouchi words which are shuffles of the
rows of ${\cal H}_{\sigma}$, and whose graphical representation
satisfy a certain property, we are able to exhibit a matrix
realization for $({\cal T},{\cal H}_{\sigma})$, for any $t\geq 1$.
These restrictions on $w({\cal T})$ include the cases already
solved.
[1] O. Azenhas and E. Marques de S\'{a}, {\it Matrix realizations of
Littlewood-Richardson sequences}, Linear and Multilinear Algebra
{\bf 27} (1990), 229-242.
[2] O. Azenhas, {\it Opposite Littlewood-Richardson sequences and
their matrix realizations}, Linear Algebra and its applications
{\bf 225} (1995), 91-116.
[3] O. Azenhas and R. Mamede, {\it Actions of the symmetric group
on sets of skew tableaux with prescribed matrix realization}, DMUC
preprint 03-25 (2003).
[4] A. Lascoux, M. P. Schutzenberger, {\it Le monoid plax\"ique},
Noncommutative Structures in Algebra and Geometric Combinatorics,
(Naples, 1978), Quad. "Ricerca Sci", vol 109, CNR, Rome, 1981.
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