Realizações matriciais de pares de quadros de Young e palavras de Yamanouchi
 
 
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.

Area(s):
Date:  2004-07-06
Start Time:   14:30
Speaker:  Ricardo Mamede (Universidade de Coimbra)
Place:  Sala 5.5
Research Groups: -Algebra and Combinatorics
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